Non-commutative Poisson Structures on quantum torus orbifolds
aa r X i v : . [ m a t h . K T ] J u l NON-COMMUTATIVE POISSON STRUCTURES ON QUANTUMTORUS ORBIFOLDS
SAFDAR QUDDUS
Abstract.
We study the Hochschild cohomology and the Gerstenhaber algebra structureon the algebraic non-commutative torus/quantum torus orbifolds resulting by the actionof finite subgroups of SL ( Z ). We also examine the Poisson structures and compute thePoisson cohomology. Introduction
Inspired by this relationship between Poisson geometry and deformation theory, JonathanBlock and Ezra Getzler [BG] and Ping Xu [Xu] independently introduced a notion of anoncommutative Poisson structure and the non-commutative Poisson cohomology was stud-ied by Xu explicitly for the smooth non-commutative 2-torus A θ . The paper of Halboutand Tang [HT] is the seminal paper for studying non-commutative Poisson structures onorbifolds or smash products. [HT] studied the Gestenhaber bracket and Poisson struc-ture(s) on the C ∞ ( M ) ⋊ G , they showed that the Gerstenhaber bracket on the orbifoldis a generalization of the classical Schouten-Nijenhuis bracket on manifold. This Gersten-haber bracket on the orbifold, called as the “twisted Schouten-Nijenhuis bracket” was usedby them to understand the Poisson structures on the orbifold by solving [Π , Π] = 0 forΠ ∈ H ( C ∞ ( M ) ⋊ G, C ∞ ( M ) ⋊ G ). This is how the non-commutative Poisson structurewas studied for A ⋊ G for A = C ∞ ( M ). We can ask the question, what about such struc-tures when A is a non-commutative (/smooth)manifold.The graded Lie bracket on Hochschild cohomology remains elusive in contrast to the cupproduct. The latter may be defined via any convenient projective resolution. But, the for-mer is defined on the bar resolution, which is useful theoretically but not computationally,and one typically computes graded Lie brackets by translating to another more convenientresolution via explicit chain maps. Such chain maps are not always easy to find. Studyingthe Grestenhaber bracket on the Hochschild cohomology also gives us an insight to thedeformations of the algebra, Hochschild cohomology of the A ⋊ G , along with the Gersten-haber bracket, is reflective of the deformation theory of modules, orbifold cohomology andPoisson cohomology. [PPTT] [LW1] [LW2] [BG].Researchers like Witherspoon, Shepler, Negron and Zhou in various collaborations havestudied the “twisted Schouten-Nijenhuis bracket” for several variant spaces and discretegroup actions, including the polynomial ring Sym ( V ) and the quantum polynomial ring S q ( V ) [SW1] [WZ] [NW]. They used appropriate Koszul resolution to study the Gersten-haber bracket structure. In the paper [WZ] the discrete group action on the quantumsymmetric polynomial algebra is considered for the special case when the action by a dis-crete group G on it is diagonal( v i χ g v i for some character χ : G → C × ). The Koszul Date : July 6, 2020.
Key words and phrases.
Gerstenhaber algebra,quantum torus, Poisson structure. esolution for the quantum symmetric polynomial is due to Wambst[W1] which he obtainedby generalizing the Connes’ Koszul resolution for non-commutative smooth 2-torus[C] toquantum symmetric algebra S q ( V ) associated to a vector space V . The quantum torus alsoknown as the quantum torus, can be thought as a quantum symmetric algebra in 2-variablesbut unlike Witherspoon and Zhou we shall be considering non-diagonal discrete group ac-tion on it arising by the restriction of the action of SL ( Z ) on the smooth non-commutativetorus A θ .In fact Gerstenhaber bracket is a generalization of the usual Schouten-brackets of multivec-tor fields [T] and we saw in [HT] that the Schouten-bracket gets twisted over orbifolds ofsmooth manifolds. Here, in this example, we shall see that over the non-commutative orb-ifolds such a twist can in fact be similarly understood over each twisted component arisingby the paracyclic or the spectral decomposition of the cohomology.Non-commutative torus was studied extensively by Connes and Rieffel in the 1980’s andarose as a quantum deformation of the algebra of smooth functions on the classical 2-torus. It can also be seen as the irrational rotational algebra [R]. Several invariants of thesame were studied, which conformed the smoothness of the “associated non-commutativespace”. The existence of non-commutative Poisson structure [Xu] on the non-commutativetorus establishes it as a preluding example to study noncommutative smooth spaces. Whatwe study here is a dense sub-algebra of it. Let θ ∈ R , consider the group algebra C of the free group on two generators U and U , with coefficients in C . Wedefine the associated quantum torus [B] as A θ := C < U ± , U ± > / < U U − λU U > for a fixed parameter λ = e πiθ ∈ C . Unless specified otherwise, we will assume in thisarticle that(0.1) λ n = 1 for all n ∈ Z The algebra A θ may be thought of as a ring of functions on a quantum torus. Geometrically,the algebras A θ , the non-commutative smooth 2-torus, arise as deformations of the ring C ∞ ( T ) of smooth functions on the two-dimensional torus T = S × S , and as such, theseare fundamental examples of noncommutative differentiable manifolds in the sense of Connes[C]. On the other hand, algebraically, A θ is just a certain norm completion of A θ .Xu studied the Poisson structure on non-commutative smooth torus A θ [Xu]. The unique2-cocycle of the non-commutative smooth torus A θ when θ satisfies the Diophantine condi-tion gave the Poisson structure. The Poisson cohomology groups were also studied therein,the non-trivial Poisson 2 cocycles give rise to formal deformations of the algebra. This ishow the formal deformations are studied and further one can ask about the index theory ofsuch spaces.We shall study the non-commutative Poisson structure on the orbifolds of A θ arising throughthe action of finite subgroups of SL ( Z ). The action of SL ( Z ) on A θ is described as follows.An element g = (cid:20) g , g , g , g , (cid:21) ∈ SL ( Z )acts on the generators U and U as described below g U = e ( πig , g , ) θ U g , U g , and ρ g U = e ( πig , g , ) θ U g , U g , .We leave it to the readers to check that the above is indeed an action of SL ( Z ) on thealgebra A θ , in fact as we shall later see the action preserves the Calabi-Yau algebraicstructure of A θ . We illustrate this action by considering the action of Z on the algebraicnoncommutative torus, A θ . We notice that the generator of Z in SL ( Z ), g = (cid:20) −
11 0 (cid:21) acts on A θ as follows (cid:20) −
11 0 (cid:21) U = U − and (cid:20) −
11 0 (cid:21) U = U .1. Gestenhaber bracket and the Poisson Structure
For any algebra A over a field k , Hochschild cohomology HH • ( A ) is the space Ext • A ⊗ A op ( A, A ),which has two compatible operations, cup product and bracket. Both operations are definedinitially on the bar resolution, a natural A ⊗ A op -free resolution of A . It is a classical factthat the Hochschild cohomology HH ∗ ( A, A ) := L •∈ Z H • ( A, A ) of an associative algebra A carries a Gerstenhaber algebra structure. A Gerstenhaber algebra is a graded associativealgebra ( H ∗ = L i ∈ Z H i , ∪ ) together with a degree − − , − ] compatiblewith the product ∪ in the sense of the following Leibniz rule(1.1) [ a ∪ b, c ] = [ a, c ] ∪ b + ( − ( | c |− | a | a ∪ [ b, c ]For an associative algebra ( A, µ ), the Hochschild co-chain groups C • ( A, A ) (cup) product ∪ : C m ( A, A ) ⊗ C n ( A, A ) → C m + n ( A, A ) defined by(1.2) ( f ∪ g )( a , ..., a m + n ) = µ ( f ( a , ..., a m ) , g ( a m +1 , ..., a m + n )) . It turns out that the Hochschild coboundary operator δ is a graded derivation with respectto the cup product. Hence, it induces a cup product ∪ on the Hochschild cohomology HH ∗ ( A, A ). Moreover, the co-chain groups C • ( A, A ) carry a degree − δ [G]. Therefore, it gives rise to a degree − HH ∗ ( A, A ). The cup product and the degree − HH ∗ ( A, A ) are compatible in the sense of (1.1) to make itinto a Gerstenhaber algebra.Using the Gerstenhaber bracket on HH ∗ ( C ∞ ( M )), one can define the Poisson structureon the manifold M , a Poisson structure on a smooth manifold M is a Lie bracket { , } : C ∞ ( M ) × C ∞ ( M ) → C ∞ ( M ) satisfying { f, gh } = { f, g } h + g { f, h } for f, g, h ∈ C ∞ ( M ).For Π ∈ H ( C ∞ ( M )) and [Π , Π] = 0, the Poisson bracket on M can be defined as { f, g } := [[Π , f ] , g ].The above characterization can be generalized to an associative algebra. Hence, the non-commutative Poisson structure of an associative algebra A can be defined as follows: Definition 1.
Let A be an associative algebra. A Poisson structure on A is an element Π ∈ H ( A, A ) such that [Π , Π] = 0 . An algebra with a Poisson structure is called a Poissonalgebra. . Poisson Structure on non-commutative Orbifolds
In this section we discuss about the Gerstenhaber brackets and Poisson structure on a gen-eral non-commutative orbifold. For a non-commutative manifold A and a discrete group G acting on it, we can consider the orbifold A ⋊ G . To understand the Gerstenhaber bracketand the possible Poisson structure on A ⋊ G we have to understand its Hochschild coho-mology, HH • ( A ⋊ G, A ⋊ G ).The Hochschild cohomology splits as [GJ](2.1) HH • ( A ⋊ G, A ⋊ G ) = M γ ∈ G HH • ( A , γ A ) G . Where γ A is set wise A with the left A -module structure defined as α · a = γ ( α ) a where α ∈ A and a ∈ γ A [Q1]. Witherspoon and Shepler [SW2] studied the same and constructedexplicit isomorphism between the two cohomology groups described above. The general,method is to decompose into the conjugacy classes of G and using an appropriate projec-tive resolution for A . Using the resolution the Gerstenhaber bracket described over thebar resolution is transferred via chain map to the convenient resolution and an explicitformulation of the Gerstenhaber bracket is then obtained. The article [SW2] had studiedthe Gestenhaber bracket over the polynomial skew group algebra and provided necessaryconditions to find possible Poisson structure for the polynomial skew group algebra.We shall understand the Gerstenhaber bracket and then find the Poisson structure(s) andfinally the associated Poisson cohomology for the non-commutative orbifolds arising fromthe action of the four finite subgroups of SL ( Z ) on the quantum torus.3. Connes’ Koszul Resolution
We briefly recall the Connes’ Koszul resolution in this section. It is a projective resolutionfor the non-commutative torus and the quantum torus. Wambst [W1] generalized it to thehigher dimensional quantum torus and later computed its homology [W2]. A θ ǫ ←− ( A θ ) e ⊗ C b ←− ( A θ ) e ⊗ < e > L ( A θ ) e ⊗ < e > b ←− ( A θ ) e ⊗ < e ∧ e > where, ( A θ ) e = A θ ⊗ ( A θ ) op ǫ ( a ⊗ b ) = ab ; b (1 ⊗ e j ) = 1 ⊗ U j − U j ⊗ b (1 ⊗ ( e ∧ e )) = ( U ⊗ − λ ⊗ U ) ⊗ e − ( λU ⊗ − ⊗ U ) ⊗ e . We shall use the above resolution to study the Hochschild cohomology HH • ( γ A θ ), whichis the cohomology of the following complex: γ A θ γ α −−→ γ A θ ⊕ γ A θ γ α −−→ γ A θ → , where the maps are as below: γ α ( ϕ ) = (( γ · U ) ϕ − ϕU , ( γ · U ) ϕ − ϕU ) ; γ α ( ϕ , ϕ ) = ( γ · U ) ϕ − λϕ U − λ ( γ · U ) ϕ + ϕ U . We leave it upon the readers to verify these maps, which can be done easily using theConnes’ Koszul resolution. . Some Homology groups of A θ ⋊ Γ LEMMA 4.1. HH ( A θ ⋊ Γ) = C Proof.
Using the resolution, we have , HH ( A θ ) = C . It is generated by the a , ⊗ ⊗ HH ( γ A θ ) is the kernel of the map γ α . The result is aroutine calculation and the invariance is also a straightforward computation. (cid:3) LEMMA 4.2.
For θ satisfying Diophantine condition, HH ( A θ ) Γ ∼ = C .Proof. We know from [Xu, Lemma 4.1] that HH ( A θ ) ∼ = C . To check its invariance underthe action of Γ, we push the cocycle into the bar complex using the map h [C] and after theaction by the generator of Γ we pull it back on to the Connes’ Koszul resolution and comparethe equivalence classes. It can be easily checked that HH ( A θ ) Γ ∼ = C . We shall explicitlycalculate for Z , other cases too have similar computation. We abide by the notations of [C].For ϕ ∈ ω A θ , let e ϕ be the corresponding element of Hom A eθ (Ω , ω A θ ). Then e ϕ ( a ⊗ b ⊗ e ∧ e )( x ) = ϕ (( ω · b ) xa ) , for all a, b, x ∈ A θ . Let ψ = k ∗ e ϕ = e ϕ ◦ k . We have ψ ( x, x , x ) = e ϕ ( k ( I ⊗ x ⊗ x ))( x ) , for all x, x , x ∈ A θ . The group ω acts on A θ in the bar complex as ω · χ ( x, x , x ) = χ ( ω · x, ω · x , ω · x ) . Further we pull the map ω ψ := ω · ψ back on to the Connes complex via the map h ∗ . Let w = h ∗ ( ω ψ ) denote the pull-back of ω ψ on the Connes’ Koszul complex. We have w ( x ) = ω ψ ( x, U , U ) − λ ω ψ ( x, U , U ) = ψ ( ω · x, U U − √ λ , U − ) − λψ ( ω · x, U − , U U − √ λ ) = e ϕ ( k ( I ⊗ U U − √ λ ⊗ U − ))( ω · x ) − λ e ϕ ( k ( I ⊗ U − ⊗ U U − √ λ ))( ω · x ).Using the results of [C] and [Q1, Section 6], we have k ( I ⊗ U U − ⊗ U − ) − λk ( I ⊗ U − ⊗ U U − ) = ( U − ⊗ U − ) . Hence, 1 √ λ ( e ϕ (( k ( I ⊗ U U − ⊗ U − ))( ω · x ) − λ e ϕ ( k ( I ⊗ U − ⊗ U U − )))( ω · x )) =1 √ λ e ϕ (( U − ⊗ U − ))( ω · x ) = √ λϕ ( U − U · ( ω · x ) · U − ).The cocycle x − , − ∈ H ( A θ , A θ ) which on the Connes’ Koszul complex is supported at the( − , −
1) is invariant under the action of Z because:1 √ λ x − , − ( U − ( ω · x ) U − ) = 1 √ λ ϕ − , − ( U − ( ω · ( x − , − U − U − )) U − ) =1 √ λ ϕ − , − ( U − x − , − U ( U U − √ λ ) − U − ) = ϕ − , − ( x − , − U − U U U − U − ) = ϕ − , − ( x − , − U − U − ) = x − , − . ence, HH ( A θ ) Z ∼ = C . (cid:3) Hochschild cohomology HH • ( − ) of A θ ⋊ Γ THEOREM 5.1.
Let Γ be any finite subgroup of SL ( Z ) then we have the following: HH ( A θ ⋊ Γ) = C , HH ( A θ ⋊ Γ) = 0 , HH • ( A θ ⋊ Γ) = 0 for • > .and HH ( A θ ⋊ Γ) ∼ = C for Γ = Z C for Γ = Z C for Γ = Z C for Γ = Z . Proof of Theorem 5.1.
The Hochschild cohomology decomposes as follows: HH • ( A θ ⋊ Γ) = M γ ∈ Γ H • ( A θ , γ A θ ) Γ . It is clear from the Koszul resolution that the cohomology HH • (( A θ ⋊ Γ)) for • > HH ( A θ ⋊ Γ) has the following decomposition: HH ( A θ ⋊ Γ) ∼ = M γ ∈ Γ H ( γ A θ ) Γ . We have HH ( A θ ) Γ = C , the proof is similar to the proof in the last section for the smoothorbifolds.To calculate HH ( γ A θ ) Γ we firstly study HH ( γ A θ ) = γ A θ /im ( γ α ) on the Connes’ Koszulcomplex. It is a straight forward computation that shows that for γ ∈ Γ, on the Koszulcomplex, the group HH ( γ A θ ) is generated by equivalence classes of elements of the form a p,q ⊗ ⊗ e ∧ e ∈ γ A θ ⊗ ( A θ ) e ⊗ Ω . where p, q ∈ { , } . For example, for g = √− ∈ Z ; HH ( γ A θ ) is two dimensional and isgenerated by a , ⊗ ⊗ e ∧ e and a , ⊗ ⊗ e ∧ e .Similar to the method we adopted, we can check that all the γ -twisted 2-cocycles are Γinvariant. In general for a commutative algebra A and some d ∈ N , HH d ( A ) ∼ = HH n − d ( A )if A is Gorenstein [V] and for non-commutative algebras that are Calabi-Yau such a duality isestablished [CYZ]. We see that the Hochschild homology of A ⋊ Γ [Q1] and the cohomologycomputed above are dual to each other in dimension. This gives us an example of an algebrawhose Van den Bergh duality is preserved upon the action of Γ. In general one can ask ifgiven a Calabi Yau algebra and a finite group acting on it, is the skew group algebra CalabiYau? Recently this phenomenon was addressed by Meur [M].The dimension of the zeroth cohomology can be concluded from the previous section. Todetermine the HH ( A θ ⋊ Γ) we firstly observe that HH ( A θ ) is two dimensional and isgenerated by the equivalence class of elements supported at φ − , and φ , − . Using themaps k and h we check the invariance of these two cocycles in the similar way that we id for the 2-cocycle above. We can easily observe that HH ( A θ ) Γ = 0 for all Γ. We nowclaim that HH ( γ A θ ) = 0 for all γ ∈ Γ. This statement is an straightforward corollary ofthe fact that HH ( A θ , γ A ∗ θ ) = 0 for all γ ∈ Γ (see [Q3, Lemma 2.6] ). (cid:3) Gestenhaber Bracket for A θ ⋊ ΓWe have described the above the method to compute the Gerstenhaber bracket by trans-porting the Gerstenhaber structure from the bar complex to convenient complex(here the γ -twisted Koszul complex) and then gain push forward to the bar complex for the explicitformula. It is to be noted that for f ∈ HH n ( A, γ A ) and g ∈ HH m ( A, µ A ), for γ , µ ∈ Γ,the Gerstenhaber bracket [ f, g ] is a n + m − < SL ( Z ), finite subgroup, the Gestenhaber bracket on HH • ( A θ ⋊ Γ) is trivial.
LEMMA 6.1.
The Gerstenhaber bracket on the non-commutative torus orbifolds A θ ⋊ Γ for Γ finite subgroups of SL ( Z ) is 0. That is: [ − , − ] : HH ∗ ( A θ ⋊ Γ) → HH ∗ ( A θ ⋊ Γ) . is the zero map.Proof. Any non-trivial 2-cocycle, Π Γ i , when paired with any of the 0-cocycles should yield a1-cocycle in HH ( A θ ⋊ Γ)(= 0). We denote by Π Γ i a 2-cocycle in A θ ⋊ Γ. Enough to showthat [Π Γ i , Π Γ i ] = 0, since [Π Γ i , Π Γ i ] is a 3-cocycle in HH ∗ ( A θ ⋊ Γ) hence 0.We note that the untwisted 2-cocycle Π = δ ∧ δ where δ and δ are the two canonicalderivations on A θ given by, δ ( U n U m ) = 2 πinU n U m and δ ( U n U m ) = 2 πimU n U m (cid:3) Poisson Structure(s) and cohomology of A θ ⋊ ΓThe Poisson cohomology for a Poisson algebra is defined as follows. For an associativePoisson algebra ( A, Π), we define a co-chain complex ( HH ∗ ( A, A ) , d Π ), d Π : HH • ( A, A ) → HH • +1 ( A, A )where d Π ( U ) = [Π , U ] , for all U ∈ HH i ( A, A ) . From the condition that [Π , Π] = 0 and the graded Jacobi identity of the Gerstenhaberbrackets, it follows that d = d Π ◦ d Π = 0. The cohomology of this complex ( HH ∗ ( A, A ) , d Π )is called Poisson cohomology of ( A, Π) and denoted by H ∗ Π ( A ).Since the Gerstenhaber bracket is 0, all the 2-cocycles are Poisson structures on A θ ⋊ Γ. HEOREM 7.1.
For all i , H Γ i ( A θ ⋊ Γ) = C , H Γ i ( A θ ⋊ Γ) = 0 and H Γ i ( A θ ⋊ Γ) ∼ = C for Γ = Z C for Γ = Z C for Γ = Z C for Γ = Z .H • Π Γ i ( A θ ⋊ Γ) = 0 for • > .Proof. The proof is straight forward consequence of Theorem 5.1 and Lemma 6.1. (cid:3) Conclusion
Although we had the machinery to compute the Gestenhaber brackets explicitly, the brack-ets are trivial for all the quantum orbifolds. The vanishing of the HH ( A θ ⋊ Γ) is thekey to isomorphism of the Hochschild cohomology and the Poisson cohomology. The pres-ence of γ -twisted 2 cocycles in the Poisson cohomology HH ( A θ ⋊ Γ) indicates interestingdeformation theory for orbifolds. It is well known that for a Poisson algebra ( A, Π) anda Poisson 2 cocycle Π ∈ HH ( A ), the group HH ( A ) is the obstruction for existenceof a deformation of Π with infinitesimal Π . So to conclude, the formal deformations inthe sense of Kontsevich[K] which was studied by Xu[Xu], Neumaier et al.[NPPT] amongstothers does exist in these non-commutative orbifolds along with that which was inheritedfrom the parent non-commutative manifold A θ .9. Acknowledgement
I acknowledge the discussion with Prof. Xiang Tang and his valuable comments.