Post on 12-Jul-2020
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
Twisted Poincaré duality for some quadratic Poisson algebras
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
Twisted Poincaré duality for some quadratic Poisson algebras
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
.
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson algebra and semi-classical limits
Poisson algebras, Poisson modules
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.
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson algebra and semi-classical limits
Poisson algebras, Poisson modules
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.
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson algebra and semi-classical limits
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).
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson algebra and semi-classical limits
Quantisation, semi-classical limit
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 .
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson algebra and semi-classical limits
Quantisation, semi-classical limit
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 .
Twisted Poincaré duality for some quadratic Poisson algebras
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.
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson (co)homology
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...).
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson (co)homology
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.
Twisted Poincaré duality for some quadratic Poisson algebras
Poisson (co)homology
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.
Twisted Poincaré duality for some quadratic Poisson algebras
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.
Twisted Poincaré duality for some quadratic Poisson algebras
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.
Twisted Poincaré duality for some quadratic Poisson algebras
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.
Twisted Poincaré duality for some quadratic Poisson algebras
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).
Twisted Poincaré duality for some quadratic Poisson algebras
Duality
Twisted Poincaré 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 .
Twisted Poincaré duality for some quadratic Poisson algebras
Duality
Twisted Poincaré 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.
Twisted Poincaré duality for some quadratic Poisson algebras
Duality
Twisted Poincaré 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).
Twisted Poincaré duality for some quadratic Poisson algebras
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.
Twisted Poincaré duality for some quadratic Poisson algebras
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 .