Birational Equivalences and Generalized Weyl Algebras
aa r X i v : . [ m a t h . K T ] S e p BIRATIONAL EQUIVALENCES AND GENERALIZED WEYL ALGEBRAS
ATABEY KAYGUNAbstract. We calculate suitably localized Hochschild homologies of various quantum groupsand Podleś spheres after realizing them as generalized Weyl algebras (GWAs). We use the factthat every GWA is birationally equivalent to a smash product with a 1-torus. We also address andsolve the birational equivalence problem, and the birational smoothness problem for GWAs.
Introduction
A birational equivalence is an algebra morphism that becomes an isomorphism after a suitablelocalization. In this paper, we show that every generalized Weyl algebra (GWA) is birationallyequivalent to a smash product with a rank-1 torus. This fact significantly simplifies their repre-sentation theory, and structure problems such as the isomorphism problem [24, 5, 38, 42, 43] andthe smoothness problem [4, 24, 41, 31, 32], provided one replaces isomorphisms with suitablenoncommutative birational equivalences. We address and solve a relative version of the birationalequivalence problem in Section 2.7, and the birational smoothness problem in Section 3.3. Wethen calculate the Hochschild homology of suitably localized examples of GWAs in Section 4.Generalized Weyl algebras are defined by Bavula [3, 4], Hodges [24] and Rosenberg [39] in-dependently under different disguises. Their representation theory resembles that of Lie al-gebras [12, 36] (see Section 2.5), their homologies are extensively studied [13, 41, 31, 32],and they found diverse uses in areas such as noncommutative resolutions of Kleinian singulari-ties [11, 6, 30] and noncommutative geometry of various quantum spheres and lens spaces [9].Apart from noncommutative resolutions of Kleinian singularities, the class is known to containthe ordinary rank-1 Weyl algebra A , the enveloping algebra U ( sl ) and its primitive quotients, thequantum enveloping algebra U q ( sl ) , the quantum monoid O q ( M ) , the quantum groups O q ( GL ) , O q ( SL ) and O q ( SU ) . We verify that the standard Podleś spheres O q ( S ) [37] and parametricPodleś spheres O q , c ( S ) of Hadfield [18] are also examples of GWAs. We finish the paper bycalculating localized Hochschild homology of all of these examples.The Hochschild homology of quantum groups O q ( GL n ) and O q ( SL n ) with coefficients in a 1-dimensional character coming from a modular pair in involution is calculated for every n > twisted generalizedWeyl algebras (TGWAs) [35, 34, 22, 23]. We conjecture that TGWAs are birationally equivalentto smash products with higher rank tori, but we leave this investigation for a future paper. The celebrated Gelfand-Kirilov Conjecture, on the other hand, states that the universal envelopingalgebra U ( g ) of a finite dimensional Lie algebra is birationally equivalent to a sufficiently highrank Weyl algebra [15]. One of the equivalent forms of the conjecture is that U ( g ) is birationallyequivalent to the smash product of a polynomial algebra with a torus. The conjecture is known tobe false in general [2, 10], but is true for a large class of Lie algebras [15, 25, 21]. The quantumanalogue of the conjecture (see [7, pp.19–21 and Sect.II.10.4] and references therein) is alsoknown to be true many instances [1, 14]. In the light of our conjecture above, we believe that theuniversal enveloping algebra U ( g ) of a rank- n semi-simple Lie algebra is birationally equivalentto the smash product of a smooth algebra with an n -torus. We also believe that the same is truefor the quantum enveloping algebras U q ( g ) and the quantum groups O q ( G ) where one replacesthe n -torus with a quantum n -torus. Plan of the article.
In Section 1 we recall some basic facts on localizations, relative homologyof algebra extensions, smash products and biproducts. In Section 2 we prove two fundamentalstructure theorems for GWAs in Sections 2.2 and 2.3. Then we state and solve birationalequivalence problem for GWAs in Section 2.7. In Section 3, we investigate the interactionsbetween homology, smash biproducts and noncommutative localizations, and in Sections 3.3and 3.4 we state and solve the birational smoothness problem for GWAs. Finally, we use ourmachinery to calculate suitably localized Hochschild homologies of various GWAs in Section 4.
Notations and conventions.
We fix an algebraically closed ground field k of characteristic 0,and we set the binomial coefficients (cid:0) nm (cid:1) = m > n or m <
0. All unadorned tensorproducts ⊗ are taken over k . We reserve k [ X ] for the free unital commutative algebra generated by a set X , while we use k { X } for the free unital algebra generated by the same set X . Throughout the paper we use T todenote the algebra of Laurent polynomials k [ x , x − ] . Acknowledgments.
This work completed while the author was on academic leave at Queen’sUniversity from Istanbul Technical University. The author is supported by the Scientific andTechnological Research Council of Turkey (TÜBİTAK) sabbatical grant 2219. The author wouldlike to thank both universities and TÜBİTAK for their support. Preliminaries
Noncommutative localizations.
Our main reference for noncommutative localizations is[29, §10].A multiplicative submonoid S ⊆ A is called a right Ore set if for every s ∈ S and u ∈ A (i) there are s ′ ∈ S and u ′ ∈ A such that su = u ′ s ′ , and(ii) if su = u ′ ∈ A such that u ′ s = S ⊆ A is a right Ore set then there is an algebra A S and a morphism of algebras ι S : A → A S such that ϕ ( S ) ⊆ A × S . The morphism ι S is universal among such S inverting morphisms whereif ϕ : A → B satisfies ϕ ( S ) ⊂ B × then there is a unique morphism of algebras ϕ ′ : A S → B with ϕ = ϕ ′ ◦ ι S .In the sequel, we are going to drop the requirement that S is a multiplicative submonoid andconsider the conditions above within the submonoid generated by S . In such cases, we are stillgoing to use the notation A S for the localization. IRATIONAL EQUIVALENCES AND GENERALIZED WEYL ALGEBRAS 3
Birational equivalences.
We call a morphism of unital associative algebras ϕ : A → A ′ as a birational equivalence if there are two Ore sets S ⊂ A and S ′ ⊂ A ′ such that ϕ ( S ) ⊆ S ′ and theextension of ϕ to the localization ϕ S : A S → A ′ S ′ is an isomorphism of unital associative algebras.This notion mimics the birational equivalences of affine varieties [17, §4.2].1.3. Smash biproducts.
Assume A and B are two unital associative algebras. A k -linear map R : B ⊗ A → A ⊗ B is called a distributive law if the following diagrams of algebras commute:(1.1) B ⊗ B ⊗ A B ⊗ R / / µ B ⊗ A (cid:15) (cid:15) B ⊗ A ⊗ B R ⊗ B / / A ⊗ B ⊗ B A ⊗ µ B (cid:15) (cid:15) B ⊗ A R / / A ⊗ BB ⊗ A ⊗ A R ⊗ B / / B ⊗ µ A O O A ⊗ B ⊗ A A ⊗ R / / A ⊗ A ⊗ B µ A ⊗ B O O B ⊗ B " " ❋❋❋❋❋❋❋❋❋ B ⊗ | | ①①①①①①①①① B ⊗ A R / / A ⊗ BA ⊗ A b b ❋❋❋❋❋❋❋❋❋ A ⊗ < < ①①①①①①①①① For notational convenience we write R ( b ⊗ a ) = R ( ) ( a ) ⊗ R ( ) ( b ) for every a ∈ A and b ∈ B .For a distributive law R : B ⊗ A → A ⊗ B there is a corresponding smash biproduct algebra A R B which is A ⊗ B as vector spaces with the multiplication ( a ⊗ b )( a ′ ⊗ b ′ ) = aR ( ) ( a ′ ) ⊗ R ( ) ( b ) b ′ for every a , a ′ ∈ A and b , b ′ ∈ B .1.4. Smash products with Hopf algebras.
Standard examples of smash biproducts come fromsmash products A H between a Hopf algebra H and a H -module algebra A where one has h ⊲ ( ab ) = ( h ( ) ⊲ a )( h ( ) ⊲ b ) for every h ∈ H and a , b ∈ A . In this case one has a distributive law of the form R : H ⊗ A → A ⊗ H by letting R ( h ⊗ a ) = ( h ( ) ⊲ a ) ⊗ h ( ) . Almost all of the smash biproducts we consider in the sequel are smash products with the Hopfalgebra of the group ring of Z , also known as the algebra of Laurent polynomials T : = k [ x , x − ] .However, the results we rely on for homology computations require the full generality of smashbiproducts.1.5. Hochschild homology.
Let A be a unital associative algebra, and let M be an A -bimodule.Consider the graded k -vector spaceCH ∗ ( A , M ) = Ê n > M ⊗ A ⊗ n together with linear maps b n : CH n ( A , M ) → CH n − ( A , M ) defined for n > b n ( m ⊗ a ⊗ a n ) = ma ⊗ a ⊗ · · · ⊗ a n + n − Õ i = (− ) i m ⊗ · · · ⊗ a i a i + ⊗ · · · ⊗ a n + (− ) n a n m ⊗ a ⊗ · · · ⊗ a n − . ATABEY KAYGUN
These maps satisfy b n b n + = n >
1, and we define H ∗ ( A , M ) = ker ( b n )/ im ( b n + ) . Weuse the notation HH ∗ ( A ) for H ∗ ( A , A ) .1.6. Amenable and smooth algebras.
An algebra A is said to have finite Hochschild homologicaldimension if hh . dim ( A ) : = sup { n ∈ N | H n ( A , M ) , , M ∈ A e - Mod } is finite. In particular, we call an algebra A (i) amenable if hh . dim ( A ) =
0, and(ii) m -smooth if hh . dim ( A ) = m +
1, for m ∈ N .For 0-smooth algebras we just use the term smooth .The particular examples of amenable algebras we use in this article are of groups ring k [ G ] overfinite groups where | G | does not divide the characteristic of k , and quotients of polynomial algebras k [ x ]/h f ( x )i where f ( x ) is a separable polynomial. For m -smooth algebras primary examples wehave in mind are the polynomial algebras k [ t i | i = , . . . , m ] and the Laurent polynomial algebras k [ t i , t − i | i = , . . . , m ] with m >
0, and their smash biproducts with amenable algebras.1.7.
Homology of smash biproducts with amenable and smooth algebras.
We recall thefollowing facts from [26]:
Proposition 1.1.
Let A and B be two algebras, and let R : B ⊗ A → A ⊗ B be an invertible distributive law. For any A R B -bimodule M and for all n > we have (1.2) H n ( A R B , M ) (cid:27) H n ( CH ∗ ( A , M ) B ) when B is amenable and (1.3) H n ( A R B , M ) (cid:27) H n ( CH ∗ ( A , M ) B ) ⊕ H n − ( CH ∗ ( A , M ) B ) when B is smooth where CH n ( A , M ) B : = CH n ( A , M ){ a ⊗ m ⊳ b − R ( ) ( a ) ⊗ R ( ) ( b ) ⊲ m | b ∈ B , a ⊗ m ∈ CH n ( A , M )} , and CH n ( A , M ) B : = { a ⊗ m ∈ CH n ( A , M ) | a ⊗ m ⊳ b = R ( ) ( a ) ⊗ R ( ) ( b ) ⊲ m , b ∈ B } . Next, let us recall the following result from [27, Prop.1.5]:
Proposition 1.2.
Assume P and Q are two unital algebras together with a left flat algebramorphism ϕ : Q → P . Let M be a P -bimodule. Then there is a spectral sequence whose firstpage is given by E i , j = H j ( Q , M ⊗ Q P ⊗ Q · · · ⊗ Q P | {z } i -times ) that converges to the Hochschild homology H ∗ ( P , M ) . One important corollary of Proposition 1.2 is that one can now remove the condition that thedistributive law is invertible from Proposition 1.1 since B ⊂ A R B is a flat extension. Corollary 1.3.
Let A and B be two algebras, and let R : B ⊗ A → A ⊗ B be any distributivelaw. Then for any A R B -bimodule M and for all n > we still have Equation (1.2) when B isamenable and Equation (1.3) when B is smooth.Proof. We set P = A R B and Q = B together with ϕ ( b ) = ⊗ b , and then we use Proposition 1.2. (cid:3) IRATIONAL EQUIVALENCES AND GENERALIZED WEYL ALGEBRAS 5
2. Generalized Weyl algebras
Algebras with automorphisms.
One standard source of distributive laws is algebras witha fixed algebra automorphism or endomorphisms. Let A be an algebra with a fixed algebraautomorphism σ ∈ Aut ( A ) . Let T = k [ Z ] = k [ x , x − ] be the group ring of the free abeliangroup on a single generator Z . Now consider the smash biproduct B : = A R T coming from thedistributive law R : T ⊗ A → A ⊗ T defined as(2.1) R ( x n ⊗ u ) = σ n ( u ) ⊗ x n for every monomial x n ∈ T with n ∈ Z and u ∈ A . Then R defines an invertible distributive law.In order to simplify the notation, we are going to write ux i for every monomial u ⊗ x i in A R T .If there are more than one automorphisms in the context, we are going to write A R ( σ ) T insteadof A R R to emphasize which automorphism we are using.2.2. A structure theorem for GWAs.
Assume A is a unital associative algebra, let a ∈ Z ( A ) and σ ∈ Aut ( A ) be fixed. Define a new algebra W a ,σ as a quotient of the free algebra generatedby A and two non-commuting indeterminates x and y subject to the following relations:(2.2) y x − a , x y − σ ( a ) , xu − σ ( u ) x , y σ ( u ) − u y for every u ∈ A . The algebra W a ,σ is called generalized Weyl algebra [3, 5].One can also realize W a ,σ as a unital subalgebra of the smash product A R T where T : = k [ x , x − ] and R : T ⊗ A → A ⊗ T is defined in Equation (2.1). For this we consider the monomorphism of k -algebras ϕ : W a ,σ → A R T given by(2.3) ϕ ( u ) = u , ϕ ( x ) = x , ϕ ( y ) = ax − for every u ∈ A . Theorem 2.1.
For every a ∈ Z ( A ) , the algebra W a ,σ is isomorphic to the unital subalgebra ofthe smash biproduct A R T generated by A , x and ax − . Hence W a ,σ is isomorphic to A R T forevery a ∈ Z ( A × ) .Proof. The result follows from the fact that the image of ϕ (as k -vector spaces) is the direct sum A ⊗ k [ x ] ⊕ ∞ Ê n = h a σ − ( a ) · · · σ − n ( a )i ⊗ Span k ( x − n − ) where h u i denotes the two sided ideal in A generated by an element u ∈ A . (cid:3) In specific cases, the fact that GWAs are subalgebras of smash products was already known [6,Lem.2.3]. However, to the best of our knowledge, the fact that one gets an isomorphism whenthe distinguished element a ∈ A is a unit, even though it implicitly follows from this embedding,is not fully taken advantage of in the literature.From now on we identify W a ,σ with im ( ϕ ) in A R T .2.3. Localizations of smash products with tori.
Let A be an algebra with a fixed automorphism σ ∈ Aut ( A ) . Assume R : T ⊗ A → A ⊗ T is the distributive law given in Equation (2.1). Let S ⊆ Z ( A ) be any multiplicative submonoid which stable under the action of σ . The proof of thefollowing Lemma is routine verification, and therefore, is omitted. Lemma 2.2.
Any multiplicative monoid S in Z ( A ) which is σ -stable is a right Ore subset in A R T , and ( A R T ) S = A S R T . ATABEY KAYGUN
Localizations of GWAs.
As before, assume A is a unital associative algebra, a ∈ Z ( A ) and σ ∈ Aut ( A ) . Recall that by Theorem 2.1 we identified the GWA W a ,σ with the subalgebra of thesmash biproduct A R T generated by the algebra A and the elements x and ax − . Then we have atower of algebra extensions of the form A R k [ x ] ⊂ W a ,σ ⊆ A R T . Theorem 2.3.
Consider the set S ⊂ Z ( A ) of the elements of the form σ m ( a n ) where n ∈ N and m ∈ Z . Then the embedding of algebras W a ,σ ⊆ A R T is a birational equivalence with respect tothe Ore set generated by S .Proof. Now, by Lemma 2.2 we have that ( A R T ) S = A S R T , and by Theorem 2.1 we see that thealgebra A S R T is itself generated by A S , x and ax − since a ∈ A S is now a unit. (cid:3) Highest weight modules of GWAs.
Assume A is unital associative with a distinguishedelement a ∈ Z ( A ) and an automorphism σ ∈ Aut ( A ) . Let V be a representation over the GWA W a ,σ . We have an (not necessarily exhaustive) increasing filtration of submodules of the form V [ ℓ ] = { v ∈ V | v ⊳ a σ − ( a ) · · · σ − ℓ ( a ) = } defined for ℓ ∈ N . Let us also define V [∞] = Ø ℓ > V [ ℓ ] . We define ht a ,σ ( V ) the height of V as the smallest integer ℓ such that V [ ℓ ] = V [∞] , and if no suchinteger exists we set ht a ,σ ( V ) = ∞ .Assume V is a finite dimensional representation. Then h = ht a ,σ ( V ) is necessarily finite. Fur-thermore, if the height filtration satisfies V [ h ] = V , then we get the analogue of a highest weightmodule for the GWA W a ,σ . Approaches for such cases can be seen in [12, 36]. Proposition 2.4.
Let S ⊆ Z ( A ) be the subset of elements of the form σ n ( a m ) with n ∈ Z and m ∈ N , and let ( W a ,σ ) S be the localization of W a ,σ at S . Assume V is an arbitrary W a ,σ -module,and let h = ht a ,σ ( V ) . Then V S : = V ⊗ W a ,σ ( W a ,σ ) S is isomorphic to ( V / V [ h ] ) S .Proof. We consider the following short exact sequence of W a ,σ -modules0 → V [ h ] → V → V / V [ h ] → ( · ) S is exact. (cid:3) Morphisms of algebra extensions.
An algebra C together with a subalgebra A is called analgebra extension. Given two extensions A ⊆ C and A ⊆ C ′ of a fixed algebra A , a morphism f : ( C , A ) → ( C ′ , A ) of extensions is a commutative triangle of algebra morphisms of the form: C f / / C ′ A _ _ ❄❄❄❄❄❄❄❄ > > ⑦⑦⑦⑦⑦⑦⑦⑦ . IRATIONAL EQUIVALENCES AND GENERALIZED WEYL ALGEBRAS 7
Isomorphisms of smash products with tori.
In this section we consider the isomorphismproblem for smash products with T = k [ x , x − ] since all isomorphism problems for GWAsbirationally reduce to isomorphism problems for such smash products. Theorem 2.5.
Assume σ and η are two algebra automorphisms of A . Then the algebra extensions A ⊆ A R ( σ ) T and A ⊆ A R ( η ) T are isomorphic if and only if η = u σ ± u − for some u ∈ A × .Proof. Assume for now that σ = u η u − or σ = u η − u − . Consider an arbitrary v ∈ A . In the firstcase define δ : A R ( σ ) T → A R ( η ) T by letting δ ( x ) = ux and we get δ ( x v ) = ux v = u η ( v ) x = σ ( v ) ux = δ ( σ ( v ) x ) which implies δ is an isomorphism of smash biproducts. The proof for the second case issimilar, and therefore, is omitted. On the opposite direction, assume δ : A R ( σ ) T → A R ( η ) T isan isomorphism of algebra extensions. The one easily see that δ restricted T yields an algebramonomorphism, and therefore, δ ( x ) = ux ± for some u ∈ A × and δ restricted to A is identity.Thus σ = u η ± u − as expected. (cid:3) Notice that given an automorphism σ ∈ Aut ( A ) and its inverse σ − extended to A R ( σ ) T are nowan inner automorphisms. From this perspective Theorem 2.5 says that given two automorphism σ and η , they define two different smash products if their outer automorphism classes are different.In particular, we have the following result: Corollary 2.6. If σ ∈ Aut ( A ) is an inner automorphism then the smash biproduct A R ( σ ) T isisomorphic to the direct product A × T .
3. Homology of GWAs
Homology of smash products with tori.
We have the following result since T is a smoothalgebra. Proposition 3.1.
Let σ ∈ Aut ( A ) and assume σ acts on CH ∗ ( A ) diagonally extending the actionon A . Let CH ∗ ( A ) T and CH ∗ ( A ) T respectively be the complex of coinvariants and invariants of σ . Then HH n ( A R T ) (cid:27) H n ( CH ∗ ( A ) T ) ⊗ T ⊕ H n − ( CH ∗ ( A ) T ) ⊗ T . Proof.
By Corollary 1.3 we get HH n ( A R T ) = H n ( CH ∗ ( A , A R T ) T ) ⊕ H n − ( CH ∗ ( A , A R T ) T ) since T is smooth. We start by splitting CH ∗ ( A , A R T ) asCH n ( A , A R T ) = CH ∗ ( A ) ⊗ T . Then the difference between the left and right actions is given by a ⊗ a ′ x m − ⊳ x − σ ( a ) ⊗ σ ( a ′ ) x ⊲ x m − = a ⊗ a ′ x m − σ ( a ) ⊗ σ ( a ′ ) x m = for every a ⊗ a ′ x m in CH ∗ ( A , A R T ) . This meansCH ∗ ( A , A R T ) T = CH ∗ ( A ) T ⊗ T and CH ∗ ( A , A R T ) T = CH ∗ ( A ) T ⊗ T . The result follows. (cid:3)
ATABEY KAYGUN
Algebraic and separable endomorphisms.
We call an algebra endomorphism σ ∈ E nd ( A ) algebraic if there is a polynomial f ( t ) ∈ k [ t ] such that f ( σ ) = E nd ( A ) . For an algebraicendomorphism σ of A , the monic polynomial f ( t ) with the minimal degree that satisfies f ( σ ) = the minimal polynomial of σ . We call an algebraic endomorphism σ ∈ E nd ( A ) as separable if the minimal polynomial of σ is separable.Notice that all endomorphisms of a finite dimensional k -algebra are algebraic. Regardless of thedimension, all automorphisms of finite order and all nilpotent non-unital endomorphisms are alsoalgebraic. If k has characteristic 0, automorphisms of finite order are separable, but nilpotentnon-unital endomorphisms are not.3.3. Algebras with separable automorphisms.
For a fixed algebraic automorphism σ ∈ Aut ( A ) ,let Spec ( σ ) be the set of unique eigen-values of σ , and let A ( λ ) be the λ -eigenspace of σ corre-sponding to λ ∈ Spec ( σ ) . Theorem 3.2.
Assume σ ∈ Aut ( A ) is separable with minimal polynomial f ( x ) , and let B be thequotient k [ x ]/h f ( x )i . Then H n ( A R T ) = H n ( CH ( )∗ ( A )) ⊗ T ⊕ H n − ( CH ( )∗ ( A )) ⊗ T and H n ( A R B ) = H n ( CH ( )∗ ( A )) ⊗ B where CH ( )∗ ( A ) is generated by homogeneous tensors of the form a ⊗ · · · ⊗ a n with a i ∈ A ( λ i ) and λ · · · λ n = for every n > .Proof. One can extend the distributive law R : T ⊗ A → A ⊗ T given in Equation (2.1) to adistributive law of the form R : B ⊗ A → A ⊗ B . Notice that since f ( x ) is separable, B is a productof a finite number of copies of k , and therefore, is amenable. Then the result for A R B immediatelyfollows from Corollary 1.3. On the other hand, CH ∗ ( A ) T = CH ∗ ( A ) B = CH ∗ ( A ) B = CH ∗ ( A ) T .Then the result for A R T follows from Proposition 3.1. (cid:3) Note that Theorem 3.2 solves the smoothness problem for smash products with T , and therefore the birational smoothness problem for all GWAs, provided that the action is implemented viaa separable automorphism. Namely, a smash product with Z via a separable automorphism issmooth if and only if the complex subcomplex of invariants CH ( )∗ ( A ) has bounded homology. Inthe next subsection we solve the birational smoothness problem for all GWAs without requiringautomorphism to be separable.3.4. Localization of GWAs in homology.
Consider the set S of elements of the form σ m ( a n ) in Z ( A ) where n ∈ N and m ∈ Z . Let k h S i be the (commutative) subalgebra of A generated by S ,and let k h S i S be its localization at S . Then we have that A S = A ⊗ k h S i k h S i S . Now let k h S i T bethe algebra of coinvariants of k h S i which is given by the following quotient k h S i T : = k h S ih σ ( s ) − s | s ∈ S i Corollary 3.3.
We have HH n (( W a ,σ ) S ) (cid:27) HH n ( A S R T ) (cid:27) H n ( CH ∗ ( A ) T ⊗ k h S i T ( k h S i T ) S ⊗ T ⊕ H n − ( CH ∗ ( A ) ⊗ k h S i k h S i S ) T ⊗ T where we view CH ∗ ( A ) as an k h S i -module and k h S i T -module on the coefficient. IRATIONAL EQUIVALENCES AND GENERALIZED WEYL ALGEBRAS 9
Proof.
By Theorem 2.3 we have ( W a ,σ ) S (cid:27) A S R T . Now, we consider the algebra extension A S ⊆ A S R T for which by [27] there is a spectral sequence whose first page is E p , q = H q ( A S , CH p ( A S R T | A S )) = H q ( A S , CH p ( T , A S R T )) that converges to HH ∗ ( A S R T ) . Since S ⊆ Z ( A ) , by [8] we know that E p , q (cid:27) H q ( A , CH p ( T , A S R T ) S ) (cid:27) H q ( A , CH p ( T , A S R T )) . Thus we have an isomorphism of the form HH ∗ ( A S R T ) (cid:27) H ∗ ( A R T , A S R T ) . Then by Proposi-tion 3.1 we get HH n (( W a ,σ ) S ) (cid:27) H n ( CH ∗ ( A , A S ) T ) ⊗ T ⊕ H n − ( CH ∗ ( A , A S ) T ) ⊗ T . Since S ⊆ Z ( A ) we get that CH ∗ ( A , A S ) = CH ∗ ( A ) ⊗ k h S i k h S i S = CH ∗ ( A ) S . On the other hand,both the coinvariants functor ( · ) T and localization functor ( · ) S are specific colimits, and colimitscommute. Then ( CH ∗ ( A ) S ) T (cid:27) ( CH ∗ ( A ) T ) S (cid:27) CH ∗ ( A ) T ⊗ k h S i T ( k h S i T ) S . The last isomorphism follows from the fact that the action of k h S i on CH ∗ ( A ) T factors through k h S i T . (cid:3)
4. Homology Calculations
The rank-1 Weyl algebra.
The ordinary rank-1 Weyl algebra A is the k -algebra definedon two non-commuting indeterminates x and y subject to the relations x y − y x = . One can define A as a GWA if we let A = k [ t ] where we set the distinguished element a = t . Wedefine σ to be the algebra automorphism of A given by f ( t ) = f ( t − ) for every f ( t ) ∈ A . Thenthe GWA W a ,σ is the ordinary Weyl algebra A . See [5, Ex.2.3].Since a = t is not a unit in A we see that W t ,σ is the proper subalgebra of k [ t ] R T generated by x and t x − where the distributive law is defined as R ( x ⊗ t ) = ( t − ) ⊗ x .Now, let S be the multiplicative system generated by elements of the form ( t − m ) where m ∈ Z .Since there is no non-constant rational function invariant under the action σ ( f ( t )) = f ( t − ) , weget that CH ∗ ( k [ t ] S ) T = CH ∗ ( k ) . Next, we see that the subalgebra generated by S is A = k [ t ] itself.Moreover, since σ ( t ) − t = k h S i T is zero, and therefore, we get HH n (( A ) S ) = ( T if n = , n > The enveloping algebra U ( sl ) . The universal enveloping algebra of sl is given by thepresentation k { E , F , H }h E H − ( H − ) E , F H − ( H + ) F , E F − F E − H i . The center of this algebra is generated by the Casimir element Ω = F E + H ( H + ) = E F + H ( H − ) . In this Subsection, we would like to write a generalized Weyl algebra isomorphic to U ( sl ) .Let A = k [ c , t ] and a = c − t ( t + ) . Define σ to be the algebra automorphism defined by σ ( f ( c , t )) = f ( c , t − ) for every f ( c , t ) ∈ A . In this case W a ,σ is generated by c , t , x and ( c − t ( t + )) x − in the smash product algebra A R T . The GWA W a ,σ is isomorphic to U ( sl ) viaan isomorphism defined as H t , E x , F
7→ ( c − t ( t + )) x − , see [13, Ex. 2.2].Let us define S to be the multiplicative system generated by elements of the form c − ( t − n )( t − n − ) , for n ∈ Z . Then ( W t ,σ ) S is isomorphic to k [ c , t ] S R T , and CH ∗ ( A S ) T = CH ∗ ( k [ c ]) . Moreover, the subalgebraof A = k [ c , t ] generated by S is A itself and since σ ( t ) − t =
1, we again get that k h S i T = HH n ( U ( sl ) S ) = ( k [ c ] ⊗ T if n = ,
20 otherwise.4.3.
Primitive quotients of U ( sl ) . One can also consider B λ : = W a ,σ /h c − λ i where W a ,σ is U ( sl ) as we defined above. These algebras are also GWAs since we can realize them using A = k [ t ] , a = λ − t ( t + ) with the σ given by t t −
1. See [5, Sect. 3].In this case, using a similar automorphism we used for U ( sl ) , we can replace S with themultiplicative system generated by elements of the form µ − ( t − n ) and µ + ( t − n ) where µ ∈ k is fixed and n ranges over Z . Then k h S i = k [ t ] and ( B λ ) S (cid:27) k [ t ] S R T . In this case, CH ∗ ( k [ t ] S ) T is CH ∗ ( k ) and k h S i T = σ ( t ) − t = HH n (( B λ ) S ) (cid:27) ( T if n = ,
20 otherwisefor every n > Quantum 2-torus.
Fix an element q ∈ k × which is not a root of unity. Let A = k [ t , t − ] andlet a = t as in the case of the ordinary Weyl algebra. But this time, let us define σ ∈ Aut ( A ) tobe the algebra automorphism given by σ ( f ( t )) = f ( qt ) for every f ( t ) ∈ A . The smash biproductalgebra A R T is the algebraic quantum 2-torus T q and the GWA W a ,σ is the quantum torus itselfsince a = t is a unit.Note that for every u ∈ A and m ∈ Z we have σ m ( u ) , u unless m = q is not a root ofunity. Thus CH ∗ ( A ) T = CH ∗ ( A ) T = CH ( )∗ ( A ) where(4.1) CH ( ) m ( A ) = Span k t n ⊗ · · · ⊗ t n m | n , . . . , n m ∈ Z with 0 = Õ i n i ! which gives us just the group homology of Z . Then by Proposition 3.1 we get HH n ( T q ) (cid:27) k ( n ) ⊗ T for every n > The quantum enveloping algebra U q ( sl ) . For a fixed q ∈ k × , the quantum envelopingalgebra of the lie algebra sl is given by the presentation k { K , K − , E , F }h K E − q E K , K F − q − FK , E F − F E = K − K − q − q − i . IRATIONAL EQUIVALENCES AND GENERALIZED WEYL ALGEBRAS 11
As before, we assume q is not a root of unity. There is an element Ω in the center of U q ( sl ) called the quantum Casimir element defined as(4.2) Ω = E F + q − K + qK − ( q − q − ) = F E + qK + q − K − ( q − q − ) . See [7, Sect.I.3]. Our first objective is to give a GWA that is isomorphic to U q ( sl ) .We start by setting A = k [ c , t , t − ] together with a = c − ( q − t + qt − ) and σ ∈ Aut ( A ) given by σ ( f ( c , t )) = f ( c , q t ) for every f ( c , t ) ∈ k [ c , t , t − ] . Define an algebramap γ : W a ,σ → U q ( sl ) given on the generators by t K , c
7→ ( q − q − ) Ω , x
7→ ( q − q − ) F , ax −
7→ ( q − q − ) E . Notice that the inverse of γ is defined easily as K t , E ax − q − q − , F xq − q − . One can show that both γ and its inverse are well-defined by showing the relations are preserved.Now, let S be the multiplicative system in A generated by the elements of the form c − ( q − n + t + q n − t − ) , for n ∈ Z . In this case too, the subalgebra of A generated by S is A itself. Then we haveCH ∗ ( A S ) T (cid:27) CH ( )∗ ( k [ c , t , t − ]) (cid:27) CH ∗ ( A ) T . On the other hand, since σ ( t ) − t = ( q − ) t and t is a unit, we get that k h S i T =
0. Thus, as inthe case of U ( sl ) we get HH n (( W a ,σ ) S ) (cid:27) HH n ( U q ( sl ) S ) (cid:27) ( k [ c ] ⊗ T if n = , , n > The quantum matrix algebra O q ( M ) . For a fixed q ∈ k × the algebra O q ( M ) of quantum2 × bc = cb , ab = q − ba , ac = q − ca , db = qbd , dc = qcd , ad − da = ( q − − q ) bc . The quantum determinant Ω = ad − q − bc = da − qbc generates the center of this algebra. See [7, pp.4–8]Now, let A = k [ u , v , w ] with the distinguished element u + q vw ∈ A where we set σ ( f ( u , v , w )) = f ( u , q − v , q − w ) for every f ( u , v , w ) ∈ A . Then the GWA W a ,σ is the subalgebra of A R T generated by A , x and ( u + q vw ) x − , and it is isomorphic to O q ( M ) via u Ω , v b , w c , x a , ( u + q vw ) x − d , and its inverse is a x , b v , c w , d
7→ ( u + q vw ) x − . Since O q ( GL ) is obtained by localizing O q ( M ) at the quantum determinant, we see that O q ( GL ) is isomorphic to ( W a ,σ ) u which itself is a GWA with A replaced by k [ u , u − , v , w ] with the remainingdatum unchanged. On the other hand, O q ( SL ) is the quotient of O q ( M ) by the two sided ideal generated by u −
1, andtherefore, is again a GWA with the same datum where this time we replace A by k [ u , v , w ]/h u − i .We also know that O q ( GL ) is isomorphic (as algebras only) to O q ( SL ) × k [ Ω ] .For the remaining of the section we are going to concentrate on O q ( SL ) only given as thesubalgebra of k [ v , w ] R T generated by v , w , x and ( + q vw ) x − .Now, let S be the Ore set generated by elements of the form 1 + q n + vw for n ∈ Z . Then O q ( SL ) S is isomorphic to k [ v , w ] S R T . In this case, since q is not a root of unity, we get thatCH ∗ ( A , A S ) T = CH ∗ ( k ) = CH ∗ ( A ) T . The subalgebra of k [ v , w ] generated by S is the polynomial algebra k [ vw ] over the indeterminate vw . Since σ ( vw ) − vw = ( q − − ) vw we get that k [ vw ] T = k Hence HH n (O q ( SL ) S ) (cid:27) k ( n ) ⊗ T for every n > Quantum group O q ( SU ) . Let us fix q ∈ k × . The algebraic quantum group O q ( SU ) is thenoncommutative *-algebra generated by two non-commuting indeterminates s and x subject tothe following relations(4.3) x ∗ x = − s ∗ s , x x ∗ = − q s ∗ s , s ∗ s = ss ∗ , xs = qsx , xs ∗ = qs ∗ x . See [18, pg.4]. One can write O q ( SU ) as a GWA W a ,σ by letting A = k [ s , s ∗ ] with thedistinguished element a ∈ A is defined as 1 − s ∗ s and σ ( f ( s , s ∗ )) = f ( qs , qs ∗ ) for every f ( s , s ∗ ) ∈ k [ s , s ∗ ] .Let S be the multiplicative system in A generated by elements of the form q n s ∗ s − n ∈ Z .Then O q ( SU ) S is isomorphic to A S R T by Theorem 2.3. If we assume that q ∈ k × is not a rootof unity we get that CH ∗ ( A S ) T = CH ∗ ( k ) = CH ∗ ( A ) T We also see that the subalgebra of k [ s , s ∗ ] generated by S is the polynomial algebra k [ ss ∗ ] , andsince σ ( ss ∗ ) − ss ∗ = ( q − ) ss ∗ we get that k h S i T = k . Then HH n (O q ( SU ) S ) (cid:27) k ( n ) ⊗ T for every n > Podleś spheres.
For a fixed q ∈ k × , the algebra of functions O q ( S ) on standard Podleśquantum spheres [37, 18] is the subalgebra of O q ( SU ) generated by elements s ∗ s , xs and s ∗ x ∗ .This means O q ( S ) is the subalgebra of the smash product k [ s , s ∗ ] R T generated by the elements s ∗ s , sx and s ∗ ( − s ∗ s ) x − . One can give a presentation for the Podleś sphere as(4.4) xt = q t x , y t = q − t y , y x = − t ( t − ) , x y = − q t ( q t − ) then we get a GWA structure if we let A = k [ t ] and where we set t = s ∗ s with a = − t ( t − ) and σ ( f ( t )) = f ( q t ) for every f ( t ) ∈ A .Let S be the multiplicative system in A generated by the set { t ( t − q n ) | n ∈ Z } then O q ( S ) S (cid:27) A S R T . Instead of this generating set one can use { t } ∪ {( t − q n ) | n ∈ Z } to get the samelocalization. Then we get that k h S i is A itself. If we assume that q ∈ k × is not a root of unity weget that CH ∗ ( A S ) T = CH ( )∗ ( k [ t , t − ]) , and CH ∗ ( A ) T = CH ∗ ( k ) . In this case k h S i T = k since σ ( t ) − t = ( q − ) t . Thus HH n (O q ( S ) S ) (cid:27) k ( n ) ⊗ T IRATIONAL EQUIVALENCES AND GENERALIZED WEYL ALGEBRAS 13 for every n > Parametric Podleś spheres.
In [18] Hadfield defines another family of Podleś spheres O q , c ( S ) given by a presentation equivalent to the following: xt = q t x , x ∗ t = q − t x ∗ , x ∗ x = c − t ( t − ) , x x ∗ = c − q t ( q t − ) . If we set A = k [ c , t ] , and let the distinguish element a ∈ A be c − t ( t − ) together with σ ( f ( c , t )) = f ( c , q t ) for every f ( c , t ) ∈ A we get a GWA structure on O q , c ( S ) similar to the GWAstructure on U ( sl ) where we changed only the algebra automorphism from σ ( f ( c , t )) = f ( c , t − ) to σ ( f ( c , t )) = f ( c , q t ) .Let S be the multiplicative system in A generated by the elements of the form c − q n t ( q n t − ) .If we assume that q ∈ k × is not a root of unity we conclude thatCH ∗ ( A S ) T = CH ∗ ( k [ c ]) = CH ∗ ( A ) T which allows us to conclude HH n (O q , c ( S ) S ) (cid:27) k ( n ) ⊗ k [ c ] ⊗ T for every n > References [1] J. Alev and F. Dumas , Sur le corps des fractions de certaines algèbres quantiques , J. Algebra, 170 (1994),pp. 229–265.[2]
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