Twisted Poincaré duality for some quadratic Poisson...

Twisted Poincaré duality for some quadratic Poisson algebras

Twisted Poincaré duality for some quadraticPoisson algebras

Letters in Mathematical Physics, 2007

S. Launois1 L. Richard2

Metz, 08-11-2007

1University of Kent2University of Edinburgh and Maxwell Institute for Mathematical Sciences


Outline

Poisson algebra and semi-classical limitsPoisson algebras, Poisson modulesQuantisation, semi-classical limit

Poisson (co)homologyPoisson cohomologyPoisson homology

DualityExample: the affine planeTwisted Poincaré dualityResults and prospects


Poisson algebra and semi-classical limits

Poisson algebras, Poisson modules

Poisson algebras

DefinitionA Poisson algebra is a commutative algebra R endowed with abilinear bracket ., . such that

I (R, ., .) is a Lie algebra;I r , . is a derivation of R for all r ∈ R.

Example

1. R = C∞(R2), coordinates (x , y), f , g = ∂f∂x

∂g∂y −

∂g∂x

∂f∂y .

Note that x , y = 1, i.e. this bracket comes from thesymplectic structure on R2.

2. [Poisson, 1809] R = C∞(R2n),f , g =

∑ni=1

∂f∂xi

∂g∂yi

− ∂g∂xi

∂f∂yi

.




Algebraic examplesNote that for a finitely generated algebra one only needs todefine the bracket on generators.

1. R = C[X , Y ], set X , Y = P ∈ R. This always defines aPoisson structure. (Poisson cohomology: Roger andVanhaecke, 2002).

2. R = C[X , Y , Z ], set Φ ∈ R. Define X , Y = ∂Φ∂Z ,

Z , X = ∂Φ∂Y , Y , Z = ∂Φ

∂X . One may check it defines aPoisson structure, Φ is called the potential of the bracket.(Poisson cohomology: Pichereau, 2006).

Poisson homology and cohomology are important invariants ofthe Poisson structure.A whole class of examples comes from the “semi-classical” limitprocess, which will be described in the sequel.




Poisson modules

DefinitionA Poisson module over R is a vector space M endowed withtwo bilinear maps . and ., .M such that

I (M, .) is a (right) module over the commutative algebra R,I (M, ., .M) is a (right) module over the Lie algebra

(R, ., .),I x .a, b = x , aM .b− x .b, aM for all a, b ∈ R and x ∈ M.I x , abM = x , aM .b + x , bM .a for all a, b ∈ R and

x ∈ M.

ExamplesLet I ⊂ R be a Poisson ideal. Then I, R/I are Poisson modules.



Quantisation, semi-classical limit

R Poisson algebra −→ A associative noncommutative algebrawith product coming from the Poisson bracket of R.

The semi-classical limit is, roughly speaking, the inverseprocess, which can be done in the following way.

From now on k = C.Let A be an algebra such that ∃h ∈ Z (A) not a zero divisor andsuch that A = A/hA is commutative (i.e. [A, A] ⊆ hA).Then seta + hA, b + hA = [a, b]/h + hA.(A, ., .) is Poisson, and called the semi-classical limit of A.A is called a quantification of A.For λ ∈ C∗, Aλ := A/(h − λ)A is called a deformation of A.

ExampleA = U(g), g = Cx ⊕ Cy ⊕ Cz, [x , y ] = z, z central.Set h = z,then A = C[X , Y ], X , Y = 1.∀q ∈ C∗, Aq ' A1(C).




Main example

M = (aij) ∈ Mn(Z) skew-symmetric.A = C[h±1] < xi , 1 ≤ i ≤ n | xixj = haij xjxi > is thequantification. (Note that h = h − 1)For q ∈ C∗, Aq = A/(h − q)A is a quantum affine space(deformation).The semi-classical limit is A = A/(1− h)A ' C[X1, . . . , Xn] withbracket as follows.Xi , Xj =

[xi ,xj ]h−1 + (h − 1)A =

(haij−1)xj xih−1 + (h − 1)A,

Xi , Xj = aijXiXj .




Formally, one could consider that Xi , Xj =[xi ,xj ]q−1

∣∣∣q=1

in the

deformation.

Remark

1. In the rest of the talk, the hypothesis thataij ∈ Z will not playany role.

2. For a monomialXα = Xα11 . . . Xαn

n we get

Xα11 . . . Xαn

n , Xi = −∑

j

aijαjXα+εi .


Poisson (co)homology

Poisson cohomology

Poisson cohomology.Now (R, ., .) denotes a Poisson algebra over C.The Poissoncohomology is given by the following complexχ∗(R) := ⊕k∈Nχk (R), with χk (R) the R-module of allskew-symmetric k -linear derivations of R;

DefinitionFor any k ∈ Z≥0,χk (R) = φ ∈ HomC(ΛkR, R) | φ(ab, a2, . . . , ak ) =aφ(b, a2, . . . , ak ) + bφ(a, a2, . . . , ak ). It is made an R-modulein the obvious way.

Proposition

1. χ0(R) = R; χ1(R) = Der(R, R).

2. If R = C[X1, . . . , Xn] then χp(R) = 0 for all p > n.



Poisson cohomology

Poisson cohomology: the differentialI The Poisson coboundary operator δk : χk (R) → χk+1(R) is

defined by

δk (P)(f0, . . . , fk ) :=k∑

i=0

(−1)i

fi , P(f0, . . . , fi , . . . , fk )

+∑

0≤i<j≤k

(−1)i+jP(fi , fj, f0, . . . , fi , . . . , fj , . . . , fk

)I Poisson cohomology group:

HPk (R) = Kerδk/Imδk−1.

RemarkThe Poisson cohomology contains important informations concerningthe Poisson structure (Casimir, derivations, deformations...).



Poisson homology

Poisson homology.The Poisson (canonical) homology of R (with value in M) isgiven by the complex CPoiss

k (R, M) := M ⊗R Ωk (R), with thefollowing

Definition

1. The R-module Ω1(R) of Kähler differential forms isgenerated by the symbols da for all a ∈ R, with relations1.1 d(ab) = adb + bda;1.2 d(a + b) = da + db;1.3 dλ = 0 for all λ ∈ C.

2. Ωk (R) = ΛkRΩ1(R) is the R-module of Kähler

k -differentials.

PropositionIf R = C[X1, . . . , Xn], then Ωk (R) ' R ⊗C ΛkV, withV = CdX1 ⊕ · · · ⊕ CdXn.



Poisson homology

Poisson homology: the differential

I The boundary operator ∂k : CPoissk (R, M) → CPoiss

k−1 (R, M) isdefined by

∂k (m ⊗ da1 ∧ · · · ∧ dak ) =

=k∑

i=1(−1)i+1m, aiM ⊗ da1 ∧ · · · ∧ dai ∧ · · · ∧ dak+

+∑

1≤i<j≤k(−1)i+jm ⊗ dai , aj ∧ da1 ∧ · · · ∧ dai ∧ · · · ∧ daj ∧ · · · ∧ dak

I Poisson homology group:

HPk (R, M) = Ker∂k/Im∂k+1.


Duality

Duality

Recall the following notations concerning our main example.

M = (aij) ∈ Mn(Z) skew-symmetric.R = C[X1, . . . , Xn] with bracket

Xi , Xj = aijXiXj .

It is the semiclassical limit of the quantum affine spaceU = C < x1, . . . , xn | xixj = qaij xjxi >, q ∈ C∗ generic.


Duality

Example: the affine plane

Affine plane: first bracket.

Consider the algebra R1 = C[X , Y ] endowed with the Poissonbracket defined by X , Y1 = 1.

Proposition

1. HP2(R1) ' C, and HPk (R1) = 0 for all k 6= 2;

2. HP0(R1) ' C, and HPk (R1) = 0 for all k ≥ 1;

3. HPk (R1) ' HP2−k (R1) for all 0 ≤ k ≤ 2.

RemarkThis kind of duality always holds in the unimodular case.


Duality

Example: the affine plane

Affine plane: second bracket.

Now consider the algebra R2 = C[X , Y ] endowed with thePoisson bracket defined by X , Y2 = XY .

Proposition

1. HP0(R2) and HP1(R2) are infinite-dimensional, andHPk (R2) = 0 for all k ≥ 2;

2. HP0(R1) ' C, HP1(R2) ' C2, HP2(R2) ' C2, andHPk (R2) = 0 for all k ≥ 3.

Main idea. This Poisson structure admits a deformation,namely the quantum affine plane, for which exists a twistedduality between the Hochschild homology and cohomology,thanks to a theorem of Van den Bergh.


Duality

Twisted Poincaré duality

Duality à la Van den Bergh for affine quantum space

I Let q ∈ C∗ be a non-root of unity.I Set U = CQ[x1, . . . , xn], the quantum affine space

parametrised by Q = (qij) ∈ Mn(C∗), with qij = qaij .I Let σ be the automorphism of U defined by σ(xi) = pixi ,

with pi =∏

j qji .I Let σU denote the U-bimodule that is U as a C-vector

space, with product twisted on the left by σ, i.e.a · u · b = σ(a)ub for all a, b ∈ U, u ∈ σU.

Theorem (Van den Bergh)

HH∗(U, σU) ≡ HHn−∗(U, U).


Duality


The Poisson module MAs a vector space, M = C[X1, . . . , Xn] = R, and M is endowedwith the following two actions of R:

I the external product “.” is just the usual product of R;I the external bracket ., .M is defined by

m, XiM :=mxi − σ(xi)m

q − 1

∣∣∣∣q=1

for all m ∈ M and i ∈ 1, . . . , n.In particular, whenm = Xα1

1 . . . Xαnn is a monomial,

Xα11 . . . Xαn

n , XiM = −∑

j

aij(αj − 1)Xα+εi .


Duality


A vector space isomorphism.

Thanks to the canonical volume form dX1 ∧ . . . ∧ dXn, the setχk (R) of all skew-symmetric k -linear derivations of R isisomorphic as a vector space to M ⊗R Ωn−k (R) via anisomorphism denoted by † and defined by:

†(P) =∑

σ∈Sk,n−k

ε(σ)P(Xσ1 , . . . , Xσk )dXσk+1 ∧ · · · ∧ dXσn

for all P ∈ χk (R). Here we denote by Sn the set of alln-permutations. For all σ ∈ Sn, we denote by ε(σ) its sign andwe set σi := σ(i). Also Sk ,n−k denotes the set of thosepermutations σ ∈ Sn such that σ1 < · · · < σk andσk+1 < · · · < σn.


Duality


Comparing homology and cohomology.

χk (R)

δk

† //

M ⊗R Ωn−k (R)

(−1)k+1∂n−k

χk+1(R)

† // M ⊗R Ωn−k−1(R)

This diagram does not commute a priori, but

PropositionFor all P ∈ χk (R), the following equality holds:

(† δ)(P) = (−1)k+1(∂ †)(P).


Duality

Results and prospects

Results

TheoremFor all k ∈ N, we have HPk (R, M) ' HPn−k (R).

Corollary (Monnier)

HPk (R) '⊕

|β|=n−kα+β∈C

CXαdXβ,

where C := γ ∈ Nn | γi = 0 or∑n

j=1 aij(γj − 1) = 0.

RemarkThe Poisson cohomology spaces ofR are canonically isomorphic tothe Hochschild cohomology spaces of its quantisation.


Duality

Results and prospects

Further readings

I Huebschmann, J.: Poisson cohomology and quantization,J. reine angew. Math. 408, 57-113 (1990).

I Chemla, S.: Poincaré duality for k-A Lie superalgebras,Bull. Soc. Math. France 122, no. 3, 371-397 (1994).

I Xu, P.: Gerstenhaber algebras and BV-algebras in Poissongeometry, Comm. Math. Phys. 200, no. 3, 545-560 (1999).

I Dolgushev, V.: The Van den Bergh duality and the modularsymmetry of a Poisson variety, preprint,arxiv.org/math.QA/0612288 .

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