aa r X i v : . [ m a t h . K T ] O c t HOMOLOGY OF QUANTUM LINEAR GROUPS
A. KAYGUN AND S. SÜTLÜAbstract. F or every n >
1, we calculate the Hochschild homology of the quantum monoids M q ( n ) , and the quantum groups GL q ( n ) and SL q ( n ) with coefficients in a 1-dimensional modulecoming from a modular pair in involution. Introduction
In this article we calculate the Hochschild homology of the quantum monoids M q ( n ) , andquantum groups GL q ( n ) and SL q ( n ) , [13, 21], with coefficients in a 1-dimensional module f − q k coming from a modular pair in involution (MPI) defined on GL q ( n ) and SL q ( n ) when q is not aroots of unity. We have an effective algorithm that calculates the explicit classes, and generatesthe corresponding Betti sequences. We also show that these homologies with coefficients in theMPI are direct summands of the Hochschild homologies of GL q ( n ) and SL q ( n ) with coefficientsin themselves, albeit with a degree shift.To achieve our goal, we introduce a general Hochschild-Serre type spectral sequence for flatalgebra extensions of the form Q ⊆ P through which calculating the homology of P reducesto calculating the homology of Q and the Q -relative homology of P . We then calculate thehomology of M q ( n ) by using the lattice of extensions M q ( a , b ) ⊆ M q ( n ) with 1 a , b n ,and related reductions in homology. Since GL q ( n ) is the localization of M q ( n ) at the quantumdeterminant, it is almost immediate to obtain the homology of GL q ( n ) from that of M q ( n ) by alocalization in homology [15, 1.1.17]. Calculating the homology of SL q ( n ) becomes immediatesince GL q ( n ) = SL q ( n ) ⊗ k [D q , D − q ] where D q is the quantum determinant [14, Proposition].The Hochschild and cyclic homology of SL q ( ) are calculated in [16, 23] using a specificresolution for SL q ( ) . The Hochschild homology of SL q ( ) with coefficients in itself twistedby the modular automorphism is calculated in [6]. The Hochschild cohomology of SL q ( n ) withcoefficients in η k η (Definition 3.1 and Proposition 3.3) is computed in [7] using the Koszulapproach similar to [23]. In [7] authors also show that Hochschild homology and cohomologyof SL q ( n ) satisfy Poincare duality. The Hochschild homology of the restricted dual O h ( G ) ofthe quantum universal enveloping algebra U h ( g ) , on the other hand, is described in [3] for asemi-simple Lie group G with its Lie algebra g . There are also results for subgroups of GL q ( n ) .Recall that for a semi-simple Lie group G , the kernel of the natural morphism G q → G is calledthe quantized Frobenius kernel of G . When q is a roots of unity with high enough order, thecohomology of the quantized Frobenius kernel of the Borel subgroup B q ( n ) of GL q ( n ) withtrivial coefficients comes from the quantum group N q ( n ) of the nilpotent part in even degrees,while it is trivial in odd degrees by [9]. The Hochschild homology of quantized Frobeniuskernels of other groups are calculated in [4, 19, 20] at roots of unity.A smash biproduct algebra (or a twisted tensor product algebra [1]) A R B involves a two-wayinteraction R : B ⊗ A → A ⊗ B between unital associative algebras A and B , as opposed to a one-way-interaction in an ordinary smash product of an algebra with a Hopf algebra. Calculating the absolute homology of a smash biproduct reduces to calculating the homology of a bisimplicialobject as we observed in [11]. In the same paper we calculated the homology of several algebrasincluding of M q ( ) using extensions of the form A ⊆ A R B . However, calculating homologiesof M q ( n ) for n > Plan of the paper.
After developing a Hochschild-Serre type spectral sequence for flat algebraextensions Q ⊆ P in Section 1, we review some basic facts and definitions about quantummonoids M q ( n ) , and groups GL q ( n ) and SL q ( n ) in Section 2. Then in Section 3 we calculatethe Hochschild homology of M q ( n ) , GL q ( n ) and SL q ( n ) for every n > f − q , n k defined using a modular pair in involution ( f − q , n , ) for the Hopfalgebras GL q ( n ) and SL q ( n ) . In Section 4, we explicitly write our calculations for the cases n = , , Notations and conventions.
We fix a ground field k and an element q ∈ k × which is not a rootsof unity. All unadorned tensor products ⊗ are over k . All algebras are assumed to be unitaland associative, but not necessarily commutative or finite dimensional over k . We use h X i todenote the two-sided ideal generated by a subset X of elements in an algebra, and Span k ( X ) for the k -vector space spanned by a subset X of elements in a vector space. For a fixed vectorspace V , we use Λ ∗ ( V ) to denote the exterior algebra over V , but used only as a vector space.For a fixed algebra A , we use CB ∗ ( A ) and CH ∗ ( A ) respectively for the bar complex and theHochschild complex of the algebra A . We use H ∗ ( A ) to denote the Hochschild homology of A .If the complexes, and therefore, homology includes coefficients other than the ambient algebra A , we will indicate this using a pair ( A , M ) . Acknowledgments.
This work completed while the first author was at the Department ofMathematics and Statistics of Queen’s University on academic leave from Istanbul TechnicalUniversity. The first author would like to thank both organizations for their support.
1. A Hochschild-Serre Type Spectral Sequence
In this section, we assume P and Q are unital associative algebras, together with a fixed morphismof algebras ϕ : Q → P .1.1. The mapping cylinder algebra.
Let Z : = P ⊕ Q be the unital associative algebra givenby the product ( p , q ) · ( p ′ , q ′ ) = ( pp ′ + p ϕ ( q ′ ) + ϕ ( q ) p ′ , qq ′ ) for any ( p , q ) , ( p ′ , q ′ ) ∈ Z , and the unit ( , ) ∈ Z . Accordingly, P Z is an ideal. Moreover,since P is a unital algebra, the homology of its bar complex vanishes in any positive degree. Assuch, P Z is an H -unital ideal, [24].Let X be a left P -module, and Y a right P -module. Then, they both can be considered as Z -modules via x ⊳ ( p , q ) : = x ⊳ p + x ⊳ ϕ ( q ) ( p , q ) ⊲ y = p ⊲ y + ϕ ( q ) ⊲ y . for any x ∈ X , and any ( p , q ) , ( p ′ , q ′ ) ∈ Z . Similarly, Y is a left Z -module. OMOLOGY OF QUANTUM LINEAR GROUPS 3
A filtration on the bar complex.
Let us now consider the bar complex CB ∗ ( X , Z , Y ) ofthe mapping cylinder algebra Z : = P ⊕ Q with coefficients in a right P -module X and a left P -module Y .We consider first the increasing filtration on CB ∗ ( X , Z , Y ) given by G ii + j = Õ n + ··· + n i = j X ⊗ P ⊗ n ⊗ Z ⊗ · · · ⊗ P ⊗ n i − ⊗ Z ⊗ P ⊗ n i ⊗ Y ⊆ CB i + j ( X , Z , Y ) . The bar differential interacts with the filtration as d ( G ii + j ) ⊆ G ii + j − . Then the associated gradedcomplex is given by(1.1) E i , j : = G ii + j / G i − i + j (cid:27) Ê n + ··· + n i = j X ⊗ P ⊗ n ⊗ Q ⊗ · · · ⊗ P ⊗ n i − ⊗ Q ⊗ P ⊗ n i ⊗ Y , with induced differentials. However, since we cannot reduce the number of Q ’s in the quotientcomplex, one can think of the quotient complex is a graded product of bar complexes withinduced differentials E i , ∗ = CB ∗ ( X , P , Q ) ⊗ Q CB ∗ ( Q , P , Q ) ⊗ Q · · · ⊗ Q CB ∗ ( Q , P , Q ) | {z } i -times ⊗ Q CB ∗ ( Q , P , Y ) where P acts on Q via 0. Since we observe that P acts on Q trivially, the E -term is given by E i , j = H i + j ( E i , ∗ ; d ) = ( Tor Pj ( X , Y ) if i = , i , . The spectral sequence then degenerates (for further details on spectral sequences we refer thereader to [17]), and we arrive at the following result.
Proposition 1.1.
Given two algebras P and Q , together with an algebra morphism ϕ : Q → P ,let Z : = P ⊕ Q be the mapping cylinder algebra, X a right P -module, and Y be a left P -module.Then, Tor Zn ( X , Y ) (cid:27) Tor Pn ( X , Y ) for all n > . A filtration on the Hochschild homology.
It is possible to adapt this setting to theHochschild homology complex. To this end, given a P -bimodule M , we start with the increasingfiltration G ii + j = Õ n + ··· + n i = j M ⊗ P ⊗ n ⊗ Z ⊗ · · · ⊗ P ⊗ n i − ⊗ Z ⊗ P ⊗ n i ⊆ CH i + j ( M , Z ) of CH ∗ ( Z , M ) . It follows from the observation b ( G ii + j ) ⊆ G ii + j − that the Hochschild complexCH ∗ ( Z , M ) is a filtered differential complex. The associated graded complex is then E i , j : = G ii + j / G i − i + j = Ê n + ··· + n i = j M ⊗ P ⊗ n ⊗ Q ⊗ · · · ⊗ P ⊗ n i − ⊗ Q ⊗ P ⊗ n i together with the induced differential b : E i , j −→ E i , j − given similar to that of (1.1) where P acts on Q by 0. Hence, the first page of the associated differential complex appears to be E i , j = H i + j ( E i , ∗ ; b ) = ( H j ( P , M ) if i = , i , . The spectral sequence then degenerates, and we arrive at the following result.
Proposition 1.2.
Given two algebras P and Q , together with an algebra morphism ϕ : Q → P ,let Z : = P ⊕ Q be the mapping cylinder algebra, X a right P -module, and Y be a left P -module.Then, H n ( Z , M ) (cid:27) H n ( P , M ) for any n > , and any P -bimodule M . A. KAYGUN AND S. SÜTLÜ
A second filtration on the bar complex.
Let P and Q be two algebras as above, but thistime we assume ϕ : Q → P is a left (or right) flat algebra morphism. In other words, P is flatas a left (resp. right) Q -module via ϕ : Q → P . Let us now consider the increasing filtration F ii + j = Õ n + ··· + n i = j X ⊗ Q ⊗ n ⊗ Z ⊗ · · · ⊗ Q ⊗ n i − ⊗ Z ⊗ Q ⊗ n i ⊗ Y ⊆ CB i + j ( X , Z , Y ) on the bar complex CB ∗ ( X , Z , Y ) . Since d ( F ii + j ) ⊆ F ii + j − , the bar complex CB ∗ ( X , Z , Y ) becomesa filtered differential complex; whose associated differential graded complex is E i , j = F ii + j / F i − i + j = Ê n + ··· + n i = j X ⊗ Q ⊗ n ⊗ P ⊗ · · · ⊗ Q ⊗ n i − ⊗ P ⊗ Q ⊗ n i ⊗ Y , together with the induced differentials d : E i , j −→ E i , j − coming from the bar complexCB ∗ ( X , Z , Y ) . As in the case of our first filtration, one can view the resulting complex as agraded multi-product of bar complexes(1.2) CB ∗ ( X , Q , P ) ⊗ P CB ∗ ( P , Q , P ) ⊗ P · · · ⊗ P CB ∗ ( P , Q , P ) | {z } i -times ⊗ Q CB ∗ ( P , Q , Y ) . In view of the assumption (that P is flat as a Q -module), the E -term is given by E i , j = H i + j ( E i , ∗ ; d ) = Tor Qj ( X ⊗ Q P ⊗ Q . . . ⊗ Q P | {z } i -times , Y ) if P is flat as a left Q -module , Tor Qj ( X , P ⊗ Q . . . ⊗ Q P | {z } i -times ⊗ Q Y ) if P is flat as a right Q -module . Keeping in mind that this spectral sequence converges to the Tor-groups of the mapping cylinderalgebra Z , which are in turn identified with the Tor-groups of the algebra P in the previoussubsection, we obtain the result which may be summarized in the following proposition. Proposition 1.3.
Given two algebras P and Q , together with the left (resp. right) flat algebramorphism ϕ : Q → P , let X a right P -module and Y be a left P -module. Then, there is aspectral sequence such that E i , j = Tor Qj ( X ⊗ Q P ⊗ Q . . . ⊗ Q P | {z } i -times , Y ) ⇒ Tor Pi + j ( X , Y ) , (cid:16) resp. E i , j = Tor Qj ( X , P ⊗ Q . . . ⊗ Q P | {z } i -times ⊗ Q Y ) ⇒ Tor Pi + j ( X , Y ) (cid:17) . A second filtration on the Hochschild homology.
We can present the arguments of theprevious subsection in terms of the Hochschild homology as well.To this end, we begin with the increasing filtration F ij + i = Õ n + ··· + n i = j M ⊗ Q ⊗ n ⊗ Z ⊗ · · · ⊗ Q ⊗ n i − ⊗ Z ⊗ Q ⊗ n i , that satisfies b ( F ii + j ) ⊆ F ii + j − . Then the associated graded complex is E i , j = F ij + i / F i − j + i = Ê n + ··· + n i = j M ⊗ Q ⊗ n ⊗ P ⊗ · · · ⊗ Q ⊗ n i − ⊗ P ⊗ Q ⊗ n i . OMOLOGY OF QUANTUM LINEAR GROUPS 5
Passing to the homology with respect to b : E i , j −→ E i , j − , an analogue of (1.2), we arrive atthe first page of the spectral sequence which is given by E i , j = H i + j ( E i , ∗ ; b ) = H j (cid:16) Q , M ⊗ Q P ⊗ Q · · · ⊗ Q P | {z } i -times (cid:17) that converges to H ∗ ( Z , M ) which is isomorphic to H ∗ ( P , M ) by Proposition 1.2. Thus, we haveproved the following proposition. Proposition 1.4.
Given two algebras P and Q , together with the left (resp. right) 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 ) .
2. Quantum Linear Groups
The algebra of quantum matrices M q ( n , m ) . Let n and m be two positive integers.Following [8, Lemma 2.10], we define M q ( n , m ) as the associative algebra on nm generators x i j where 1 i n and 1 j m . These generators are subject to the following relations x j ℓ x i ℓ = q x i ℓ x j ℓ for all 1 i < j n and 1 ℓ m , (2.1) x ℓ j x ℓ i = q x ℓ i x ℓ j for all 1 i < j m and 1 ℓ n , (2.2) x ℓ i x k j = x k j x ℓ i for all 1 k < ℓ n and 1 i < j m , (2.3) x ki x ℓ j − x ℓ j x ki = ( q − − q ) x k j x ℓ i for all 1 k < ℓ n and 1 i < j m . (2.4)For convenience, we are going to use M q ( n ) for M q ( n , n ) . We also use the following convention:for a n and b m when we write M q ( a , b ) ⊆ M q ( n , m ) we mean that we use the subalgebragenerated by x i j for 1 i a and 1 j b in M q ( n , m ) . Notice that these generators aresubject to the same relations, and therefore, the canonical map M q ( a , b ) → M q ( n , m ) is injective.It follows from (2.1) and (2.2) that all column or row subalgebras(2.5) Col ℓ : = h x i ℓ | i n i Row ℓ : = h x ℓ j | j n i are isomorphic to the quantum affine n -space k nq which is defined as the k -algebra(2.6) M q ( , n ) (cid:27) M q ( n , ) (cid:27) k nq : = k { x , . . . , x n }/h x j x i − q x j x i | i < j i . See [5, Subsect. 3.1] for the multiparametric version. Next, we note from [21, Thm. 3.5.1] and[13, Prop. 9.2.6] that B = Ö i , j n x t ij i j | t i j > is a vector space basis of M q ( n ) , with respect to any fixed order of the generators. A. KAYGUN AND S. SÜTLÜ
The bialgebra structure on M q ( n ) . The algebra M q ( n ) of quantum matrices is a bialgebrawhose comultiplication ∆ : M q ( n ) → M q ( n ) ⊗ M q ( n ) is given by ∆ ( x i j ) : = Õ k x ik ⊗ x k j and whose counit ε : M q ( n ) → k is given by ε ( x i j ) = δ i j . The quantum determinant.
Let S n be the group of permutations of the set { , . . . , n } , andlet ℓ ( σ ) ∈ N be the length of σ ∈ S n . Let also I : = { i , . . . , i m } and J : = { j , . . . , j m } be twosubsets of { , . . . , n } such that i < . . . < i m and j < . . . < j m . Then, the element D I J : = Õ σ ∈ S m (− q ) ℓ ( σ ) x i σ ( ) j . . . x i σ ( m ) j m = Õ σ ∈ S m (− q ) ℓ ( σ ) x i j σ ( ) . . . x i m j σ ( m ) ∈ M q ( m ) is called the quantum m-minor determinant as defined in [13, Sect. 9.2.2] and [21, Sect. 4.1].On one extreme we have D I J = x i j for I = { i } and J = { j } . On the other extreme, if we let I = J = { , . . . , n } we get the quantum determinant which is denoted by D q . The quantumdeterminant is in the center M q ( n ) . Moreover, if q is not a root of unity, then the center of M q ( n ) is generated by the quantum determinant. For this result see [13, Prop. 9.9], [21, Thm. 4.6.1],or [18].2.4. The quantum general linear group GL q ( n ) . The quantum group GL q ( n ) is obtained byadjoining D − q to the bialgebra M q ( n ) . More precisely, GL q ( n ) = M q ( n )[ t ]h t D q − i . Let us note from this definition that M q ( n ) is a subalgebra of GL q ( n ) . In terms of generators andrelations, GL q ( n ) is the algebra generated by n + x i j and t with i , j ∈ { , . . . , n } ,satisfying the same relations as (2.1) - (2.4), and D q t = t D q = , (2.7) x i j t = t x i j . (2.8)On the other hand, GL q ( n ) is the localization M q ( n ) D q of M q ( n ) with respect to D q as in [21,Sect. 5.3], and [15, Prop. 1.1.17]. As such, the bialgebra structure on M q ( n ) extends uniquelyto GL q ( n ) [21, Lemma 5.3.1]. Furthermore, GL q ( n ) is a Hopf algebra with the antipode S : GL q ( n ) → GL q ( n ) given by S ( x i j ) : = (− q ) j − i A ji D − q , S (D − q ) : = D q , where A i j : = D I J with I = { , . . . , n } − { i } , and J = { , . . . , n } − { j } . The matrix (cid:2) q i − j A i j (cid:3) i , j n is called the quantum cofactor matrix of (cid:2) x i j (cid:3) i , j n , and the Hopf algebra GL q ( n ) is called the quantum general linear group . OMOLOGY OF QUANTUM LINEAR GROUPS 7
The quantum special linear group SL q ( n ) . Next, we recall briefly the quantum versionof the special linear group. It is given as the quotient space SL q ( n ) : = GL q ( n )hD q − i = M q ( n )hD q − i , which happens to be a Hopf algebra with the bialgebra structure induced from GL q ( n ) , or from M q ( n ) , and the antipode S : SL q ( n ) → SL q ( n ) is given by S ( x i j ) : = q j − i A ji induced from GL q ( n ) . The Hopf algebra SL q ( n ) is called the quantum special linear group .Actually, one can write GL q ( n ) as a direct product of SL q ( n ) and the Laurent polynomial ringover the quantum determinant k [D q , D − q ] . Proposition 2.1. [14, Proposition]
There is an isomorphism of algebras of the form GL q ( n ) (cid:27) SL q ( n ) ⊗ k [D q , D − q ] . Modular pairs of involution for GL q ( n ) and SL q ( n ) . Finally, we are going to see thatboth Hopf algebras GL q ( n ) and SL q ( n ) admit a modular pair in involution (MPI). Let us recallfrom [2] that an Hopf algebra H is said to admit an MPI if there is an algebra homomorphism δ : H → k and a group-like element σ ∈ H such that δ ( σ ) = S δ ( h ) = σ h σ − where S δ ( h ) : = δ ( h ( ) ) S ( h ( ) ) for any h ∈ H . Proposition 2.2.
Let f − q , n : GL q ( n ) → k (resp. f − q , n : SL q ( n ) → k ) be given by f − q , n ( x i j ) : = δ i j q ( n + )− i . Then, ( f − q , n , ) is a MPI for the Hopf algebra GL q ( n ) , (resp. for the Hopf algebra SL q ( n ) .)Proof. We will give the proof for GL q ( n ) . The proof for the case of SL q ( n ) is similar, andtherefore, is omitted. It is given in [21, Lemma 5.4.1] that f q , n : GL q ( n ) → k given by f q , n ( x i j ) = δ i j q i −( n + ) is an algebra homomorphism, i.e. a character. Then, its convolution inverse f − q , n : GL q ( n ) → k is also a character. The claim then follows from the observation that e S f − q , n = f q , n ∗ S ∗ f − q , n since S = f − q , n ∗ Id ∗ f q , n by [21, Thm. 5.4.2] where ∗ is the convolution multiplication on theset of characters of Hopf algebras. (cid:3) As for GL q ( n ) , there is a second choice of MPI. Proposition 2.3.
Let f − q , n : GL q ( n ) → k be as before, and let D − q ∈ GL q ( n ) be the quantumdeterminant. Then, ( f − q , n , D − q ) is a modular pair in involution for the Hopf algebra GL q ( n ) .Proof. We have, for any x ∈ GL q ( n ) , e S f − q , n ( x ) = e S f − q , n (cid:16) D − q f − q , n ( x ( ) ) S ( x ( ) ) (cid:17) = e S f − q , n ( S ( x ( ) )) f − q , n ( x ( ) ) S (D − q ) = D − q f − q , n ( S ( x ( ) )) S ( x ( ) ) f − q , n ( x ( ) )D q = D − q x D q = x . A. KAYGUN AND S. SÜTLÜ
Furthermore, we have f q , n (D q ) = f q , n Õ σ ∈ S n (− q ) ℓ ( σ ) x σ ( ) . . . x n σ ( n ) ! = Õ σ ∈ S n (− q ) ℓ ( σ ) f q , n ( x σ ( ) ) . . . f q , n ( x n σ ( n ) ) (2.9) = q n ( n + )− ( + ··· + n ) = , and hence f − q , n (D − q ) = (cid:3)
3. Hochschild Homology of M q ( n ) Homology of quantum matrices M q ( n , m ) . Given a sequence ( q , . . . , q n ) of scalars in k ,and let α : M q ( n ) → k be the character given by α ( x i j ) = δ i j q i . Accordingly, the counit ε ( x i j ) = δ i j , the characters f q , n and f − q , n of Proposition 2.2, namely, f q , n ( x i j ) = δ i j q i − − n and f − q , n ( x i j ) = δ i j q n − i + and finally the character η : M q ( n , m ) → k given by(3.1) η ( x i j ) = , for 1 i n and 1 j m correspond to the sequences ε ↔ ( , . . . , ) , (3.2) f q , n ↔ ( q − n + , q − n + , . . . , q n − ) , (3.3) f − q , n ↔ ( q n − , q n − , . . . , q − n + ) , (3.4) η ↔ ( , . . . , ) . (3.5)Let, now, α, β : M q ( n , m ) → k be two characters given by two sequences of scalars as definedabove. Let also α k β denote the M q ( n , m ) -bimodule k with the actions given by(3.6) x i j ⊲ = α ( x i j ) and 1 ⊳ x i j = β ( x i j ) for any x i j ∈ M q ( n , m ) . In addition, the absence of a subscript such as α k or k α indicates thatthe action on the unspecified side is given by the counit.The following result gives us the license to consider only the 1-dimensional bimodules whoseright action is given by the counit. Proposition 3.1.
Given any characters α and β defined by a sequence of scalars ( q a , . . . , q a n ) and ( q b , . . . , q b n ) as defined above, there is an automorphism θ α : M q ( n , m ) → M q ( n , m ) so thatthe action of M q ( n , m ) twisted by θ α on β k α reduces to α − β k .Proof. We define(3.7) θ ( x i j ) = q − a i x i j OMOLOGY OF QUANTUM LINEAR GROUPS 9 and observe that the relations (2.1) through (2.4) are invariant under this action. The action of M q ( n , m ) twisted by θ α is defined as1 ◭ x i j = α ( θ α ( x i j )) = δ i j q − a i q a i = δ i j and the left action is given as x i j ◮ = β ( θ α ( x i j )) = δ i j q b i q − a i for every generator x i j . (cid:3) Lemma 3.2.
The Hochschild homology of the quantum affine n -space M q ( n , ) (cid:27) M q ( , n ) withcoefficients in η k η is given by H ℓ ( M q ( , n ) , η k η ) (cid:27) k ⊕ ( n ℓ ) . Proof.
Setting Q : = k [ x ] ⊆ M q ( , n ) = : P , we have H ∗ ( M q ( , n ) , η k η ) ⇐ E i , j = H j ( k [ x ] , η k η ⊗ k [ x ] M q ( , n ) ⊗ k [ x ] . . . ⊗ k [ x ] M q ( , n ) | {z } i -many ) (cid:27) H j ( k [ x ] , η k η ⊗ M q ( , n − ) ⊗ · · · ⊗ M q ( , n − ) | {z } i -many ) , where the k [ x ] action is still given by η on the coefficient complex. Thus, the E -term of thespectral sequence splits as E i , j = H j ( k [ x ] , η k η ) ⊗ M q ( , n − ) ⊗ · · · ⊗ M q ( , n − ) | {z } i -many (cid:27) CH i ( M q ( , n − ) , η k η ) ⊗ H j ( k [ x ] , η k η ) since the action of M q ( , n − ) on H j ( k [ x ] , k ) is again given by η . On the other hand for k [ x ] we have H j ( k [ x ] , η k η ) = ( k if j = ,
10 otherwise.Then we see that H ℓ ( M q ( , n ) , η k η ) (cid:27) H ℓ ( M q ( , n − ) , η k η ) ⊕ H ℓ − ( M q ( , n − ) , η k η ) . The result follows from recursion. (cid:3)
Let Λ ∗ ( X ) denote the exterior algebra generated by a set X of indeterminates. The followingresult follows from an easy dimension counting. Proposition 3.3.
We have isomorphisms of vector spaces of the form H ℓ ( M q ( n , m ) , η k η ) (cid:27) Λ ℓ ( x i j | i n , j m ) for every m , n > and ℓ > .Proof. Let prove this by induction on n . For n = n . Consider the extension M q ( n , m ) ⊆ M q ( n + , m ) with the canonical embedding. Then by Proposition 1.4 we get H ∗ ( M q ( n + , m ) , η k η ) ⇐ E i , j = H j ( M q ( n , m ) , M q ( n + , m ) ⊗ M q ( n , m ) · · · ⊗ M q ( n , m ) M q ( n + , m ) | {z } i -times ) (cid:27) H j ( M q ( n , m ) , CH i ( M q ( , m ) , η k η )) (cid:27) H j ( M q ( n , m ) , η k η ) ⊗ CH i ( M q ( , m ) , η k η ) since M q ( n , m ) acts by η on the coefficient complex. Accordingly, H ℓ ( M q ( n + , m ) , η k η ) (cid:27) Ê i + j = ℓ H j ( M q ( n , m ) , η k η ) ⊗ H i ( M q ( , m ) , η k η ) , and thereforedim k H ℓ ( M q ( n + , m ) , η k η ) = Õ ℓ + ℓ = ℓ dim k H ℓ ( M q ( n , m ) , η k η ) · dim k H ℓ ( M q ( , m ) , η k η ) = Õ ℓ + ℓ = ℓ (cid:18) nm ℓ (cid:19) (cid:18) m ℓ (cid:19) = (cid:18) ( n + ) m ℓ (cid:19) as we wanted to show. (cid:3) Homology of M q ( n ) . In this subsection we compute the Hochschild homology of thealgebra M q ( n ) of quantum matrices with coefficients in α = ( q a , . . . , q a n ) , where a , . . . , a n ∈ Z .To this end, we begin with the extension Row n ⊆ M q ( n ) that yields, in the relative complex, α k ⊗ Row n M q ( n ) ⊗ Row n · · · ⊗ Row n M q ( n ) | {z } i -times (cid:27) α k ⊗ M q ( n − , n ) ⊗ · · · ⊗ M q ( n − , n ) | {z } i -times . Hence, the E -page of the spectral sequence is E i , j (cid:27) H j ( Row n , CH i ( M q ( n − , n ) , α k )) . We then note that the elements x ni ∈ Row n , for i , n , act on the coefficient complex via η ,whereas x nn act via a scalar q a n on the left. On the right, the action of x nn is via another scalardetermined by the total degree of the terms in x in in CH i ( M q ( n − , n ) , α k ) for 1 i n − E i , j (cid:27) Ê a H j ( Row n , q an k q − a ) ⊗ CH ( a ) i ( M q ( n − , n ) , α k ) , where CH ( a )∗ denotes the subcomplex of terms whose total degree in x in , for i = , . . . , n − a ∈ Z . Let us remark also that since these terms act by η , the graded subspaceCH ( a )∗ ( M q ( n − , n ) , α k ) of CH ∗ ( M q ( n − , n ) , α k ) is indeed a subcomplex.For the homology of the row algebra this time, we use the lattice of extensions Row n ( a , b ) ⊆ Row n , where Row n ( a , b ) is the subalgebra of Row n generated by x na , . . . , x nb . Thus, we mayexpress H j ( Row n , q an k q − a ) = Ê c H c ( Row n ( b + , n ) , q an k q − a + c − j ) ⊗ Λ j − c ( x n , . . . , x nb ) OMOLOGY OF QUANTUM LINEAR GROUPS 11 for every 1 b n −
1. In particular, H j ( Row n , q an k q − a ) = Ê c H c ( Row n ( n , n ) , q an k q − a + c − j ) ⊗ Λ j − c ( x n , . . . , x nn − ) . Since Row n ( n , n ) = k [ x nn ] , the direct sum above has only two non-zero terms: those with c = c =
1. Therefore, H j ( Row n , q an k q − a ) = (cid:16) H ( k [ x nn ] , q an k q − j − a ) ⊗ Λ j ( x n , . . . , x nn − ) (cid:17) ⊕ (cid:16) H ( k [ x nn ] , q an k q − j − a + ) ⊗ Λ j − ( x n , . . . , x nn − ) (cid:17) . On the other hand, we observe that the homology is zero unless q an k q − a + c − j is symmetric. Thus, H j ( Row n , q an k q − a ) = Λ jq ( x n , . . . , x nn − ) if a = − a n − j > , Λ j − ( x n , . . . , x nn − ) x nn if a = − a n − j + > , E -page of the spectral sequence reduces to H ℓ ( M q ( n ) , α k )⇐ E i , j (cid:27) CH (− a n − j ) i ( M q ( n − , n ) , α k ⊗ Λ j ( x n , . . . , x n , n − ))⊕ CH (− a n − j + ) i ( M q ( n − , n ) , α k ⊗ Λ j − ( x n , . . . , x n , n − ) x nn ) . Now, let Col n ( a , b ) be the subalgebra generated by x an , . . . , x bn in M q ( n − , n ) . Then, in viewof Proposition 1.4 we have H ( a ) i ( M q ( n − , n ) , α k ⊗ Λ j ( x n , . . . , x nn ))⇐ E r , s = H ( a ) s ( Col n ( , n − ) , CH r ( M q ( n − ) , α k ⊗ Λ j ( x n , . . . , x nn ))) . Since Col n ( , n − ) acts on the coefficient complex via η , the E -page splits, and we arrive at H ( a ) i ( M q ( n − , n ) , α k ⊗ Λ j ( x n , . . . , x nn )) (cid:27) H i − a ( M q ( n − ) , α k ⊗ Λ j ( x n , . . . , x nn ) ⊗ Λ a ( x n , . . . , x n − , n )) . From these we get the E -page E i , j (cid:27) H i + j + a n ( M q ( n − ) , α k ⊗ Λ j ( x n , . . . , x n , n − ) ⊗ Λ − a n − j ( x n , . . . , x n − , n ))⊕ H i + j + a n − ( M q ( n − ) , α k ⊗ Λ j − ( x n , . . . , x n , n − ) x nn ⊗ Λ − a n − j + ( x n , . . . , x n − , n )) . Therefore, H ℓ ( M q ( n ) , α k ) (cid:27) Ê j , s H ℓ + a n − s ( M q ( n − ) , α k ⊗ Λ j ( x n , . . . , x n , n − ) ⊗ Λ − a n − j ( x n , . . . , x n − , n )) ⊗ Λ s ( x nn ) . Now, consider the subspace S ∗ n of Λ ∗ ( x i j | i , j n ) generated by x in and x n j with 1 i n − j n −
1. Also, we use S ∗ n ( b ) to denote the homogeneous vector subspace of S ∗ n ofterms whose total degree over terms of type x in and x ni is b . We observe that H ℓ ( M q ( n ) , α k ) (cid:27) Ê s H ℓ + a n − s ( M q ( n − ) , α k ⊗ S ∗ n − (− a n )) ⊗ Λ s ( x nn ) (cid:27) Ê β, s H ℓ + a n − s ( M q ( n − ) , α k β − ) ⊗ S ∗ n − ( b , . . . , b n − , − a n ) ⊗ Λ s ( x nn ) (cid:27) Ê β, s H ℓ + a n − s ( M q ( n − ) , αβ k ) ⊗ S ∗ n − ( b , . . . , b n − , − a n ) ⊗ Λ s ( x nn ) . The sum is taken over all β = ( q b , . . . , q b n ) , where the multi-degree ( b , . . . , b n ) indicates thatwe consider the k -vector space spanned by monomials Γ that has the total degree deg i ( Γ ) = b i in terms of x si and x it for all s , t = , . . . , n and i = , . . . , n , and we set(3.8) deg i ( x i j ) = deg i ( x ji ) = − j < i , j = i , j > i . It follows at once that b n = − a n , since there are no indices j > n . Proceeding the computationrecursively, we arrive at the following result. Theorem 3.4.
Fix a sequence of non-zero scalars α = ( q a , . . . , q a n ) , and let us define Λ ∗( α ) ( x i j | i , j n ) as the subspace of differential forms with multi-degree ( a , . . . , a n ) where the a i is the total degree– in the sense of (3.8) – of terms involving the indeterminates x si and x it for s , t = , . . . , n , and i = , . . . , n . Then H ℓ + | α | ( M q ( n ) , α k ) (cid:27) Ê s Λ ℓ − s ( α ) ( x i j | i , j n ) ⊗ Λ s ( x , . . . , x nn ) as vector spaces for every n > , ℓ > . Homologies of GL q ( n ) and SL q ( n ) with coefficients in f − q , n k . We are going to derive thehomology of GL q ( n ) and SL q ( n ) from that of M q ( n ) by using the localization of the Hochschildhomology.Let us first recall the localization of the homology, [15, Prop. 1.1.17]. Proposition 3.5.
Given an algebra A , and a multiplicative subset S ⊆ A so that ∈ S and < S ,and an A -bimodule M , there are the following canonical isomorphisms: H ∗ ( A , M ) S (cid:27) H ∗ ( A , M S ) (cid:27) H ∗ ( A S , M S ) , where M S : = Z ( A ) S ⊗ Z ( A ) M , Z ( A ) denotes the center of A , and Z ( A ) S stands for the localization of Z ( A ) at S . Now, in view of the fact that GL q ( n ) is the localization of M q ( n ) at S = {D nq | n > } , we obtainthe Hochschild homology of GL q ( n ) readily from the above localization result. Theorem 3.6.
We have H ℓ ( M q ( n ) , f − q , n k ) = H ℓ ( GL q ( n ) , f − q , n k ) (cid:27) H ℓ ( SL q ( n ) , f − q , n k ) ⊕ H ℓ − ( SL q ( n ) , f − q , n k ) for every ℓ > and for every n > . OMOLOGY OF QUANTUM LINEAR GROUPS 13
Proof.
Let us recall from [18, Thm. 1.6], see also [22], that Z ( M q ( n )) = k [D q ] , and that GL q ( n ) = M q ( n ) S for the multiplicative system S = {D nq | n > } generated by thequantum determinant. Accordingly, we have Z ( M q ( n )) S = k [D q , D − q ] , and f − q , n k S = k [D − q ] ⊗ f − q , n k is the GL q ( n ) -bimodule so that the M q ( n ) -bimodule structure concentrated on f − q , n k , and the k [D − q ] -bimodule structure is on k [D − q ] . Then we have H ∗ ( GL q ( n ) , f − q , n k S ) (cid:27) k [D − q ] ⊗ H ∗ ( GL q ( n ) , f − q , n k ) so that the GL q ( n ) -bimodule structure on f − q , n k is determined by the trivial action of D − q . Onthe other hand, H ∗ ( M q ( n ) , f − q , n k ) S (cid:27) k [D − q ] ⊗ H ∗ ( M q ( n ) , f − q , n k ) , where the GL q ( n ) -bimodule structure is given in such a way that the M q ( n ) -bimodule structureis on H ∗ ( M q ( n ) , f − q , n k ) , and the D − q -action structure is concentrated on k [D − q ] . Proposition 3.5then yields the first claim. The second part the computation follows from [14, Proposition] and[11, Theorem 2.7]. (cid:3) Homologies of GL q ( n ) and SL q ( n ) with coefficients in themselves. Let H be a Hopfalgebra with an invertible antipode, and let X be an arbitrary H -bimodule. Proposition 3.7.
There is an isomorphism of graded vector spaces H ∗ ( H , X ) (cid:27) Tor H ∗ ( ad ( X ) , k ) where we define the adjoint action of H on X as x h : = S − ( h ( ) ) x h ( ) for every x ∈ X and h ∈ H .Proof. Let CB ∗ ( H ) be the bar complex of H viewed as an algebra and we will use CB ∗ ( X , H , Y ) to denote X ⊗ H CB ∗ ( H ) ⊗ H Y to denote the two sided complex with coefficients in a left H -module Y and right H -module X . Let us write an isomorphism of complexes ρ ∗ : CH ∗ ( H , X ) → CB ∗ ( ad ( X ) , H , k ) as ρ n ( x ⊗ h ⊗ · · · ⊗ h n ) = h , ( ) · · · h n , ( ) x ⊗ h , ( ) ⊗ · · · ⊗ h n − , ( ) ⊗ h n , ( ) It is straight-forward but tedious exercise that ρ ∗ is an isomorphism of pre-simplicial k -modules,and therefore, an isomorphism of complexes. (cid:3) The isomorphism given in Proposition 3.7 is a well-known isomorphism used in (co)homologyof Hopf algebras, and usually referred as
MacLane Isomorphism . This is an extension of thesame result for a group algebras, and universal enveloping algebras.Let us use f − n , q H to denote the regular representation of H = SL q ( n ) or H = GL q ( n ) twisted onthe left by the automorphism θ f − n , q defined in Proposition 3.1. Theorem 3.8.
The canonical map D q : f − q , n k → ad ( f − q , n GL q ( n )) induces a split injection inhomology of the form H m ( GL q ( n ) , f − q , n k ) → H m ( GL q ( n ) , f − n , q GL q ( n )) (cid:27) H m + ( GL q ( n )) and H m ( SL q ( n ) , f − q , n k ) → H m ( SL q ( n ) , f − q , n SL q ( n )) (cid:27) H m + ( SL q ( n )) for every m > . Proof.
We use Proposition 3.7 for H = GL q ( n ) with coefficients in the twisted module f − q , n H . Theadjoint module ad ( f − q , n H ) splits as a direct sum of irreducible submodules with multiplicities, andone of those modules is the module f − q , n k . In the case H = GL q ( n ) or H = SL q ( n ) , the quantumdeterminant D q lies inside that submodule since D q is in the center. Then Tor H ∗ ( f − q , n k , k ) is adirect summand of H ∗ ( H , f − q , n H ) . The rest follows from an untwist as in [12, Section 2.9] but forHochschild homology. (cid:3)
4. Explicit calculations M q ( ) , GL q ( ) and SL q ( ) . The character f − q , is given by the sequence ( q , q − ) . Then, | α | = H ℓ ( M q ( ) , ( q , q − ) k ) = Ê s Λ ℓ − s ( q , q − ) ( x , x ) ⊗ Λ s ( x , x ) . One can write 4 different exterior product between x and x , anddeg (( )) = ( , ) deg (( x )) = deg (( x )) = ( , − ) (4.1) deg (( x , x )) = ( , − ) from which we only take the degree ( , − ) -terms of exterior degree 1. Thus(4.2) H ℓ ( M q ( ) , ( q , q − ) k ) = ℓ = ℓ > , Span k (( x ) , ( x )) if ℓ = , Span k (( x , x ) , ( x , x ) , ( x , x ) , ( x , x )) if ℓ = , Span k (( x , x , x ) , ( x , x , x )) if ℓ = . The Betti numbers of the homology are given in Figure 1. m k H m ( M q ( ) , f − q , k ) k H m ( GL q ( ) , f − q , k ) k H m ( SL q ( ) , f − q , k ) Figure 1.
The Betti numbers for M q ( ) , GL q ( ) and SL q ( ) M q ( ) , GL q ( ) and SL q ( ) . The character f − q , is given by ( q , , q − ) and | α | =
0. Theexterior degree 1 terms are deg (( x )) = deg (( x )) = ( , − , ) deg (( x )) = deg (( x )) = ( , , − ) (4.3) deg (( x )) = deg (( x )) = ( , , − ) and we need terms of degree signature ( , , − ) . We must solve a system of Z -linear equations ( α , α , α ) − −
10 1 − = ( , , − ) , OMOLOGY OF QUANTUM LINEAR GROUPS 15 where α i ∈ { , , } . The only solutions are ( α , α , α ) = ( , , ) or ( α , α , α ) = ( , , ) . For the first solution, there is only one term of exterior degree 4: ( x , x , x , x ) . On theother hand, for the second solution there are 8 such terms of exterior degree 3. Then we use theexterior algebra on x , x and x to promote these terms to higher degrees. In short, we have: H + ℓ ( M q ( ) , ( q , , q − ) k ) = Span k (cid:16) ( x , x , x ) , ( x , x , x ) , ( x , x , x ) , ( x , x , x ) , ( x , x , x ) , ( x , x , x ) , ( x , x , x ) , ( x , x , x ) (cid:17) ⊗ Λ ℓ ( x , x , x ) (4.4) ⊕ Span k (( x , x , x , x )) ⊗ Λ ℓ − ( x , x , x ) . The Betti numbers of the homology are given in Figure 2. m k H m ( M q ( ) , f − q , k ) k H m ( GL q ( ) , f − q , k ) k H m ( SL q ( ) , f − q , k ) Figure 2.
The Betti numbers for M q ( ) , GL q ( ) and SL q ( ) M q ( ) , GL q ( ) and SL q ( ) . The character f − q , is now given by the sequence α = ( q , q , q − , q − ) with | α | =
0. The exterior degree 1 terms aredeg (( x )) = deg (( x )) = ( , − , , ) deg (( x )) = deg (( x )) = ( , , − , ) deg (( x )) = deg (( x )) = ( , , − , ) deg (( x )) = deg (( x )) = ( , , , − ) (4.5) deg (( x )) = deg (( x )) = ( , , , − ) deg (( x )) = deg (( x )) = ( , , , − ) and we need the total multi-degree ( , , − , − ) . Thus we solve ( α , α , α , α , α , α ) − − −
10 1 − −
10 0 1 − = ( , , − , − ) again with the restriction that α i ∈ { , , } . The Betti numbers of this case are given in Figure 3. m k H m ( M q ( ) , f − q , k ) k H m ( GL q ( ) , f − q , k ) k H m ( SL q ( ) , f − q , k ) Figure 3.
The Betti numbers for M q ( ) , GL q ( ) and SL q ( ) References [1] A. Cap, H. Schichl, and J. Vanžura. On twisted tensor products of algebras.
Comm. Algebra , 23(12):4701–4735, 1995.[2] A. Connes and H. Moscovici. Cyclic cohomology and Hopf algebras.
Lett. Math. Phys. , 48(1):97–108, 1999.Moshé Flato (1937–1998).[3] P. Feng and B. Tsygan. Hochschild and cyclic homology of quantum groups.
Comm. Math. Phys. , 140(3):481–521, 1991.[4] V. Ginzburg and S. Kumar. Cohomology of quantum groups at roots of unity.
Duke Math. J. , 69(1):179–198,1993.[5] J. A. Guccione and J. J. Guccione. Hochschild and cyclic homology of Ore extensions and some examples ofquantum algebras. K -Theory , 12(3):259–276, 1997.[6] T. Hadfield and U. Krähmer. Twisted homology of quantum SL ( ) . K -Theory , 34(4):327–360, 2005.[7] T. Hadfield and U. Krähmer. On the Hochschild homology of quantum SL ( N ) . C. R. Math. Acad. Sci. Paris ,343(1):9–13, 2006.[8] J. Hong and O. Yacobi. Quantum polynomial functors.
J. Algebra , 479:326–367, 2017.[9] J. Hu. Cohomology of quantum general linear groups.
J. Algebra , 213(2):513–548, 1999.[10] A. Kaygun. Noncommutative fibrations.
Comm. Algebra , 47(8):3384–3398, 2019.[11] A. Kaygun and S. Sütlü. On the Hochschild homology of smash biproducts. arXiv:1809.08873 , 2018.[12] A. Kaygun and S. Sütlü. Hopf-cyclic cohomology of quantum enveloping algebras.
J. Noncommut. Geom. ,10(2):429–446, 2016.[13] A. Klimyk and K. Schmüdgen.
Quantum groups and their representations . Texts and Monographs in Physics.Springer-Verlag, 1997.[14] T. Levasseur and J. T. Stafford. The quantum coordinate ring of the special linear group.
J. Pure Appl. Algebra ,86(2):181–186, 1993.[15] J. L. Loday.
Cyclic homology , volume 301 of
Grundlehren der Mathematischen Wissenschaften [FundamentalPrinciples of Mathematical Sciences] . Springer-Verlag, second edition, 1998. Appendix E by Maria O. Ronco,Chapter 13 by the author in collaboration with Teimuraz Pirashvili.[16] T. Masuda, Y. Nakagami, and J. Watanabe. Noncommutative differential geometry on the quantum SU ( ) . I.An algebraic viewpoint. K -Theory , 4(2):157–180, 1990.[17] J. McCleary. A User’s Guide to Spectral Sequences . Cambridge University Press, 2001.[18] M. Noumi, H. Yamada, and K. Mimachi. Finite-dimensional representations of the quantum group GL q ( n ; C ) and the zonal spherical functions on U q ( n − )\ U q ( n ) . Japan. J. Math. (N.S.) , 19(1):31–80, 1993.[19] B. Parshall and J. P. Wang. Cohomology of infinitesimal quantum groups. I.
Tohoku Math. J. (2) , 44(3):395–423, 1992.[20] B. Parshall and J. P. Wang. Cohomology of quantum groups: the quantum dimension.
Canad. J. Math. ,45(6):1276–1298, 1993.[21] B. Parshall and J.P. Wang. Quantum linear groups.
Mem. Amer. Math. Soc. , 89(439):vi+157, 1991.[22] N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev. Quantization of Lie groups and Lie algebras.
Algebra i Analiz , 1(1):178–206, 1989.[23] M. Rosso. Koszul resolutions and quantum groups.
Nuclear Phys. B Proc. Suppl. , 18B:269–276 (1991), 1990.Recent advances in field theory (Annecy-le-Vieux, 1990).[24] M. Wodzicki. Excision in cyclic homology and in rational algebraic K -theory. Ann. of Math. (2) , 129(3):591–639, 1989.
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