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Subfactors from Penn on. Vaughan Jones, Vanderbilt March 30 2019 Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 1 / 33

Transcript of VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon....

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Subfactors from Penn on.

Vaughan Jones,Vanderbilt

March 30 2019

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 1 / 33

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Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 2 / 33

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PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OFTHE AMERICAN MATHEMATICAL SOCIETYHELD AT QUEENS UNIVERSITYKINGSTON, ONTARIOJULY 14-AUGUST 2, 1980EDITED BYRICHARD V. KADISONPrepared by the American Mathematical Societywith partial support from National Science Foundation grant MCS79-27061

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 3 / 33

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Cohomological invariants for groups of outer automorphisms algebrasCOLIN E. SUTHERLANDAutomorphism groups and invariant statesERLING ST0RMERCompact ergodic groups of automorphismsMAGNUS B. LANDSTADActions of discrete groups on factorsV. F. R. JONESErgodic theory and von Neumann algebrasCALVIN C. MOORETopologies on measured groupoidsARLAN RAMSAY

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 4 / 33

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The Seminar

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 5 / 33

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Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 6 / 33

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Subfactors: definition.

For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.The novelty is that this dimMpHq is actually a real number ě 0 or 8.Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.We see that all dimensions in this case are given by the numbers k

n wherek is a non-negative integer (or infinity).Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

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Subfactors: definition.For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).

This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.The novelty is that this dimMpHq is actually a real number ě 0 or 8.Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.We see that all dimensions in this case are given by the numbers k

n wherek is a non-negative integer (or infinity).Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

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Subfactors: definition.For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.

The novelty is that this dimMpHq is actually a real number ě 0 or 8.Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.We see that all dimensions in this case are given by the numbers k

n wherek is a non-negative integer (or infinity).Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

Page 10: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Subfactors: definition.For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.The novelty is that this dimMpHq is actually a real number ě 0 or 8.

Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.We see that all dimensions in this case are given by the numbers k

n wherek is a non-negative integer (or infinity).Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

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Subfactors: definition.For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.The novelty is that this dimMpHq is actually a real number ě 0 or 8.Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).

The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.We see that all dimensions in this case are given by the numbers k

n wherek is a non-negative integer (or infinity).Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

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Subfactors: definition.For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.The novelty is that this dimMpHq is actually a real number ě 0 or 8.Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.

We see that all dimensions in this case are given by the numbers kn where

k is a non-negative integer (or infinity).Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

Page 13: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Subfactors: definition.For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.The novelty is that this dimMpHq is actually a real number ě 0 or 8.Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.We see that all dimensions in this case are given by the numbers k

n wherek is a non-negative integer (or infinity).

Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

Page 14: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Subfactors: definition.For the purposes of this talk a II1 factor M should be thought of as a"field" (albeit noncommutative and with a ton of zero-divisors).This is because of von Neumann’s result that a vector space H on whichM acts is completely characterised as an M-module by its dimensiondimMpHq.The novelty is that this dimMpHq is actually a real number ě 0 or 8.Some intituion can be gained by considering the case of M “ MnpCq(which has all the attributes of a II1 factor except infinite dimensionality).The smallest M “ MnpCq module has dimension n over C and thusdimension 1

n as measured by M.We see that all dimensions in this case are given by the numbers k

n wherek is a non-negative integer (or infinity).Thus the II1 factors are infinite dimensional versions of MnpCq thedimensions of whose modules "fill in the gaps" of those of M “ MnpCq.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 7 / 33

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That was something of a crash course in II1 factors and a bitimpressionistic.

If you want an example you can take the group PSLp2,Zq ă PSL2pRq.The "holomorphic discrete series" of PSL2pRq is the natural unitary actionof PSL2pRq on L2 holomorphic functions on the upper half plane withmeasure dxdy

y2´n .The "commutant" PSL2pZq1 of the action of PSL2pZq is a II1 factor andthe dimension of the Hilbert space is something like 1

n´1 .Radulescu.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 8 / 33

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That was something of a crash course in II1 factors and a bitimpressionistic.If you want an example you can take the group PSLp2,Zq ă PSL2pRq.The "holomorphic discrete series" of PSL2pRq is the natural unitary actionof PSL2pRq on L2 holomorphic functions on the upper half plane withmeasure dxdy

y2´n .

The "commutant" PSL2pZq1 of the action of PSL2pZq is a II1 factor andthe dimension of the Hilbert space is something like 1

n´1 .Radulescu.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 8 / 33

Page 17: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

That was something of a crash course in II1 factors and a bitimpressionistic.If you want an example you can take the group PSLp2,Zq ă PSL2pRq.The "holomorphic discrete series" of PSL2pRq is the natural unitary actionof PSL2pRq on L2 holomorphic functions on the upper half plane withmeasure dxdy

y2´n .The "commutant" PSL2pZq1 of the action of PSL2pZq is a II1 factor andthe dimension of the Hilbert space is something like 1

n´1 .

Radulescu.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 8 / 33

Page 18: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

That was something of a crash course in II1 factors and a bitimpressionistic.If you want an example you can take the group PSLp2,Zq ă PSL2pRq.The "holomorphic discrete series" of PSL2pRq is the natural unitary actionof PSL2pRq on L2 holomorphic functions on the upper half plane withmeasure dxdy

y2´n .The "commutant" PSL2pZq1 of the action of PSL2pZq is a II1 factor andthe dimension of the Hilbert space is something like 1

n´1 .Radulescu.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 8 / 33

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A II1 factor always has an identity. If M is a II1 factor a subfactor of N isa subfactor

containing the same identity as that of M.Thus one may define the real number rM : Ns “ dimNpMq.

As an example one may take Γ ă PSL2pZq a large subgroup so that thecommutant Γ1 of Γ is also II1 factor.One then has

rΓ1 : PSL2pZq1s “ rPSL2pZq : Γs

In this and similar ways it is easy to get all non-negative integers as indicesof II1 factors. But it is supposed to be a real number.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 9 / 33

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A II1 factor always has an identity. If M is a II1 factor a subfactor of N isa subfactor containing the same identity as that of M.

Thus one may define the real number rM : Ns “ dimNpMq.

As an example one may take Γ ă PSL2pZq a large subgroup so that thecommutant Γ1 of Γ is also II1 factor.One then has

rΓ1 : PSL2pZq1s “ rPSL2pZq : Γs

In this and similar ways it is easy to get all non-negative integers as indicesof II1 factors. But it is supposed to be a real number.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 9 / 33

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A II1 factor always has an identity. If M is a II1 factor a subfactor of N isa subfactor containing the same identity as that of M.Thus one may define the real number rM : Ns “ dimNpMq.

As an example one may take Γ ă PSL2pZq a large subgroup so that thecommutant Γ1 of Γ is also II1 factor.One then has

rΓ1 : PSL2pZq1s “ rPSL2pZq : Γs

In this and similar ways it is easy to get all non-negative integers as indicesof II1 factors. But it is supposed to be a real number.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 9 / 33

Page 22: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

A II1 factor always has an identity. If M is a II1 factor a subfactor of N isa subfactor containing the same identity as that of M.Thus one may define the real number rM : Ns “ dimNpMq.

As an example one may take Γ ă PSL2pZq a large subgroup so that thecommutant Γ1 of Γ is also II1 factor.

One then has

rΓ1 : PSL2pZq1s “ rPSL2pZq : Γs

In this and similar ways it is easy to get all non-negative integers as indicesof II1 factors. But it is supposed to be a real number.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 9 / 33

Page 23: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

A II1 factor always has an identity. If M is a II1 factor a subfactor of N isa subfactor containing the same identity as that of M.Thus one may define the real number rM : Ns “ dimNpMq.

As an example one may take Γ ă PSL2pZq a large subgroup so that thecommutant Γ1 of Γ is also II1 factor.One then has

rΓ1 : PSL2pZq1s “ rPSL2pZq : Γs

In this and similar ways it is easy to get all non-negative integers as indicesof II1 factors. But it is supposed to be a real number.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 9 / 33

Page 24: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

A II1 factor always has an identity. If M is a II1 factor a subfactor of N isa subfactor containing the same identity as that of M.Thus one may define the real number rM : Ns “ dimNpMq.

As an example one may take Γ ă PSL2pZq a large subgroup so that thecommutant Γ1 of Γ is also II1 factor.One then has

rΓ1 : PSL2pZq1s “ rPSL2pZq : Γs

In this and similar ways it is easy to get all non-negative integers as indicesof II1 factors.

But it is supposed to be a real number.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 9 / 33

Page 25: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

A II1 factor always has an identity. If M is a II1 factor a subfactor of N isa subfactor containing the same identity as that of M.Thus one may define the real number rM : Ns “ dimNpMq.

As an example one may take Γ ă PSL2pZq a large subgroup so that thecommutant Γ1 of Γ is also II1 factor.One then has

rΓ1 : PSL2pZq1s “ rPSL2pZq : Γs

In this and similar ways it is easy to get all non-negative integers as indicesof II1 factors. But it is supposed to be a real number.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 9 / 33

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The following result was proved when I was at Penn in 1981-1982:

Theorem: If rM : Ns ă 4 it is 4 cos2 π{n for n “ 3, 4, 5, 6, ....4 and allthese values occur. If ą 4 any real number possible.Picture:

1 2 3 4

....

2.6180339... =4cos /5π2

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 10 / 33

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The following result was proved when I was at Penn in 1981-1982:Theorem: If rM : Ns ă 4 it is 4 cos2 π{n for n “ 3, 4, 5, 6, ....4 and allthese values occur. If ą 4 any real number possible.

Picture:

1 2 3 4

....

2.6180339... =4cos /5π2

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 10 / 33

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The following result was proved when I was at Penn in 1981-1982:Theorem: If rM : Ns ă 4 it is 4 cos2 π{n for n “ 3, 4, 5, 6, ....4 and allthese values occur. If ą 4 any real number possible.Picture:

1 2 3 4

....

2.6180339... =4cos /5π2

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 10 / 33

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About the proof I will only say that it involves a detailed understanding ofwhat has become known as the Temperley-Lieb algebra TL(n ` 1, τ)

This is the *-algebra generated by projections e1, e2, e3, ¨ ¨ ¨ en withrelations:

e2i “ ei “ e˚i

eiej “ ejei for |i ´ j | ě 2

eiei˘1ei “ τei

Which possesses a trace functional tr completely defined by the formula

trpxen`1q “ τ trpxq

if x is a word on e1, e2, ¨ ¨ ¨ en.

This algebra is loaded with structure and was the cause of a lot ofdiscussion at the time at Penn.We talked about ei ei oh!.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 11 / 33

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About the proof I will only say that it involves a detailed understanding ofwhat has become known as the Temperley-Lieb algebra TL(n ` 1, τ)

This is the *-algebra generated by projections e1, e2, e3, ¨ ¨ ¨ en withrelations:

e2i “ ei “ e˚i

eiej “ ejei for |i ´ j | ě 2

eiei˘1ei “ τei

Which possesses a trace functional tr completely defined by the formula

trpxen`1q “ τ trpxq

if x is a word on e1, e2, ¨ ¨ ¨ en.

This algebra is loaded with structure and was the cause of a lot ofdiscussion at the time at Penn.We talked about ei ei oh!.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 11 / 33

Page 31: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

About the proof I will only say that it involves a detailed understanding ofwhat has become known as the Temperley-Lieb algebra TL(n ` 1, τ)

This is the *-algebra generated by projections e1, e2, e3, ¨ ¨ ¨ en withrelations:

e2i “ ei “ e˚i

eiej “ ejei for |i ´ j | ě 2

eiei˘1ei “ τei

Which possesses a trace functional tr completely defined by the formula

trpxen`1q “ τ trpxq

if x is a word on e1, e2, ¨ ¨ ¨ en.

This algebra is loaded with structure and was the cause of a lot ofdiscussion at the time at Penn.We talked about ei ei oh!.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 11 / 33

Page 32: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

About the proof I will only say that it involves a detailed understanding ofwhat has become known as the Temperley-Lieb algebra TL(n ` 1, τ)

This is the *-algebra generated by projections e1, e2, e3, ¨ ¨ ¨ en withrelations:

e2i “ ei “ e˚i

eiej “ ejei for |i ´ j | ě 2

eiei˘1ei “ τei

Which possesses a trace functional tr completely defined by the formula

trpxen`1q “ τ trpxq

if x is a word on e1, e2, ¨ ¨ ¨ en.

This algebra is loaded with structure and was the cause of a lot ofdiscussion at the time at Penn.We talked about ei ei oh!.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 11 / 33

Page 33: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

About the proof I will only say that it involves a detailed understanding ofwhat has become known as the Temperley-Lieb algebra TL(n ` 1, τ)

This is the *-algebra generated by projections e1, e2, e3, ¨ ¨ ¨ en withrelations:

e2i “ ei “ e˚i

eiej “ ejei for |i ´ j | ě 2

eiei˘1ei “ τei

Which possesses a trace functional tr completely defined by the formula

trpxen`1q “ τ trpxq

if x is a word on e1, e2, ¨ ¨ ¨ en.

This algebra is loaded with structure and was the cause of a lot ofdiscussion at the time at Penn.

We talked about ei ei oh!.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 11 / 33

Page 34: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

About the proof I will only say that it involves a detailed understanding ofwhat has become known as the Temperley-Lieb algebra TL(n ` 1, τ)

This is the *-algebra generated by projections e1, e2, e3, ¨ ¨ ¨ en withrelations:

e2i “ ei “ e˚i

eiej “ ejei for |i ´ j | ě 2

eiei˘1ei “ τei

Which possesses a trace functional tr completely defined by the formula

trpxen`1q “ τ trpxq

if x is a word on e1, e2, ¨ ¨ ¨ en.

This algebra is loaded with structure and was the cause of a lot ofdiscussion at the time at Penn.We talked about ei ei oh!.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 11 / 33

Page 35: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Today I would like to say something about what has happened since then.

I will split the talk up somehwat artificially into three parts:

The internal theory of subfactors.

Interactions with other parts of mathematics.

Interactions with physics.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 12 / 33

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Today I would like to say something about what has happened since then.I will split the talk up somehwat artificially into three parts:

The internal theory of subfactors.

Interactions with other parts of mathematics.

Interactions with physics.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 12 / 33

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Internal theory:One may sum up the internal theory developments with the followingpicture:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 13 / 33

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Ocneanu Popa Wenzl Goodman de la Harpe Haagerup Asaeda Izumi BischGraham Lehrer Morrison Peters Snyder Bigelow Xu Grossman Liu PenneysTener Evans Gannon Etingof Nikschych Ostrik Yasuda Calegari JobsWolfram

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 14 / 33

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What are all the stick insects?

It is important to note that subfactors are inextricably connected tobimodules (or correspondences in the sense of Connes).

A bimodule is a left and right (commuting) module over a pair of algebras.

MHN

.

It can be quite confusing because for a subfactor N Ď M, M is anN ´M,M ´M,M ´N and N ´N bimodule whereas typically in bimoduletheory one considers a single MHM and takes the tensor powers over M-bk

MH.N−M bimodules x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x N−N bimodules

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 15 / 33

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What are all the stick insects?

It is important to note that subfactors are inextricably connected tobimodules (or correspondences in the sense of Connes).

A bimodule is a left and right (commuting) module over a pair of algebras.

MHN

.

It can be quite confusing because for a subfactor N Ď M, M is anN ´M,M ´M,M ´N and N ´N bimodule whereas typically in bimoduletheory one considers a single MHM and takes the tensor powers over M-bk

MH.N−M bimodules x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x N−N bimodules

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 15 / 33

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What are all the stick insects?

It is important to note that subfactors are inextricably connected tobimodules (or correspondences in the sense of Connes).

A bimodule is a left and right (commuting) module over a pair of algebras.

MHN

.

It can be quite confusing because for a subfactor N Ď M, M is anN ´M,M ´M,M ´N and N ´N bimodule whereas typically in bimoduletheory one considers a single MHM and takes the tensor powers over M-bk

MH.N−M bimodules x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x N−N bimodules

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 15 / 33

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What are all the stick insects?

It is important to note that subfactors are inextricably connected tobimodules (or correspondences in the sense of Connes).

A bimodule is a left and right (commuting) module over a pair of algebras.

MHN

.

It can be quite confusing because for a subfactor N Ď M, M is anN ´M,M ´M,M ´N and N ´N bimodule whereas typically in bimoduletheory one considers a single MHM and takes the tensor powers over M-bk

MH.N−M bimodules x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x N−N bimodules

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 15 / 33

Page 43: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

What are all the stick insects?

It is important to note that subfactors are inextricably connected tobimodules (or correspondences in the sense of Connes).

A bimodule is a left and right (commuting) module over a pair of algebras.

MHN

.

It can be quite confusing because for a subfactor N Ď M, M is anN ´M,M ´M,M ´N and N ´N bimodule whereas typically in bimoduletheory one considers a single MHM and takes the tensor powers over M-bk

MH.

N−M bimodules x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x N−N bimodules

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 15 / 33

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What are all the stick insects?

It is important to note that subfactors are inextricably connected tobimodules (or correspondences in the sense of Connes).

A bimodule is a left and right (commuting) module over a pair of algebras.

MHN

.

It can be quite confusing because for a subfactor N Ď M, M is anN ´M,M ´M,M ´N and N ´N bimodule whereas typically in bimoduletheory one considers a single MHM and takes the tensor powers over M-bk

MH.N−M bimodules x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x N−N bimodules

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 15 / 33

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induction/restriction

N−M bimodules x x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x x N−N bimodules

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 16 / 33

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x

The principal graph:

x

x x x x

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 17 / 33

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Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 18 / 33

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Interactions with other parts of mathematics:

A knot is a smooth closed curve in 3 dimensional space. A link is acollection of knots.They can be represented by two dimensional picturesthus:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 19 / 33

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Interactions with other parts of mathematics:

A knot is a smooth closed curve in 3 dimensional space. A link is acollection of knots.

They can be represented by two dimensional picturesthus:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 19 / 33

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Interactions with other parts of mathematics:

A knot is a smooth closed curve in 3 dimensional space. A link is acollection of knots.They can be represented by two dimensional picturesthus:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 19 / 33

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Braids are collections of curves in 3 dimensional space connecting two barsthus:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 20 / 33

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Braids with n strings form a group Bn under concatenation. This group hasa simple algebraic presentation (Artin) on generators σi , i “ 1, 2, ¨ ¨ ¨ n ´ 1

Here are σ1 and σ2 in B3:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 21 / 33

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Braids with n strings form a group Bn under concatenation. This group hasa simple algebraic presentation (Artin) on generators σi , i “ 1, 2, ¨ ¨ ¨ n ´ 1Here are σ1 and σ2 in B3:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 21 / 33

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Braids with n strings form a group Bn under concatenation. This group hasa simple algebraic presentation (Artin) on generators σi , i “ 1, 2, ¨ ¨ ¨ n ´ 1Here are σ1 and σ2 in B3:

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 21 / 33

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The presenting relations are

σiσi`1σi “ σi`1σiσi`1

σjσi “ σiσj if |i ´ j | ě 2

Recall our projections ei :

ei “ e˚i “ ei

eiei˘1ei “ τei where τ´1 “ rM : Ns

eiej “ ejei if |i ´ j | ě 2

Represent Bn in II1 factor by

σi Ñ te1 ´ p1´ ei q

with rM : Ns “ 2` t ` t´1.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 22 / 33

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The presenting relations are

σiσi`1σi “ σi`1σiσi`1

σjσi “ σiσj if |i ´ j | ě 2

Recall our projections ei :

ei “ e˚i “ ei

eiei˘1ei “ τei where τ´1 “ rM : Ns

eiej “ ejei if |i ´ j | ě 2

Represent Bn in II1 factor by

σi Ñ te1 ´ p1´ ei q

with rM : Ns “ 2` t ` t´1.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 22 / 33

Page 57: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

The presenting relations are

σiσi`1σi “ σi`1σiσi`1

σjσi “ σiσj if |i ´ j | ě 2

Recall our projections ei :

ei “ e˚i “ ei

eiei˘1ei “ τei where τ´1 “ rM : Ns

eiej “ ejei if |i ´ j | ě 2

Represent Bn in II1 factor by

σi Ñ te1 ´ p1´ ei q

with rM : Ns “ 2` t ` t´1.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 22 / 33

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The connection between braids and links:In 1984 while still at Penn, I visited Joan Birman at Columbia.Recall our braid:

We can “close” it thus:

And remove the bars to obtain: A LINK!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 23 / 33

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The connection between braids and links:In 1984 while still at Penn, I visited Joan Birman at Columbia.Recall our braid:

We can “close” it thus:

And remove the bars to obtain: A LINK!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 23 / 33

Page 60: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

The connection between braids and links:In 1984 while still at Penn, I visited Joan Birman at Columbia.Recall our braid:

We can “close” it thus:

And remove the bars to obtain: A LINK!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 23 / 33

Page 61: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

The connection between braids and links:In 1984 while still at Penn, I visited Joan Birman at Columbia.Recall our braid:

We can “close” it thus:

And remove the bars to obtain: A LINK!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 23 / 33

Page 62: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

The connection between braids and links:In 1984 while still at Penn, I visited Joan Birman at Columbia.Recall our braid:

We can “close” it thus:

And remove the bars to obtain:

A LINK!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 23 / 33

Page 63: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

The connection between braids and links:In 1984 while still at Penn, I visited Joan Birman at Columbia.Recall our braid:

We can “close” it thus:

And remove the bars to obtain:

A LINK!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 23 / 33

Page 64: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

The connection between braids and links:In 1984 while still at Penn, I visited Joan Birman at Columbia.Recall our braid:

We can “close” it thus:

And remove the bars to obtain: A LINK!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 23 / 33

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All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....) Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.e.g. for trefoil knot

VK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 66: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....) Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.e.g. for trefoil knot

VK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 67: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....)

Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.e.g. for trefoil knot

VK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 68: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....) Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.

(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.e.g. for trefoil knot

VK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 69: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....) Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q

(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.e.g. for trefoil knot

VK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 70: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....) Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.

e.g. for trefoil knotVK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 71: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....) Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.e.g. for trefoil knot

VK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 72: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

All links can be obtained in this way (Alexander) and Joan told me of aresult of A.A. Markov which says exactly what different braids give thesame link. Then a miracle occurred: Markov’s theorem correspondsexactly (!!) to our condition

tracepxen`1q “ τ tracepxq if x is a word on 1, e1, e2, ¨ ¨ ¨ en

(which in another miracle had already been called a Markov conditionafter A.A. Markov.....) Putting the ingredients together we get aninvariant of knots/links, called VLptq by the following procedure:(i)Take L and represent it as the closure of a braid α.(ii) Represent the braid α in the algebra of the ei ’s via σi ÞÑ tei ´ p1´ ei q(iii) Take the trace of the braid in the algebra, multiply by a simple fudgefactor and voilà.e.g. for trefoil knot

VK ptq “ t ` t3 ´ t4

So 1984 was a bad year for Orwell characters, a great year for me (atPenn)!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 24 / 33

Page 73: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Many questions arose from this work. One is somewhat notoriously stillopen: Is there a knot K , different from the unknot, for which VK ptq?

The analogous question for links has been answered known thanks toThistlethwaite and his computer.

But it was Witten who in 1988 really opened the subject up with afunctional integral interpretation of VLptq which led immediately to ageneralisation of VLptq, at least for t a root of unity, to the case where Llives in an arbitrary closed 3 manifold.

First proved by Reshetikhin and Turaev. Then many others.

Thus began the subject of "quantum topology".

The understanding of the structure and even the mathematical nature ofthese invariants is an ongoing body of research with strong interactionswith physics, in particular conformal field theory.(Kontsevich, Bar Natan, Lawrence, Le, Ohtsuki, Garoufalidis, Aganagic,Vafa, Gukov.....)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 25 / 33

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Many questions arose from this work. One is somewhat notoriously stillopen: Is there a knot K , different from the unknot, for which VK ptq?

The analogous question for links has been answered known thanks toThistlethwaite and his computer.

But it was Witten who in 1988 really opened the subject up with afunctional integral interpretation of VLptq which led immediately to ageneralisation of VLptq, at least for t a root of unity, to the case where Llives in an arbitrary closed 3 manifold.

First proved by Reshetikhin and Turaev. Then many others.

Thus began the subject of "quantum topology".

The understanding of the structure and even the mathematical nature ofthese invariants is an ongoing body of research with strong interactionswith physics, in particular conformal field theory.(Kontsevich, Bar Natan, Lawrence, Le, Ohtsuki, Garoufalidis, Aganagic,Vafa, Gukov.....)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 25 / 33

Page 75: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Many questions arose from this work. One is somewhat notoriously stillopen: Is there a knot K , different from the unknot, for which VK ptq?

The analogous question for links has been answered known thanks toThistlethwaite and his computer.

But it was Witten who in 1988 really opened the subject up with afunctional integral interpretation of VLptq which led immediately to ageneralisation of VLptq, at least for t a root of unity, to the case where Llives in an arbitrary closed 3 manifold.

First proved by Reshetikhin and Turaev. Then many others.

Thus began the subject of "quantum topology".

The understanding of the structure and even the mathematical nature ofthese invariants is an ongoing body of research with strong interactionswith physics, in particular conformal field theory.(Kontsevich, Bar Natan, Lawrence, Le, Ohtsuki, Garoufalidis, Aganagic,Vafa, Gukov.....)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 25 / 33

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Many questions arose from this work. One is somewhat notoriously stillopen: Is there a knot K , different from the unknot, for which VK ptq?

The analogous question for links has been answered known thanks toThistlethwaite and his computer.

But it was Witten who in 1988 really opened the subject up with afunctional integral interpretation of VLptq which led immediately to ageneralisation of VLptq, at least for t a root of unity, to the case where Llives in an arbitrary closed 3 manifold.

First proved by Reshetikhin and Turaev. Then many others.

Thus began the subject of "quantum topology".

The understanding of the structure and even the mathematical nature ofthese invariants is an ongoing body of research with strong interactionswith physics, in particular conformal field theory.(Kontsevich, Bar Natan, Lawrence, Le, Ohtsuki, Garoufalidis, Aganagic,Vafa, Gukov.....)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 25 / 33

Page 77: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Many questions arose from this work. One is somewhat notoriously stillopen: Is there a knot K , different from the unknot, for which VK ptq?

The analogous question for links has been answered known thanks toThistlethwaite and his computer.

But it was Witten who in 1988 really opened the subject up with afunctional integral interpretation of VLptq which led immediately to ageneralisation of VLptq, at least for t a root of unity, to the case where Llives in an arbitrary closed 3 manifold.

First proved by Reshetikhin and Turaev. Then many others.

Thus began the subject of "quantum topology".

The understanding of the structure and even the mathematical nature ofthese invariants is an ongoing body of research with strong interactionswith physics, in particular conformal field theory.(Kontsevich, Bar Natan, Lawrence, Le, Ohtsuki, Garoufalidis, Aganagic,Vafa, Gukov.....)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 25 / 33

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Interactions with other parts of mathematics:

Tensor categories, planar algebras.

Bimodules over a II1 (or other) factor form a tensor category-whateverthat means.

If a subcategory contains only finitely many isomorphism classes of simpleobjects (=irreducible bimodules) it is called a "fusion category.

The main suspects in the study of fusion categories are Etingof Nikschychand Ostrik.

There are many examples of fusion categories related to finite groups butit is fair to say that the most interesting ones have arisen as the N ´ Nbimodules coming from a subfactor with finite principal graph.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 26 / 33

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Interactions with other parts of mathematics:

Tensor categories, planar algebras.

Bimodules over a II1 (or other) factor form a tensor category-whateverthat means.

If a subcategory contains only finitely many isomorphism classes of simpleobjects (=irreducible bimodules) it is called a "fusion category.

The main suspects in the study of fusion categories are Etingof Nikschychand Ostrik.

There are many examples of fusion categories related to finite groups butit is fair to say that the most interesting ones have arisen as the N ´ Nbimodules coming from a subfactor with finite principal graph.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 26 / 33

Page 80: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with other parts of mathematics:

Tensor categories, planar algebras.

Bimodules over a II1 (or other) factor form a tensor category-whateverthat means.

If a subcategory contains only finitely many isomorphism classes of simpleobjects (=irreducible bimodules) it is called a "fusion category.

The main suspects in the study of fusion categories are Etingof Nikschychand Ostrik.

There are many examples of fusion categories related to finite groups butit is fair to say that the most interesting ones have arisen as the N ´ Nbimodules coming from a subfactor with finite principal graph.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 26 / 33

Page 81: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with other parts of mathematics:

Tensor categories, planar algebras.

Bimodules over a II1 (or other) factor form a tensor category-whateverthat means.

If a subcategory contains only finitely many isomorphism classes of simpleobjects (=irreducible bimodules) it is called a "fusion category.

The main suspects in the study of fusion categories are Etingof Nikschychand Ostrik.

There are many examples of fusion categories related to finite groups butit is fair to say that the most interesting ones have arisen as the N ´ Nbimodules coming from a subfactor with finite principal graph.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 26 / 33

Page 82: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with other parts of mathematics:

Tensor categories, planar algebras.

Bimodules over a II1 (or other) factor form a tensor category-whateverthat means.

If a subcategory contains only finitely many isomorphism classes of simpleobjects (=irreducible bimodules) it is called a "fusion category.

The main suspects in the study of fusion categories are Etingof Nikschychand Ostrik.

There are many examples of fusion categories related to finite groups butit is fair to say that the most interesting ones have arisen as the N ´ Nbimodules coming from a subfactor with finite principal graph.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 26 / 33

Page 83: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with other parts of mathematics:

Tensor categories, planar algebras.

Bimodules over a II1 (or other) factor form a tensor category-whateverthat means.

If a subcategory contains only finitely many isomorphism classes of simpleobjects (=irreducible bimodules) it is called a "fusion category.

The main suspects in the study of fusion categories are Etingof Nikschychand Ostrik.

There are many examples of fusion categories related to finite groups butit is fair to say that the most interesting ones have arisen as the N ´ Nbimodules coming from a subfactor with finite principal graph.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 26 / 33

Page 84: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with other parts of mathematics:

Tensor categories, planar algebras.

Bimodules over a II1 (or other) factor form a tensor category-whateverthat means.

If a subcategory contains only finitely many isomorphism classes of simpleobjects (=irreducible bimodules) it is called a "fusion category.

The main suspects in the study of fusion categories are Etingof Nikschychand Ostrik.

There are many examples of fusion categories related to finite groups butit is fair to say that the most interesting ones have arisen as the N ´ Nbimodules coming from a subfactor with finite principal graph.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 26 / 33

Page 85: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

In general these tensor categories admit a diagrammatic calculus which inthe case of subfactors was formalised rigorously as "planar algebras".

Theyare a major tool in the construction and analysis of subfactors.

The original motivation for planar algebras included the desire togeneralise a beautiful pictorial representation, due to L. Kauffman, of theTL algebra in which one uses the braid picture but with the followingtwo-dimesional pictures replacing the crossings:

Ei “

Checking the ei relations is then a very pleasant exercise. (The τ factorappears in a different place!)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 27 / 33

Page 86: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

In general these tensor categories admit a diagrammatic calculus which inthe case of subfactors was formalised rigorously as "planar algebras".Theyare a major tool in the construction and analysis of subfactors.

The original motivation for planar algebras included the desire togeneralise a beautiful pictorial representation, due to L. Kauffman, of theTL algebra in which one uses the braid picture but with the followingtwo-dimesional pictures replacing the crossings:

Ei “

Checking the ei relations is then a very pleasant exercise. (The τ factorappears in a different place!)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 27 / 33

Page 87: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

In general these tensor categories admit a diagrammatic calculus which inthe case of subfactors was formalised rigorously as "planar algebras".Theyare a major tool in the construction and analysis of subfactors.

The original motivation for planar algebras included the desire togeneralise a beautiful pictorial representation, due to L. Kauffman, of theTL algebra in which one uses the braid picture but with the followingtwo-dimesional pictures replacing the crossings:

Ei “

Checking the ei relations is then a very pleasant exercise. (The τ factorappears in a different place!)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 27 / 33

Page 88: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

In general these tensor categories admit a diagrammatic calculus which inthe case of subfactors was formalised rigorously as "planar algebras".Theyare a major tool in the construction and analysis of subfactors.

The original motivation for planar algebras included the desire togeneralise a beautiful pictorial representation, due to L. Kauffman, of theTL algebra in which one uses the braid picture but with the followingtwo-dimesional pictures replacing the crossings:

Ei “

Checking the ei relations is then a very pleasant exercise. (The τ factorappears in a different place!)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 27 / 33

Page 89: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

In general these tensor categories admit a diagrammatic calculus which inthe case of subfactors was formalised rigorously as "planar algebras".Theyare a major tool in the construction and analysis of subfactors.

The original motivation for planar algebras included the desire togeneralise a beautiful pictorial representation, due to L. Kauffman, of theTL algebra in which one uses the braid picture but with the followingtwo-dimesional pictures replacing the crossings:

Ei “

Checking the ei relations is then a very pleasant exercise. (The τ factorappears in a different place!)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 27 / 33

Page 90: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum groups.

It was observed during a seminar at Penn (at which Kadison was present),by Mitch Baker, Bob Powers and myself, that the structure of the TLalgebra is (for generic τ) the SAME as that of the SUp2q-invariant CARalgebra!

This is now understood as a "quantum Schur-Weyl duality" in whichSUp2q is deformed to the Jimbo-Drinfeld-Woronowicz quantum versionSUqp2q , together with its representations, and TLpn, 2` q ` q´1q is thecommutant of the tensor product action of SUqp2q on the tensor powers ofthe 2-dimensional representation.

This has all been vastly generalised, but it should be stressed that therepresentation of the e 1i s in the tensor power was in fact discovered in thesubfactor context, by Pimsner and Popa!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 28 / 33

Page 91: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum groups.

It was observed during a seminar at Penn (at which Kadison was present),by Mitch Baker, Bob Powers and myself, that the structure of the TLalgebra is (for generic τ) the SAME as that of the SUp2q-invariant CARalgebra!

This is now understood as a "quantum Schur-Weyl duality" in whichSUp2q is deformed to the Jimbo-Drinfeld-Woronowicz quantum versionSUqp2q , together with its representations, and TLpn, 2` q ` q´1q is thecommutant of the tensor product action of SUqp2q on the tensor powers ofthe 2-dimensional representation.

This has all been vastly generalised, but it should be stressed that therepresentation of the e 1i s in the tensor power was in fact discovered in thesubfactor context, by Pimsner and Popa!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 28 / 33

Page 92: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum groups.

It was observed during a seminar at Penn (at which Kadison was present),by Mitch Baker, Bob Powers and myself, that the structure of the TLalgebra is (for generic τ) the SAME as that of the SUp2q-invariant CARalgebra!

This is now understood as a "quantum Schur-Weyl duality" in whichSUp2q is deformed to the Jimbo-Drinfeld-Woronowicz quantum versionSUqp2q , together with its representations, and TLpn, 2` q ` q´1q is thecommutant of the tensor product action of SUqp2q on the tensor powers ofthe 2-dimensional representation.

This has all been vastly generalised, but it should be stressed that therepresentation of the e 1i s in the tensor power was in fact discovered in thesubfactor context, by Pimsner and Popa!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 28 / 33

Page 93: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum groups.

It was observed during a seminar at Penn (at which Kadison was present),by Mitch Baker, Bob Powers and myself, that the structure of the TLalgebra is (for generic τ) the SAME as that of the SUp2q-invariant CARalgebra!

This is now understood as a "quantum Schur-Weyl duality" in whichSUp2q is deformed to the Jimbo-Drinfeld-Woronowicz quantum versionSUqp2q , together with its representations, and TLpn, 2` q ` q´1q is thecommutant of the tensor product action of SUqp2q on the tensor powers ofthe 2-dimensional representation.

This has all been vastly generalised, but it should be stressed that therepresentation of the e 1i s in the tensor power was in fact discovered in thesubfactor context, by Pimsner and Popa!

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 28 / 33

Page 94: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.

The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.The ei relations come from the nearest neighbour interaction between thespins.(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 95: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.The ei relations come from the nearest neighbour interaction between thespins.(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 96: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.

The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.The ei relations come from the nearest neighbour interaction between thespins.(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 97: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.The ei relations come from the nearest neighbour interaction between thespins.(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 98: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.The ei relations come from the nearest neighbour interaction between thespins.(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 99: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.

The ei relations come from the nearest neighbour interaction between thespins.(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 100: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.The ei relations come from the nearest neighbour interaction between thespins.

(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 101: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Interactions with physics.The reason the ei algebra is called the TL algebra is because tworepresentations of it were used by Temperley and Lieb in proving amathematical equivalence between two models in 2-D statisticalmechanics.

Lieb’s "ice type" model and the Potts model.The models are on a squarelattice and are analysed by a "transfer matix" which corresponds to addinga row of spins to the lattice.

Roughly speaking each ei corresponds to adding a single spin to thetransfer matrix which will thus have the form

pxe1 ` yqpxe2 ` yqpxe3 ` yq.....pxen ` yq

Taking the trace corresponds to periodic boundary conditions, althoughthese were not the boundary conditions that T and L used-for their oneswhat they showed was that the calculation (of....) can be done ENTIRELYusing the ei relations.The ei relations come from the nearest neighbour interaction between thespins.(David Evans.)

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 29 / 33

Page 102: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum spin chains, tensor networks.

If quantum spins are arranged in a one dimensional array with interactionsbetween nearest neighbours, the Hilbert space for the spin chain is bnC2.

On which the e 1i s act. In a nearest neighbour fashion.

The operatorn

ÿ

i“1

ei

has been used as a Hamiltonian governing the time evoloution of aquantum spin chain of length n. Some think (e.g. Pasquier and Saleur)that the resulting physics can be rescaled to obtain a quantum field theorywith conformal symmetry.

Subfactors allow one to construct many other "anyonic" spin chains withother potential Hamiltonians.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 30 / 33

Page 103: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum spin chains, tensor networks.

If quantum spins are arranged in a one dimensional array with interactionsbetween nearest neighbours, the Hilbert space for the spin chain is bnC2.

On which the e 1i s act. In a nearest neighbour fashion.

The operatorn

ÿ

i“1

ei

has been used as a Hamiltonian governing the time evoloution of aquantum spin chain of length n. Some think (e.g. Pasquier and Saleur)that the resulting physics can be rescaled to obtain a quantum field theorywith conformal symmetry.

Subfactors allow one to construct many other "anyonic" spin chains withother potential Hamiltonians.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 30 / 33

Page 104: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum spin chains, tensor networks.

If quantum spins are arranged in a one dimensional array with interactionsbetween nearest neighbours, the Hilbert space for the spin chain is bnC2.

On which the e 1i s act.

In a nearest neighbour fashion.

The operatorn

ÿ

i“1

ei

has been used as a Hamiltonian governing the time evoloution of aquantum spin chain of length n. Some think (e.g. Pasquier and Saleur)that the resulting physics can be rescaled to obtain a quantum field theorywith conformal symmetry.

Subfactors allow one to construct many other "anyonic" spin chains withother potential Hamiltonians.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 30 / 33

Page 105: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum spin chains, tensor networks.

If quantum spins are arranged in a one dimensional array with interactionsbetween nearest neighbours, the Hilbert space for the spin chain is bnC2.

On which the e 1i s act. In a nearest neighbour fashion.

The operatorn

ÿ

i“1

ei

has been used as a Hamiltonian governing the time evoloution of aquantum spin chain of length n. Some think (e.g. Pasquier and Saleur)that the resulting physics can be rescaled to obtain a quantum field theorywith conformal symmetry.

Subfactors allow one to construct many other "anyonic" spin chains withother potential Hamiltonians.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 30 / 33

Page 106: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum spin chains, tensor networks.

If quantum spins are arranged in a one dimensional array with interactionsbetween nearest neighbours, the Hilbert space for the spin chain is bnC2.

On which the e 1i s act. In a nearest neighbour fashion.

The operatorn

ÿ

i“1

ei

has been used as a Hamiltonian governing the time evoloution of aquantum spin chain of length n.

Some think (e.g. Pasquier and Saleur)that the resulting physics can be rescaled to obtain a quantum field theorywith conformal symmetry.

Subfactors allow one to construct many other "anyonic" spin chains withother potential Hamiltonians.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 30 / 33

Page 107: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Quantum spin chains, tensor networks.

If quantum spins are arranged in a one dimensional array with interactionsbetween nearest neighbours, the Hilbert space for the spin chain is bnC2.

On which the e 1i s act. In a nearest neighbour fashion.

The operatorn

ÿ

i“1

ei

has been used as a Hamiltonian governing the time evoloution of aquantum spin chain of length n. Some think (e.g. Pasquier and Saleur)that the resulting physics can be rescaled to obtain a quantum field theorywith conformal symmetry.

Subfactors allow one to construct many other "anyonic" spin chains withother potential Hamiltonians.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 30 / 33

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Quantum spin chains, tensor networks.

If quantum spins are arranged in a one dimensional array with interactionsbetween nearest neighbours, the Hilbert space for the spin chain is bnC2.

On which the e 1i s act. In a nearest neighbour fashion.

The operatorn

ÿ

i“1

ei

has been used as a Hamiltonian governing the time evoloution of aquantum spin chain of length n. Some think (e.g. Pasquier and Saleur)that the resulting physics can be rescaled to obtain a quantum field theorywith conformal symmetry.

Subfactors allow one to construct many other "anyonic" spin chains withother potential Hamiltonians.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 30 / 33

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Conformal (and other) quantum field theory.

No doubt the richest and most exciting interaction of subfactors withphysics is the connection with conformal quantum field theory in 2dimensions.Need to go back to Haag-Kastler who proposed that one study QFT usingalgebras of local observables.

To each nice region O of space time there is a von Neumann algebraApOq of observables localised in O.

The association O ÑApOq satisfies several very general axioms, whichprove in particular that the ApOq are type III1 factors.

But the one that is of most interest to us is causality which asserts that, ifO and O1 are regions so that light cannot get from O to O1 then

ApOq commutes with ApO1q

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 31 / 33

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Conformal (and other) quantum field theory.

No doubt the richest and most exciting interaction of subfactors withphysics is the connection with conformal quantum field theory in 2dimensions.

Need to go back to Haag-Kastler who proposed that one study QFT usingalgebras of local observables.

To each nice region O of space time there is a von Neumann algebraApOq of observables localised in O.

The association O ÑApOq satisfies several very general axioms, whichprove in particular that the ApOq are type III1 factors.

But the one that is of most interest to us is causality which asserts that, ifO and O1 are regions so that light cannot get from O to O1 then

ApOq commutes with ApO1q

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 31 / 33

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Conformal (and other) quantum field theory.

No doubt the richest and most exciting interaction of subfactors withphysics is the connection with conformal quantum field theory in 2dimensions.Need to go back to Haag-Kastler who proposed that one study QFT usingalgebras of local observables.

To each nice region O of space time there is a von Neumann algebraApOq of observables localised in O.

The association O ÑApOq satisfies several very general axioms, whichprove in particular that the ApOq are type III1 factors.

But the one that is of most interest to us is causality which asserts that, ifO and O1 are regions so that light cannot get from O to O1 then

ApOq commutes with ApO1q

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 31 / 33

Page 112: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Conformal (and other) quantum field theory.

No doubt the richest and most exciting interaction of subfactors withphysics is the connection with conformal quantum field theory in 2dimensions.Need to go back to Haag-Kastler who proposed that one study QFT usingalgebras of local observables.

To each nice region O of space time there is a von Neumann algebraApOq of observables localised in O.

The association O ÑApOq satisfies several very general axioms, whichprove in particular that the ApOq are type III1 factors.

But the one that is of most interest to us is causality which asserts that, ifO and O1 are regions so that light cannot get from O to O1 then

ApOq commutes with ApO1q

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 31 / 33

Page 113: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Conformal (and other) quantum field theory.

No doubt the richest and most exciting interaction of subfactors withphysics is the connection with conformal quantum field theory in 2dimensions.Need to go back to Haag-Kastler who proposed that one study QFT usingalgebras of local observables.

To each nice region O of space time there is a von Neumann algebraApOq of observables localised in O.

The association O ÑApOq satisfies several very general axioms, whichprove in particular that the ApOq are type III1 factors.

But the one that is of most interest to us is causality which asserts that, ifO and O1 are regions so that light cannot get from O to O1 then

ApOq commutes with ApO1q

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 31 / 33

Page 114: VaughanJones, Vanderbilt March302019deturck/kadison-conference/jones.pdf · SubfactorsfromPennon. VaughanJones, Vanderbilt March302019 Vaughan Jones, Vanderbilt Subfactors from Penn

Conformal (and other) quantum field theory.

No doubt the richest and most exciting interaction of subfactors withphysics is the connection with conformal quantum field theory in 2dimensions.Need to go back to Haag-Kastler who proposed that one study QFT usingalgebras of local observables.

To each nice region O of space time there is a von Neumann algebraApOq of observables localised in O.

The association O ÑApOq satisfies several very general axioms, whichprove in particular that the ApOq are type III1 factors.

But the one that is of most interest to us is causality which asserts that, ifO and O1 are regions so that light cannot get from O to O1 then

ApOq commutes with ApO1q

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 31 / 33

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Or:ApOq ĎApO1q1

A Subfactor !Doplicher, Haag and Roberts gave a much deeper analysis of this situationwhich involved bimodules(=endomorphisms).If spacetime is the circle S1 and ApOq and ApO1q are complementaryintervals I and I c we have the picture:

cI I

c(I)A A(I )

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 32 / 33

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Or:ApOq ĎApO1q1

A Subfactor !

Doplicher, Haag and Roberts gave a much deeper analysis of this situationwhich involved bimodules(=endomorphisms).If spacetime is the circle S1 and ApOq and ApO1q are complementaryintervals I and I c we have the picture:

cI I

c(I)A A(I )

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 32 / 33

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Or:ApOq ĎApO1q1

A Subfactor !Doplicher, Haag and Roberts gave a much deeper analysis of this situationwhich involved bimodules(=endomorphisms).

If spacetime is the circle S1 and ApOq and ApO1q are complementaryintervals I and I c we have the picture:

cI I

c(I)A A(I )

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 32 / 33

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Or:ApOq ĎApO1q1

A Subfactor !Doplicher, Haag and Roberts gave a much deeper analysis of this situationwhich involved bimodules(=endomorphisms).If spacetime is the circle S1 and ApOq and ApO1q are complementaryintervals I and I c we have the picture:

cI I

c(I)A A(I )

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 32 / 33

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Or:ApOq ĎApO1q1

A Subfactor !Doplicher, Haag and Roberts gave a much deeper analysis of this situationwhich involved bimodules(=endomorphisms).If spacetime is the circle S1 and ApOq and ApO1q are complementaryintervals I and I c we have the picture:

cI I

c(I)A A(I )

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 32 / 33

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cI I

c(I)A A(I )

A. Wassermann and I proposed, and he proved, that the subfactor in thiscase can indeed, in the WZW models, be an interesting one, in particularobtaining subfactors of index 4 cos2 π{n for all integers n ě 3.

But we saw there are lots of interesting subfactors around, beginning withthe Haagerup, which are unlikely to be obtainable from WZW in any way.

Over ten years ago Evans and Gannon saw structure in the Haagerupwhich suggested that there is actually a CFT which produces it!Unfortunately this construction has not yet been achieved.We are left withthe intriguing question, which is no doubt the main question in theexternal theory of subfactors:

Does EVERY (finite depth) subfactor arise in Conformal Field Theory?

This may take some time-perhaps even a good chunk of the 21st century.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 33 / 33

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cI I

c(I)A A(I )

A. Wassermann and I proposed, and he proved, that the subfactor in thiscase can indeed, in the WZW models, be an interesting one, in particularobtaining subfactors of index 4 cos2 π{n for all integers n ě 3.

But we saw there are lots of interesting subfactors around, beginning withthe Haagerup, which are unlikely to be obtainable from WZW in any way.

Over ten years ago Evans and Gannon saw structure in the Haagerupwhich suggested that there is actually a CFT which produces it!Unfortunately this construction has not yet been achieved.We are left withthe intriguing question, which is no doubt the main question in theexternal theory of subfactors:

Does EVERY (finite depth) subfactor arise in Conformal Field Theory?

This may take some time-perhaps even a good chunk of the 21st century.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 33 / 33

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cI I

c(I)A A(I )

A. Wassermann and I proposed, and he proved, that the subfactor in thiscase can indeed, in the WZW models, be an interesting one, in particularobtaining subfactors of index 4 cos2 π{n for all integers n ě 3.

But we saw there are lots of interesting subfactors around, beginning withthe Haagerup, which are unlikely to be obtainable from WZW in any way.

Over ten years ago Evans and Gannon saw structure in the Haagerupwhich suggested that there is actually a CFT which produces it!Unfortunately this construction has not yet been achieved.We are left withthe intriguing question, which is no doubt the main question in theexternal theory of subfactors:

Does EVERY (finite depth) subfactor arise in Conformal Field Theory?

This may take some time-perhaps even a good chunk of the 21st century.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 33 / 33

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cI I

c(I)A A(I )

A. Wassermann and I proposed, and he proved, that the subfactor in thiscase can indeed, in the WZW models, be an interesting one, in particularobtaining subfactors of index 4 cos2 π{n for all integers n ě 3.

But we saw there are lots of interesting subfactors around, beginning withthe Haagerup, which are unlikely to be obtainable from WZW in any way.

Over ten years ago Evans and Gannon saw structure in the Haagerupwhich suggested that there is actually a CFT which produces it!Unfortunately this construction has not yet been achieved.

We are left withthe intriguing question, which is no doubt the main question in theexternal theory of subfactors:

Does EVERY (finite depth) subfactor arise in Conformal Field Theory?

This may take some time-perhaps even a good chunk of the 21st century.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 33 / 33

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cI I

c(I)A A(I )

A. Wassermann and I proposed, and he proved, that the subfactor in thiscase can indeed, in the WZW models, be an interesting one, in particularobtaining subfactors of index 4 cos2 π{n for all integers n ě 3.

But we saw there are lots of interesting subfactors around, beginning withthe Haagerup, which are unlikely to be obtainable from WZW in any way.

Over ten years ago Evans and Gannon saw structure in the Haagerupwhich suggested that there is actually a CFT which produces it!Unfortunately this construction has not yet been achieved.We are left withthe intriguing question, which is no doubt the main question in theexternal theory of subfactors:

Does EVERY (finite depth) subfactor arise in Conformal Field Theory?

This may take some time-perhaps even a good chunk of the 21st century.

Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 33 / 33

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cI I

c(I)A A(I )

A. Wassermann and I proposed, and he proved, that the subfactor in thiscase can indeed, in the WZW models, be an interesting one, in particularobtaining subfactors of index 4 cos2 π{n for all integers n ě 3.

But we saw there are lots of interesting subfactors around, beginning withthe Haagerup, which are unlikely to be obtainable from WZW in any way.

Over ten years ago Evans and Gannon saw structure in the Haagerupwhich suggested that there is actually a CFT which produces it!Unfortunately this construction has not yet been achieved.We are left withthe intriguing question, which is no doubt the main question in theexternal theory of subfactors:

Does EVERY (finite depth) subfactor arise in Conformal Field Theory?

This may take some time-perhaps even a good chunk of the 21st century.Vaughan Jones, Vanderbilt Subfactors from Penn on. March 30 2019 33 / 33