From Schur-Weyl duality to quantum symmetric pairs · Schur-Weyl duality. . . . . . . Quantization....
Transcript of From Schur-Weyl duality to quantum symmetric pairs · Schur-Weyl duality. . . . . . . Quantization....
. . . . . .Schur-Weyl duality
. . . . . . .Quantization
. . . . . . . . .q-Schur duality
. . . . . . .BLM construction
. . . . . . .Quantum symmetric pairs
From Schur-Weyl duality to quantum symmetric pairs
Chun-Ju Lai
Max Planck Institute for Mathematics in Bonn
Dec 8, 2016
. . . . . .Schur-Weyl duality
. . . . . . .Quantization
. . . . . . . . .q-Schur duality
. . . . . . .BLM construction
. . . . . . .Quantum symmetric pairs
Outline
..1 Schur-Weyl duality
..2 Quantization
..3 q-Schur duality
..4 BLM construction
..5 Quantum symmetric pairs
. . . . . .Schur-Weyl duality
. . . . . . .Quantization
. . . . . . . . .q-Schur duality
. . . . . . .BLM construction
. . . . . . .Quantum symmetric pairs
Background
• GLn := GLn(C) = general linear group of Cn
• Schur’s 1901 dissertation on the polynomial representations of GLn:− Each polynomial representation V is semi-simple
(i.e. V =⊕
irreducible modules)− Each polynomial representation is homogeneous for some degree d ∈ N− There is a correspondence{
polynomial representationof GLn of degree d
}1:1←→
{representation of
symmetric group Sd
}{irreducibles} ↔ {irreducibles}↔
{partitions λ = (λ1, . . . , λn) ⊢ d}
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Background
• Weyl’s 1926 research on representation of semi-simple Lie group− based on repn theory of Lie algebra, integration over compact form− has no counterpart to Sd in Schur’s method
• In 1927, Schur re-derived his 1901 dissertation by showing the doublecentralizer property for (GLn,Sd), which was publicized by Weyl in his1939 book “The Classic Groups”.
• The methods used in the classical Schur-Weyl are still important inrepresentation theory today
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. . . . . . .Quantum symmetric pairs
Dual actions
• gln := gln(C): general linear Lie algebraV := Cn natural representation of gln⇒ action on V ⊗d by the Leibniz rule, e.g.,
g · (v1 ⊗ v2) = (g · v1)⊗ v2 + v1 ⊗ (g · v2)
so that its exponential etg acts group-like –
etg · (v1 ⊗ v2) = etg · v1 ⊗ etg · v2
• Sd has a (right) action on V ⊗d by permuting tensor factors, e.g.,
(v1 ⊗ v2) · (12) = v2 ⊗ v1
glnψ↷ V ⊗d
φ↶ Sd
general linear Lie algebra tensor space symmetric group
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Example: commutivity
Here’s an example showing(g(v1 ⊗ v2)
)(12) = g
((v1 ⊗ v2)(12)
)V ⊗ V
(12) - V ⊗ V
v1 ⊗ v2 - v2 ⊗ v1
⟳
gv1 ⊗ v2 + v1 ⊗ gv2?
- gv2 ⊗ v1 + v2 ⊗ gv1?
V ⊗ V
g
?
(12)- V ⊗ V
g
?
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Double Centralizer Property
• We have ψ : gln → End(V ⊗d), φ : Sd → End(V ⊗d) and
glnψ↷ V ⊗d
φ↶ Sd
general linear Lie algebra tensor space symmetric group
Schur-Weyl duality (1927)
..1 The actions of gln and Sd on the tensor space V ⊗d commute
..2 The algebras generated by the actions of gln and Sd in End(V ⊗d) arecentralizing algebras of each other, i.e.,
Endφ(A)(V⊗d) = ψ(B), Endψ(B)(V
⊗d) = φ(A),
where A = C[Sd] = group algebra, B = U(gln) = univ env algebra
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Double centralizer property
• The double centralizer property leads to
Corollary
There is a decomposition
V ⊗d =⊕λ⊢d
Vλ ⊗ Lλ,
where {Vλ} = Irrep(Sd);{Lλ} are distinct irreducibles of gln or 0.
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Quantization
U(gln)univ env algebra
↷ (Cn)⊗d ↶ C[Sd]group algebra of sym group
Uq(gln)quantum group
quantization
?
............↷ (Q(v)n)⊗d ↶ H(Sd)
Hecke algebra
quantization
?
............
Here the quantized algebras are over Q(v) with v = q1/2: indeterminate
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Hecke algebra
• Recall that the symmetric group Sd is generated by simple reflections
s1, s2, . . . , sd−1
subject to the braid relations and s2i = 1.
• Hecke algebra H(Sd) is a Q(v)-algebra generated by
T1, T2, . . . , Td−1
subject to braid relations and the Hecke relations
(Ti + 1)(Ti − q) = 0. (1)
• Specializing q → 1, (1) recovers s2i = 1
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Hecke algebra
• H(Sd) has a linear basis (called standard basis)
{Tw | w ∈ Sd}
• H(Sd) has a involution ¯ : H(Sd)→ H(Sd) sending
v 7→ v−1, Tw 7→ T−1w−1
• H(Sd) has a bar-invariant basis {Cw}w∈Sdcalled the Kazhdan-Lusztig(KL)
basis
• The famous Kazhdan-Lusztig theory (’79) offered a solution the the difficultproblem of determining irreducible character problem for category O ofsemisimple Lie algebras using the KL basis
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Quantum groups
• Recall that the universal enveloping algebra U(gln) is an associative algebragenerated by
{ei, fi, dj , d−1j }, subject to
Chevalley/Serre relations (e.g. e21e2 + e2e21 = 2e1e2e1)
• Around 1985, Drinfeld and Jimbo introduced the quantum groupU = Uq(gln) as a Q(v)-algebra generated by
{Ei, Fi, Dj , D−1j }, subject to
q-Chevellay/Serre relations (e.g. E21E2 + E2E
21 = (v + v−1)E1E2E1)
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Quantum groups
• U provides solutions to the Yang-Baxter equation
• U is a Hopf algebra. Particularly it has a comultiplication ∆ : U→ U⊗U
• There is a triangular decomposition U = U−U0U+
• There is a modified (i.e. idempotented) quantum group U:
Uquantum group
replacing 1 with idempotents−−−−−−−−−−−−−−−−−⇀↽−−−−−−−−−−−−−−−−−take certain infinite sum
Umodified quantum group
{U-modules} 1:1↔ {weight modules of U}
• U is viewed as a preadditive category in categorification
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Canonical basis
• U has a sheaf theoretic interpretation
⇒ bar involution ¯ : U→ U from the Verdier duality
• U− admits canonical basis, i.e., U− has a standard basis {ai}i∈I that is“unitriangular” and “integral”:
ai ∈ ai +∑j<i
Z[v, v−1]aj ,
and a (unique) canonical basis {bi}i∈I that is “bar-invariant” and“positive”:
bi = bi ∈ ai +∑j<i
v−1N[v−1]aj
• This is analogous to (dual) Kazhdan-Lusztig basis for Hecke algebra
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Canonical basis
• Canonical basis = Kashiwara’s global crystal basis, i.e., for any weight λ,
{bivλ} is a basis of irreducible integrable U-module L(λ) := U−vλ
• Canonical basis {bi} has positivity, i.e.
bibj ∈∑k
N[v, v−1]bk
• U does not admit canonical basis, while U does
• Existence of canonical basis has connection/application in− Categorification− Algebraic combinatorics− Category O− Cluster algebra− Geometric representation theory
. . . . . .Schur-Weyl duality
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. . . . . . .BLM construction
. . . . . . .Quantum symmetric pairs
Outline
..1 Schur-Weyl duality
..2 Quantization
..3 q-Schur duality
..4 BLM construction
..5 Quantum symmetric pairs
. . . . . .Schur-Weyl duality
. . . . . . .Quantization
. . . . . . . . .q-Schur duality
. . . . . . .BLM construction
. . . . . . .Quantum symmetric pairs
q-Schur duality of type A
Our goal here is to describe the quantized Schur-Weyl duality below inexamples:
Uq(gln)quantum group
ψ↷ V⊗d
φ↶ H(Sd)
Hecke algebra
We start with describing first a tensor space V⊗d admitting natural actionsfrom both sides
. . . . . .Schur-Weyl duality
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Natural representation V of Uq(gln)
• V :=n∑i=1
Q(v)vi: natural representation of Uq(gln), e.g. n = 3
v1F1⇄E1
v2F2⇄E2
v3
D3 ⟳×1
⟳×1
⟳×q
D−12 ⟳×1
⟳×q−1
⟳×1
• Uq(gln) acts on V⊗d via comultiplication, e.g.
∆(F2) = D−12 D3 ⊗ F2 + F2 ⊗ 1
Hence
F2(v3 ⊗ v2) = D−12 D3v3 ⊗ F2v2 +���F2v3 ⊗ v2 = qv3 ⊗ v3F2(v2 ⊗ v3) = D−12 D3v2 ⊗���F2v3 + F2v2 ⊗ v3 = v3 ⊗ v3
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Hecke algebra action
• For H(S2), we have the following equalities in one-line notation(1 = [12], s1 = [21])
T[12]T1 = T[21]T[21]T1 = qT[12] + (q − 1)T[21]
• This leads to a natural Hecke algebra action on V⊗2 respecting themultiplication rule, e.g.
(va ⊗ vb)T1 =
vb ⊗ va if b > a
qvb ⊗ va + (q − 1)va ⊗ vb if b < a
qva ⊗ va if b = a
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Example: commutivity
Here’s an example showing(F2(v3 ⊗ v2)
)T1 = F2
((v3 ⊗ v2)T1
)V⊗ V
T1 - V⊗ V
v3 ⊗ v2 - (q − 1)v3 ⊗ v2 + qv2 ⊗ v3
⟳
qv3 ⊗ v3?
- q2v3 ⊗ v3 = (q2 − q)v3 ⊗ v3 + qv3 ⊗ v3?
V⊗ V
F2
?
T1- V⊗ V
F2
?
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. . . . . . .Quantum symmetric pairs
q-Schur duality
Uq(gln)quantum group
ψ↷ V⊗d
φ↶ H(Sd)
Hecke algebra
q-Schur duality of type A (Jimbo’86)
The algebras Uq(gln) and H(Sd) satisfy the double centralizer property:
Endφ(A)(V⊗d) = ψ(B), Endψ(B)(V⊗d) = φ(A),
where A = H(Sd), B = Uq(gln)
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. . . . . . .Quantum symmetric pairs
q-Schur algebra
• In other words, there is a surjection, for each d, from Uq(gln) to theq-Schur algebra
Sn,d = EndH(Sd)(V⊗d),
and hence we have
U(gln)quantum group↠
Sn,dq-Schur algebra
↷ V⊗dtensor space
↶ H(Sd)Hecke algebra
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q-Schur duality for other types
• It is known that Uq(gln) is a type A quantum group. There are (q-)Schurduality for other types, e.g.,
Onorthogonal group↠
Son,d
orthogonal Schur algebra
↷ V ⊗d ↶ BdBrauer algebra
Uq(son)quantum group↠
Son,d
orthogonal q-Schur algebra
↷ V⊗d ↶ BMWdBirman-Murakami-Wenzl algebra
• In these pictures the “types” are associated to the classical/quantum groups
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q-Schur duality for other types
• If we associate the “types” to the Hecke algebra instead, we obtain anotherfamily of (q-)Schur duality:
SXn,d := EndHX
d(V⊗d)
q-Schur algebra of type X
↷ V⊗d ↶ HXd
type X Hecke algebra
• Question: Is there a bottom-top construction that recovers the quantumgroup from the Schur algebras?
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BLM construction
• The answer is Yes, the construction for type A is provided in 1990 byBeilinson-Lusztig-MacPherson (BLM) geometrically using partial flags and aso-called stabilization procedure. We can paraphrase it algebraically asbelow:
KAn
stabilization algebra of type A
⇒
SAn,d := EndHA
d(V⊗d)
q-Schur algebra of type A
↷ V⊗d ↶ HAd := H(Sd)
type A Hecke algebra
• Stabilization algebra KAn ≈ inverse limit lim←−
d∈NSAn,d
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BLM construction
KAn
stabilization algebra of type A⇒
SAn,d := EndHA
d(V⊗d)
q-Schur algebra of type A
↷ V⊗d ↶ HAd := H(Sd)
type A Hecke algebra
• Canonical bases for SAn,d(d ∈ N) lift “compatibly” to canonical basis of KA
n
• KAn ≃ U(gln)
⇒ A concrete realization of canonical basis of U(gln)
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BLM construction
• A BLM construction (for type X) produces from a family of q-Schur algebra{SX
n,d | d ∈ N} an Q(v)-algebra KXn that enjoys similar properties
KXn
stabilization algebra of type X
⇒
SXn,d := EndHX
d(V⊗d)
q-Schur algebra of type X
↷ V⊗d ↶ HXd
type X Hecke algebra
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New “quantum groups”
• In type B/C, there are two types of BLM constructions due toBao-Kujawa-Li-Wang (’14) associated to involutions (type ı, ȷ) of theDynkin diagram of type A
..
...1
...2
..· · · ...
n2 − 1
...
n2
. . . . . .
...n
...
n− 1
..· · · ...n2 + 2
...n2 + 1
............. ..
. ...1
..· · · ...
n−12
. . . . ...
n+12
. ...n
..· · · ...n+32
.........
type ȷ : n even type ı : n odd
B KBCn is NOT the modified Drinfeld quantum group of type B/C; it is a
modified quantization of certain product gl• × sl•
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Applications of new “quantum groups”
• Depending on parity of n, KBCn is isomorphic to involutive quantum groups
Uın or Uȷ
n introduced by Bao-Wang (’13). The canonical basis theory forUın,U
ȷn
− gives a new formulation of KL theory for Lie algebra of type B/C− establishes for the first time KL theory for Lie superalgebra osp(2m+ 1|2n)
(i.e. super type B/C)
⇒ reformulation of KL theory for Lie algebra of type D and for Li superalgebraosp(2m|2n) (i.e. super type D) by Bao (’16)
• We expect similar applications for other types (e.g. affine classical)
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BLM-type constructions
• Here’s a summary on what are known
BLM construction Geometric Algebraic(method) (dimension counting on flags) (combinatorics)Affine A Ginzburg-Vasserot (’93) Du-Fu (’14)
Lusztig (’99)Finite B/C BKLW (’14)Finite D Fan-Li (’14)Affine C Fan-Lai-Li-Luo-Wang1 (’16) FLLLW2 (’16)
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Coideal subalgebras
• It is worth mentioning that Uın,U
ȷn are coideal subalgebras of U(gln), i.e.,
the comultiplication ∆ : U(gln)→ U(gln)⊗U(gln) satisfies that
∆(Uın) ⊂ Uı
n ⊗U(gln)
∆(Uȷn) ⊂ Uȷ
n ⊗U(gln)
B A coideal subalgebra is different from a Hopf subalgebra B of U s.t.∆(B) ⊂ B⊗B
. . . . . .Schur-Weyl duality
. . . . . . .Quantization
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. . . . . . .BLM construction
. . . . . . .Quantum symmetric pairs
Outline
..1 Schur-Weyl duality
..2 Quantization
..3 q-Schur duality
..4 BLM construction
..5 Quantum symmetric pairs
. . . . . .Schur-Weyl duality
. . . . . . .Quantization
. . . . . . . . .q-Schur duality
. . . . . . .BLM construction
. . . . . . .Quantum symmetric pairs
Symmetric pairs
• A symmetric pair (g, gθ) consists of a Lie algebra g and its fixed-pointsubalgebra gθ with respect to an involution
θ : g→ g
• It plays an important role in the study of real reductive groups
• Classification of symmetric pairs of finite type= Classification of real simple Lie algebras↔ Satake diagrams (of finite type)
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Quantum symmetric pairs
• The quantum symmetric pair(QSP) is introduced by Letzter (’02) for finitetype; and by Kolb (’14) for symmetrizable Kac-Moody as a quantization ofthe symmetric pair (g, gθ)
• The pair (U,B) is a QSP if U = U(g) and B = B(θ) is certain coidealsubalgebra of U, i.e., the comultiplication ∆ : U→ U⊗U satisfies that
∆(B) ⊂ B⊗U
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Examples
The following are examples of QSP:
• (Uq(gln), twisted Yangian) from Yang-Baxter equation
• (Uq(g),B) where g is of classical type, and B is constructed based onsolutions of the reflection equation
RKRK = KRKR
• (Uq(sl2), q-Onsager algebra) from Ising model
• Quantum algebras from BLM construction. In other words, quantumalgebras arising from Schur-Weyl duality
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QSP from BLM construction
Recall that
{SXn,d | d ∈ N} =========
stabilization⇒ KX
n =========take inf sum
⇒ KXn
The resulting quantum algebras are:
type quantum algebra KXn
Finite A Uq(gln)Finite B/C KBC
n ≃ Uın,U
ȷn
Affine A Uq(gln)
Affine C KCn (four variants)
Proposition
The pairs (U(gln),Uın), (U(gln),U
ȷn) and all four variants of (U(gln),K
Cn) are
quantum symmetric pairs.
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Applications and future work
• A canonical basis theory for QSP of Satake diagrams finite type is developedrecently by Bao-Wang (’16). It may lead to similar application as in(U(gln),K
Xn) with further study
? canonical basis theory for affine QSP
? BLM construction from {SXn,d} when X is of affine B/D, finite and affine
exceptional type
? BLM-type construction for other Schur-type dualities (e.g., for Braueralgebras, partition algebras)
? BLM-type construction for Howe duality
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Thank you for your attention