Quantum function algebras from finite-dimensional Nichols algebras
aa r X i v : . [ m a t h . QA ] A ug Quantum function algebras from finite-dimensionalNichols algebras
Marco Andr´es Farinati ∗ Gast ´on Andr´es Garc´ıa † Abstract
We describe how to find quantum determinants and antipode formulas from braidedvector spaces using the FRT-construction and finite-dimensional Nichols algebras. It im-proves the construction of quantum function algebras using quantum grassmanian alge-bras. Given a finite-dimensional Nichols algebra B , our method provides a Hopf algebra H such that B is a braided Hopf algebra in the category of H -comodules. It also serves assource to produce Hopf algebras generated by cosemisimple subcoalgebras, which are ofinterest for the generalized lifting method. We give several examples, among them quan-tum function algebras from Fomin-Kirillov algebras associated with the symmetric groupon three letters. Introduction
Let k be a field and V a finite-dimensional k -vector space. A map c ∈ Aut( V ⊗ V ) is called abraiding if it satisfies the braid equation ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ) in End( V ⊗ V ⊗ V ) . (1)In such a case, the pair ( V, c ) is called a braided vector space.Given a braided vector space ( V, c ) , Faddeev, Reshetikhin and Takhtajan [FRT] introduceda method, the FRT-construction for short, to construct a coquasitriangular bialgebra A ( c ) suchthat V is an A ( c ) -comodule and c is a morphism of A ( c ) -comodules. By the very definition, thisbialgebra is universal with such properties. Besides, it turns out that the category A ( c ) M of left A ( c ) -comodules is braided monoidal. Notice that if V = 0 then A ( c ) is never a Hopf algebra:suppose the contrary and write S for the antipode. Since A ( c ) = ⊕ n ∈ N A ( c ) n is graded innon-negative degrees and generated by comatrix elements t ji , ǫ ( t ii ) = X k t ki S ( t ik ) ∈ M n> A ( c ) n , one gets a contradiction. In the trivial example given by τ = the flip map in k n , the FRT-construction yields the coordinate affine ring O (M n ) on n × n matrices over k , and one needsto localize the commutative algebra on the determinant in order to obtain the Hopf algebra O (GL n ) . In general, the abelianization of the FRT-construction gives the bialgebra O (End( c )) , ∗ Partially supported by CONICET, UBACyT † Partially supported by CONICET, ANPCyT, Secyt.2010
Mathematics Subject Classification:
Keywords:
Quantum function algebras, Nichols algebras, quantum determinants. f of V such that f ⊗ f com-mutes with c . If one could localize the FRT-construction and get a Hopf algebra H ( c ) , then onewould have a surjective map H ( c ) ։ O (Aut( c )) , that is, a quantum group much larger thanthe ”classical” automorphism group of the braiding. In general, by [Sch, Lemma 3.2.9] (see [T]for a review) in case the braiding c is rigid there exists a coquasitriangular Hopf algebra H ( c ) associated with ( V, c ) satisfying a universal property: V ∈ H ( c ) M with a certain comodulestructure map λ and if B is a coquasitriangular bialgebra such that V ∈ B M with comodulestructure λ B : V → B ⊗ V , then there exists a coquasitriangular bialgebra map f : H ( c ) → B such that λ B = ( f ⊗ id) λ . Furthermore, by [Sch, Lemma 3.2.11], the Hopf algebra H ( c ) isgenerated as algebra by elements { t ji , u ji } ≤ i,j ≤ n satisfying X k,ℓ c kℓij t rk t sℓ = X k,ℓ t ki t ℓj c rskℓ , and n X k =1 u ki t jk = δ ji = n X k =1 t ki u jk . (2)The coalgebra structure is given by ∆( t ji ) = P nk =1 t ki ⊗ t jk , ε ( t ji ) = δ ji and ∆( u ji ) = P nk =1 u jk ⊗ u ki , ε ( u ji ) = δ ji . Moreover, one has that S H ( c ) ( t ji ) = u ji for all ≤ i, j ≤ n . Note that, since H ( c ) iscoquasitriangular, the square of the antipode is an inner automorphism, and as a consequence,the antipode and all its powers are defined on the generators t ji , u ji . The comodule category H ( c ) M is the one generated by V and V ∗ , and in general the map A ( c ) → H ( c ) needs not to beinjective (see example in Subsection 3.6.1).In this paper we consider the following 3-step problem: given a finite-dimensional rigidbraided vector space ( V, c ) , ( a ) find a ”quantum determinant” for the FRT-construction A ( c ) , ( b ) prove that the localization H ( c ) = A ( c )[ D − ] of A ( c ) at the quantum determinant is aHopf algebra, ( c ) prove that H ( c ) ≃ H ( c ) .In Subsection 2.1, we introduce a method for finding a quantum determinant associated witha rigid solution of the braid equation. Two of our main results, Theorem 2.19 and Theorem2.21, give sufficient conditions to ensure the existence, and a concrete way to compute it, of agroup-like element D ∈ A = A ( c ) , such that D is normal in A and, under certain conditions,the localization on D is a Hopf algebra H ( c ) . Moreover, our proof yields an explicit formulafor the antipode. Finally, we show in Corollary 2.22 that H ( c ) is isomorphic to the universalcoquasitriangular Hopf algebra H ( c ) associated with ( V, c ) . In this way, we obtain a realizationof H ( c ) as a localization of A ( c ) .As a classical motivation of this problem one can mention the famous work of Y. Manin[M], see also [M2], where the author introduces two operations • and ◦ on quadratic algebras,interpreted as internal tensor products, and proves that the internal end ( A ) = A ! • A of aquadratic algebra A is always a bialgebra, recovering some remarkable examples such as thequantum function algebra M q (2) . The problem of finding quantum determinant is presentin this work, introducing what Manin calls a quantum grasmannian algebra (qga) in [M], ora Frobenius quantum space (Fqs) in [M2], where a ”volume form” plays a crucial role. Thedefinition of a qga, or a Fqs, assures the existence of a group-like element that is the naturalcandidate for a quantum determinant, but the problem of finding the antipode (or even toprove its existence) remains open.In [H], Hayashi constructed quantum determinants for multiparametric quantum defor-mations of O (SL n ) , O (GL n ) , O (SO n ) , O (O n ) and O (Sp n ) , inverting all group-like elements in2 given quasitriangular bialgebra, and showing that the ending result is a Hopf algebra. Todefine the quantum determinants, qga’s are considered for the deformations of the classicalexamples. The idea of considering quantum exterior algebras (qea) is also present in the work ofFiore [F], where the author defines quantum determinants for the quantum function algebras SO q ( N ) , O q ( N ) , and Sp q ( N ) , which are defined through (a quotient of) the FRT-construction,by means of the coaction of these on a volume element. This is where the quantum determi-nant comes into (co)action. More generally, qea’s and quantum determinants appear in thework of Etingof, Schedler and Soloviev [ESS] as universal objects associated with the exterioralgebra when considering set-theoretical (involutive) solutions to QYBE’s. All quantum deter-minants appearing in this way should be central. Nevertheless, we found an example that thismight not be the case, see Subsection 3.2.Motivated by the results in [ESS], the definition of the qga and the quantum exterior alge-bras, and properties of the Nichols algebra associated with a rigid solution of the braid equation,in these notes we introduce certain class of graded connected algebras extending Manin’s defi-nition of Fqs, see Definition 2.1, that enable us not only to consider volume elements and provethe existence of a quantum determinant, but also to find an explicit formula for the natural can-didate of the antipode in the FRT-construction, localized at the quantum determinant.These qga’s defined by Hayashi and the quantum exterior algebras considered by Fiore areall quadratic. In general, for a given braiding, there is no quadratic qga, but still there mightbe a finite-dimensional Nichols algebra associated with it. As a consequence, our method stillapply in this case, see example in Subsection 3.5.Quantum determinants are intensively studied in the literature as the classical problem ofdefining the determinant of a matrix with non-commutative entries, and because they also givea way to construct new examples of quantum groups, see for example [M2], [KL], [PW], [ER],[CWW], [JZ], [JoZ], [KKZ] and references therein. It is worth to mention that in the work onquantum determinants by Etingof and Retakh [ER], the existence of formulas with ”quantumminors” is considered. In our approach, the existence and concrete formulas for these ”minors”emerge clearly.Another features of the procedure to find quantum determinants are the following: givena finite-dimensional Nichols algebra B , the method provides a Hopf algebra H such that B isa braided Hopf algebra in the braided category of left H -comodules. It also gives families ofHopf algebras generated by simple subcoalgebras. Finite-dimensional quotients of these kindof Hopf algebras are of interest in the classification program of finite-dimensional complexHopf algebras by means of the generalized Lifting Method, see for example [AC], [GJG].The paper is organized as follows. In Section 1 we recall the FRT-construction and the defi-nition of the Nichols algebra associated with a braided vector space. In Section 2, we introducethe method for finding quantum determinants and ”quantum cofactor formulas”, proving ourmain results Theorems 2.19, 2.21 and Corollary 2.22. Finally, we illustrate our contribution withseveral examples, including cases where the determinant is not central, and quantum functionalgebras from Fomin-Kirillov algebras associated with the symmetric groups on three letters. Acknowledgments
We want to thank Peter Schauenburg for answering all our questions, together with the clearestexample in each case, and Chelsea Walton for many hints and suggestions. We also thank thereferee for the careful reading of our manuscript, and for the comments and suggestions thathelped us to improve the presentation. In particular, the argument used in the introduction toshow that A ( c ) is never a Hopf algebra is due to her/him.3 Preliminaries: the A B c In this section we give the definitions and basic properties of the FRT-construction and Nicholsalgebras, and recall known results that are needed for our construction.Throughout the notes, k denotes an arbitrary field. We use the standard conventions forHopf algebras and write ∆ , ε and S for the coproduct, counit and antipode, respectively. Wealso use Sweedler’s notation ∆( h ) = h (1) ⊗ h (2) for the comultiplication. Given a bialgebra A , the category of finite-dimensional left A -comodules is denoted by A M . The readers arereferred to [Ra] for further details on the basic definitions of Hopf algebras. A ( c ) In this subsection, we follow [LR]. Let ( V, c ) be a finite-dimensional braided vector space andfix { x i } ni =1 a basis of V . Write { x ∗ i } ni =1 for the basis of V ∗ dual to { x i } ni =1 . Recall that a solutionof the braid equation c is rigid , if the map c ♭ : V ∗ ⊗ V → V ⊗ V ∗ given by c ♭ ( f ⊗ x ) = P ni =1 (ev ⊗ id ⊗ id)( f ⊗ c ( x ⊗ x i ) ⊗ x ∗ i ) is invertible.Let C = End( V ) ∗ be the coalgebra linearly spanned by the matrix coefficients { t ji } ≤ i,j ≤ n .Then, V has a natural left C -comodule structure. Note that, as C ∼ = M n ( k ) ∗ ∼ = V ⊗ V ∗ thesegenerators are induced by the basis { x i } ni =1 , via the correspondence t ji ↔ x i ⊗ x ∗ j . The coalgebrastructure is given by ∆( t ji ) = n X k =1 t ki ⊗ t jk , ε ( t ji ) = δ ij for all ≤ i, j ≤ n, (3)and V is a (left) C -comodule by setting λ ( x i ) = n X j =1 t ji ⊗ x j for all ≤ i ≤ n. Write
T C for the tensor algebra of C . Extending as algebra maps the comultiplication and thecounit of C to T C , the latter becomes a bialgebra and V ⊗ V is a (left) T C -comodule. In general,a linear map c : V ⊗ V → V ⊗ V is not necessarily T C -colinear. Actually, if one consider thedifference of the two possible compositions in the following diagram, computed in the basis { x i ⊗ x j } i,j , one gets x i ⊗ x j ❴ (cid:15) (cid:15) ✩ - - V ⊗ V c / / λ (cid:15) (cid:15) V ⊗ V λ (cid:15) (cid:15) P k,ℓ c kℓij x k ⊗ x ℓ ❴ (cid:15) (cid:15) P k,ℓ t ki t ℓj ⊗ x k ⊗ x ℓ ✖ / / T C ⊗ ( V ⊗ V ) id ⊗ c / / T C ⊗ ( V ⊗ V ) P r,s,k,ℓ c kℓij t rk t sℓ ⊗ x r ⊗ x s P k,ℓ,r,s t ki t ℓj c rskℓ ⊗ x r ⊗ x s where the coefficients c kℓij are defined by the equality c ( x i ⊗ x j ) = P kl c kℓij x k ⊗ x ℓ . Hence, onearrives naturally at the following definition: Definition 1.1. [FRT] The FRT-construction (or universal quantum semigroup) for ( V, c ) is the k -algebra A = A ( c ) generated by the elements { t ji } ≤ i,j ≤ n , satisfying the following relations: P k,ℓ c kℓij t rk t sℓ = P k,ℓ t ki t ℓj c rskℓ ∀ ≤ i, j, r, s ≤ n. (4)4t is well-known that A ( c ) is a bialgebra with comultiplication and counit determined by(3), which satisfies a universal property: the map λ : V → A ( c ) ⊗ V equips V with the structureof a left comodule over A ( c ) such that the map c becomes a comodule map. If A is anotherbialgebra coacting on V via a linear map λ ′ such that c is A -colinear, then there exists a uniquebialgebra morphism f : A ( c ) → A such that λ ′ = ( f ⊗ id V ) λ . Remark 1.2.
Let V be a finite-dimensional k -vector space, c ∈ End( V ⊗ V ) and A = A ( c ) . For n ≥ , the linear map given by c k := id V ⊗ k − ⊗ c ⊗ id V n − k − : V ⊗ n → V ⊗ n is A -colinear. Thatis, the comodule map λ : V ⊗ n → A ⊗ V ⊗ n satisfies that λc k = (id A ⊗ c k ) λ for all ≤ k ≤ n − . Proof.
This follows from the fact that c is A -colinear and the category of A -comodules is tenso-rial.It is well-known that if c satisfies the braid equation, then A = A ( c ) is a coquasitriangularbialgebra, that is, there exists a convolution-invertible bilinear map r : A × A → k satisfying ( CQT r ( ab, c ) = r ( a, c (1) ) r ( b, c (2) )( CQT r ( a, bc ) = r ( a (2) , b ) r ( a (1) , c )( CQT r ( a (1) , b (1) ) a (2) b (2) = b (1) a (1) r ( a (2) , b (2) ) This map is uniquely determined by r ( t ki , t ℓj ) = c kℓji for all ≤ i, j, k, ℓ ≤ n. Remark 1.3. ( a ) The first two conditions say that for any group-like element D , the maps r ( D, − ) , r ( − , D ) : A → k are algebra maps. ( b ) The last condition can be express by the equality r ∗ m = m op ∗ r . Moreover, on a = t rj and b = t si , it reads X k,ℓ r ( t kj , t ℓi ) t rk t sℓ = X k,ℓ t ℓi t kj r ( t rk , t sℓ ) , that is, X k,ℓ c kℓji t rk t sℓ = X k,ℓ t ℓi t kj c rslk . Conditions ( CQT and ( CQT say that r is determined by the values of r on genera-tors, so it extends to the tensor algebra; ( CQT says that r descends to A . ( c ) For a group-like element D , ( CQT gives a commutation rule: r ( D, b (1) ) Db (2) = b (1) D r ( D, b (2) ) for all b ∈ A ( d ) The category A M is braided with c M,N : M ⊗ N → N ⊗ M given by c ( m ⊗ n ) = r ( m ( − , n ( − ) n (0) ⊗ m (0) for all M, N ∈ A M . A stronger result than the commutation rule above is due to Hayashi and holds for anycoquasitriangular bialgebra.
Lemma 1.4. [H, Theorem 2.2] Let A be a coquasitriangular bialgebra. For any group-like element g ∈ A , there is a bialgebra automorphism J g : A → A given by J g ( a ) = r ( a (1) , g ) a (2) r − ( a (3) , g ) suchthat ga = J g ( a ) g for all a ∈ A. xample 1.5. Let X be a set and s : X × X → X × X a set-theoretical solution of the braidequation, that is s satisfies ( s × id X )(id X × s )( s × id X ) = (id X × s )( s × id X )(id X × s ) . For x, y, a, b ∈ X , let z, t, u, v ∈ X be such that ( z, t ) = s ( x, y ) , and s ( u, v ) = ( a, b ) . Let V = k X be the k -vector space linearly spanned by the elements of X and let c be the linearization of s .Then, the set of equations for the corresponding FRT-construction on ( V, c ) is t ux t vy = t az t bt In particular, for the flip solution τ ( x, y ) = ( y, x ) on a finite set X = { x , . . . , x n } , we have that t bx t ay = t ay t bx ; in other words, A ( τ ) = O (M n ) . This is not a Hopf algebra, but if one consider theelement in A given by the usual determinant D := det n = X σ ∈ S n ( − ℓ ( σ ) t σ (1) · · · t nσ ( n ) , then the localization on D is the Hopf algebra A ( τ )[ D − ] = O (GL n ) . We will generalize thisconstruction for nontrivial examples. Remark 1.6.
Note that, since (4) is homogeneous, A ( c ) = A ( qc ) for all = q ∈ k . Also, if c isinvertible, then A ( c ) = A ( c − ) . B Let ( V, c ) be a braided vector space. The braid group B n = h τ , . . . , τ n − | τ i τ j = τ j τ i , τ i +1 τ i τ i +1 = τ i τ i +1 τ i , for ≤ i ≤ n − and j = i ± i acts on V ⊗ n via ρ n : B n → GL( V ⊗ n ) with ρ n ( τ i ) = c i = id V ⊗ i − ⊗ c ⊗ id V n − i − : V ⊗ n → V ⊗ n .Using the Matsumoto (set-theoretical) section from the symmetric group S n to B n : M : S n → B n , ( i, i + 1) τ i , for all ≤ i ≤ n − , one can define the quantum symmetrizer QS n : V ⊗ n → V ⊗ n by QS n = X σ ∈ S n ρ n ( M ( σ )) ∈ End( V ⊗ n ) . For example QS = id + c , and QS = id + c ⊗ id + id ⊗ c + (id ⊗ c )( c ⊗ id) + ( c ⊗ id)(id ⊗ c ) + ( c ⊗ id)(id ⊗ c )( c ⊗ id) . The Nichols algebra associated with ( V, c ) is the quotient of the tensor algebra T V by thehomogeneous ideal J = M n ≥ Ker QS n , or equivalently, B ( V, c ) := ⊕ n Im( QS n ) . In particular, B ( V, c ) is a graded algebra. Note that B ( V, c ) = k , B ( V, c ) = V and B ( V, c ) = ( V ⊗ V ) / (Ker(id + c )) .There are several equivalent definitions of the Nichols algebra associated with ( V, c ) , eachof them particularly useful for different purposes. For more details, see [A].6 roposition 1.7. The Nichols algebra B ( V, c ) is an A ( c ) -comodule algebra.Proof. By Remark 1.2, we have that c k is A ( c ) -colinear, which implies that QS n is an A ( c ) -comodule map. Thus, Ker QS n is an A ( c ) -subcomodule of V ⊗ n for all n ≥ . Hence, taking thequotient module J defines an A ( c ) -comodule structure on B ( V, c ) =
T V / J .Nichols algebras are a key ingredient in the classification of finite-dimensional pointedHopf algebras and there is extensive literature covering the problem of finding finite-dimen-sional Nichols algebras. If the Nichols algebra is finite-dimensional and the braiding is rigid,then special features arise. These properties guide us to make a general construction thatmotivates the definition of ”weakly graded-Frobenius algebra” that is the core of next section. In this section we introduce a method for finding a quantum determinant associated with arigid solution of the braid equation (and additional assumptions), and prove our main resultsin Theorems 2.19, 2.21 and Corollary 2.22.
The following definition extends the notion of Frobenius quantum space introduced by Maninin [M2, § loc. cit. , we use it to define quantum determinants, to establish quantumCramer and Lagrange identities, and to produce categorical dual objects. Definition 2.1.
Let A be a bialgebra and V ∈ A M . An A -comodule algebra B is called a weakly graded-Frobenius (WGF) algebra for A and V if the following conditions are satisfied:(WGF1) B is an N -graded A -comodule algebra, that is B = L n ≥ B n , λ ( B n ) ⊆ A ⊗ B n , where λ : B → A ⊗ B is the structure map, and B n · B m ⊆ B n + m for all n, m ≥ ;(WGF2) B is connected (i.e. B = k ) and B = V as A -comodules;(WGF3) dim k B < ∞ and dim k B top = 1 , where top = max { n ∈ N : B n = 0 } ;(WGF4) the multiplication induces non-degenerate bilinear maps B × B top − → B top , B top − × B → B top . Some remarks are in order: ( i ) Let A , A ′ be bialgebras and let B a WGF-algebra for A and V . If f : A ′ → A is abialgebra map, then B is also a WGF-algebra for A ′ . ( ii ) Let A be a bialgebra and ( V, c ) a braided vector space. If V ∈ A M is such that c is A -colinear and B is a WGF-algebra for A and V , then the universal property of A ( c ) determinesa unique bialgebra map f : A ( c ) → A . Consequently B is a WGF-algebra for A ( c ) and V . Inthis case, B is directly associated with the braided vector space ( V, c ) . For short, we say that B is a WGF-algebra for A ( c ) . ( iii ) A finite-dimensional graded algebra B = L n ≥ B n with B = k is called graded-Frobenius (GF) if there exists p ∈ N such that dim B p = 1 , B p + j = 0 for j > and the multiplication B j × B p − j → B p is non-degenerate for all j with ≤ j ≤ p . For instance any finite-dimensionalgraded connected Hopf algebra in the category of Yetter-Drinfeld modules over a Hopf algebra H is GF, see [N] and [AG1, § iv ) Let ( V, c ) be a finite-dimensional rigid braided vector space and let B = B ( V, c ) be theNichols algebra associated with it. If dim k B < ∞ , then by Proposition 1.7, the very definitionof Nichols algebra and ( iii ) above, it follows that B is a GF-algebra and hence a WGF-algebrafor A ( c ) . In this way, the theory of Nichols algebras provides plenty of examples that are notnecesarily quadratic, nor N -homogeneous. ( v ) One can easily give examples of WGF-algebras that are not GF by adding to a GF-algebra some elements in intermediate degrees with zero products, but these examples areartificial in the sense that they do not occur naturally from the data ( V, c ) . However, given analgebra B , it is in general a difficult task to check whether or not it is a Nichols algebra: oneshould also care about the coalgebra structure, verify that it is generated in degree one andthere are no primitive elements of degree bigger than one. But for our pourposes, the onlyproperty that we need from the algebra B is just part of the definition of graded Frobenius,and this is easy to check in examples. For this reason, we decide to extend the notion from GFto WGF, even though the only (no artificial) examples that we have are already GF. As a matterof example, concerning the Foming-Kirillov algebras, for the (known to be) finite dimensionalones, their Hilbert polynomials were known much before we knew they were Nichols algebras,see for example [FK], [MS], [AG2], [Gr] and [GGI]. ( vi ) It is known that there are plenty of examples of braided Hopf algebras in positivecharacteristic that are not Nichols algebras (e.g. they failed to have all primitive elements indegree one). To the best of our knowledge, there are no examples in characteristic zero ofgraded connected finite-dimensional braided Hopf algebras that are not Nichols algebras. ( vii ) The conditions in (WGF4) appeared in [M2], related to involutive solutions of theQYBE (thus the corresponding c is a symmetry) and in [Gu], related to Hecke-type solutions.It is known that in both cases the quantum exterior algebras are Nichols algebras, thus thisDefinition generalizes [M2, Gu].Fix a braided vector space ( V, c ) and let A = A ( c ) be the bialgebra given by the FRT-construction associated with ( V, c ) . The existence of a weakly graded-Frobenius algebra B for A allows not only to define a quantum determinant for A , but also to give an explicit for-mula for the antipode. We begin with the definition of a quantum determinant associated with B . Note that our definition is consistent with quantum (homological) determinants definedpreviously by other authors, see for example [M2], [JoZ], [KKZ], [CWW]. Definition 2.2.
Let B be a weakly graded-Frobenius algebra for A and write B top = k b forsome = b ∈ B . We call such an element a volume element for B . Since B top is an A -subcomodule, we have that the coaction on b equals λ ( b ) = D ⊗ b for some group-like element D ∈ A . We call this element D the quantum determinant in A associated with B .Note that D ∈ G ( A ) is independent of the scalar multiple of b . Example 2.3.
Consider the braiding c = − τ on an n -dimensional space V . Then A ( − τ ) = O (M n ) and B ( V, c ) = Λ V is a left O (M n ) -comodule algebra. If { x · · · , x n } is a basis of V , thenone may take b = x ∧ · · · ∧ x n ∈ Λ n V . In this case, D = X σ ∈ S n ( − ℓ ( σ ) t σ (1) · · · t nσ ( n ) is given by the usual determinant. Notation 2.4.
Let { x , . . . , x n } be a basis of V . Since by assumption the multiplication B × B top − → B top = k b is non-degenerate, there exists a basis of B top − , say { ω , . . . , ω n } ∈ B top − , such that x i ω j = δ ji b ∈ B top . ≤ i, j ≤ n , we define the elements T ji ∈ A by the equality λ ( ω i ) = X j T ij ⊗ ω j for all ≤ i ≤ n. It is easy to check that ∆( T ij ) = P nk =1 T ik ⊗ T kj and ε ( T ji ) = δ ji for all ≤ i, j ≤ n . Example 2.5.
Consider the braiding c = − τ on V ⊗ V as in Example 2.3 above. Then, theelements w j = ( − i +1 x ∧ · · · b x j · · · ∧ x n give a ”dual basis” with respect to { x , . . . , x n } andthe volume form b = x ∧ · · · ∧ x n .Next we generalize the formula when expanding a determinant by a row using minors: Proposition 2.6.
The following formula holds in A ( c ) : n X k =1 t ki T jk = δ ji D for all ≤ i, j ≤ n. (5) Proof.
Using the fact that { x i } ≤ i ≤ n and { w j } ≤ j ≤ n are dual bases with respect to the multipli-cation, that is x i ω j = δ ji b for all ≤ i, j ≤ n , by the comodule structure on B we get that λ ( δ ji b ) = δ ji D ⊗ b = λ ( x i ω j ) = λ ( x i ) λ ( ω j )= X k,ℓ t ki T jℓ ⊗ x k ω ℓ = X k,ℓ t ki T jℓ ⊗ δ ℓk b = X k t ki T jk ⊗ b . Example 2.7.
For M ∈ M n ( k ) , let Cof( M ) be the ( n × n ) -matrix whose ( i, j ) -entry is the ij -minor. For c = − τ , Proposition 2.6 is nothing else than the well-known fact M · Cof( M ) t = det( M ) I ∀ M ∈ M n ( k ) . In this subsection we prove our main theorem. We begin by introducing a Hopf algebra as-sociated with the quantum determinant and give some properties of its category of finite-dimensional left comodules. For the rest of this subsection, we fix a finite-dimensional rigidbraided vector space ( V, c ) and assume there exists a weakly graded-Frobenius algebra B for A . We write D for the quantum determinant and b for the volume element. By Lemma 1.4,we know that there exists an automorphism J := J D ∈ Aut( A ) associated with the quantumdeterminant D such that Da = J ( a ) D for all a ∈ A . Definition 2.8.
Let A be a k -algebra and D a non-zero element in A . We define the localization of A in D as a pair ( H, ι ) , where H is a k -algebra and ι : A → H is an algebra map that satisfiesthe following universal property: for any algebra map f : A → B such that f ( D ) is invertiblein B , there exists a unique algebra map ¯ f : H → B such that ¯ f ◦ ι = f ; i.e. the followingdiagram commutes A ι / / ∀ f s.t. f ( D ) invertible (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ H ∃ ! ¯ f ~ ~ ⑥ ⑥ ⑥ ⑥ B We call ι : A → H the canonical map . By the universal property above, it follows that thelocalization, if it exists, is unique up to isomorphism. We denote it as H = A [ D − ] , if noconfusion arises. 9 emark 2.9. The localization of a bialgebra with respect to a group-like element D alwaysexists in the following sense: If A is a k -algebra one can always consider the polynomial algebrain one indeterminate k [ x ] and the free product A ∗ k [ x ] . It has the universal property thatgiven a k -algebra map f : A → B and b ∈ B , then there exists a unique k -algebra map Φ : A ∗ k [ x ] such that Φ | A = f and Φ( x ) = b .Now, given an element D ∈ A , one can consider the two-sided ideal J generated by x D − and Dx − , and define A [ D − ] := ( A ∗ k [ x ]) /J . If f : A → B is an algebra map such that f ( D ) = s is invertible, then one can consider s − ∈ B and define Φ : A ∗ k [ x ] → B by Φ | A = f and Φ( x ) = s − . This map satisfy Φ( x D −
1) = 0 = Φ( Dx − , so, it induces an algebramap on the quotient. In other words, H := A ∗ k [ x ] /J satisfies the universal property.The only problem that one can face is that maybe J is not a proper ideal. If J = A ∗ k [ x ] (e.g. if D = 0 then J = h i ) then H is the zero algebra, and one has in H .For a counitary bialgebra A and a nonzero group-like element D ∈ A one has the advan-tage that ε ( D ) = 1 (by counitarity) and A ∗ k [ x ] has a unique counitary bialgebra structuredetermined by ∆ | A ⊗ A = ∆ A , ∆( x ) = x ⊗ x , ε | A = ε A and ε ( x ) = 1 . One can easily seethat ε ( x D −
1) = 0 = ε ( Dx − , so J is included in the kernel of the counit. In particular ( A ∗ k [ x ]) /J is a non-zero k -algebra. Example 2.10.
Let G = F = F ( x, y ) the free group on two elements x and y , H = k [ G ] thegroup algebra, and A ⊂ H the k subalgebra generated by x , x − and y . Taking D = y , theinclusion A → H has the universal property of the localization of A in D .In the above example we see that the localization is not necesarily a “calculus of fractions”,in the sense that not every element in H is of the form y − n a or ay − n for a ∈ A ; that is, y do notsatisfy the Ore condition in k h x ± , y i ⊂ k [ F ] . Nevertheless, our situation is much simpler: Remark 2.11.
Due tu Hayashi’s result (see Lemma 1.4), for a coquasitriangular bialgebra A anda (non-zero) group-like element D , there exists an automorphism J D : A → A such that Da = J D ( a ) D for all a ∈ A . So, the multiplicative set { D n } n ∈ N satisfies the Ore condition and everyelement in A [ D − ] can be written as D − n a (or aD − n ) for some a ∈ A , n ∈ N . In particular, thelocalization A [ D − ] as defined above coincides with the Ore-localization corresponding to themultiplicative set { D n } n ∈ N . In particular, A [ D − ] is a coquasitriangular bialgebra.We introduce now the localization of A ( c ) in the quantum determinant. Definition 2.12.
Let H ( c ) be the k -algebra generated by the elements { t ji } i,j and D − satisfyingthe relations (4) and DD − = 1 = D − D. (6)It easy to see that H ( c ) is indeed a localization of A ( c ) in D . For this reason, we write indis-tinctly H ( c ) = A [ D − ] ; the canonical map is denoted by ι : A ( c ) → H ( c ) . See Section 3 forexamples. The next result follows from [H, Theorem 3.1]. We give its proof for completeness. Lemma 2.13. H ( c ) is a coquasitriangular bialgebra.Proof. Let A ′ be the algebra generated by the same elements but satisfying only (4) and t ji D − = D − J ( t ji ) for all ≤ i, j ≤ n. Then, H ( c ) = A ′ /J , where J is the two-sided ideal generated by the relation (6). In particular,we have that ι : A → H ( c ) factorizes through A ′ . Note that since J is a bialgebra map, one hasthat aD − = D − J ( a ) for all a ∈ A . 10y defining ∆( D − ) = D − ⊗ D − and ε ( D − ) = 1 , we may endow A ′ with a bialgebrastructure: since J is a bialgebra map, one has that ∆( aD − − D − J ( a )) = a (1) D − ⊗ a (2) D − − D − J ( a (1) ) ⊗ D − J ( a (2) )= a (1) D − ⊗ a (2) D − − D − J ( a (1) ) ⊗ a (2) D − + D − J ( a (1) ) ⊗ a (2) D − − D − J ( a (1) ) ⊗ D − J ( a (2) )= (cid:0) a (1) D − − D − J ( a (1) ) (cid:1) ⊗ a (2) D − + D − J ( a (1) ) ⊗ (cid:0) a (2) D − − D − J ( a (2) ) (cid:1) , for all a ∈ A . So, ∆ is well-defined on A ′ . Also ε is well-defined since, by the explicit descrip-tion of J (see Lemma 1.4), we have ε ( aD − ) = ε ( a ) = ε ( D − J ( a )) for all a ∈ A .To show that H ( c ) is a bialgebra, it is enough to show that J is also a coideal. This followsby a direct computation since both D and D − are group-like elements. Finally, the coquasi-triangular structure is defined extending the coquasitriangular structure on A by r ( D − , a ) = r − ( D, a ) and r ( a, D − ) = r − ( a, D ) for all a ∈ A . It is well-defined thanks to (CQT1)-(CQT2). Remark 2.14.
In the category H ( c ) M , the comodule B top is invertible, that is, there exists an H ( c ) -comodule M such that B top ⊗ M ≃ k ≃ M ⊗ B top ; in particular, B top and M are one-dimensional. Indeed, consider the one-dimensional vector space k D − with generator D − and whose left H ( c ) -comodule structure is given by λ ( D − ) = D − ⊗ D − . Since the (diagonal)coaction on k b ⊗ k D − is trivial, i. e. λ ( b ⊗ D − ) = DD − ⊗ ( b ⊗ D − ) = 1 ⊗ ( b ⊗ D − ) , it turns out that k b ⊗ k D − ∼ = k as H ( c ) -comodules. Similarly k D − ⊗ B top ∼ = k . Definition 2.15.
Let V ∗ and ∗ V be the H ( c ) -comodules given by V ∗ := B top − ⊗ k D − , ∗ V := k D − ⊗ B top − Using that the multiplication m B of B gives non-degenerate colinear maps V ⊗ B top − → B top = k b , B top − ⊗ V → B top = k b , we may define evaluation maps ev ℓ : ∗ V ⊗ V → k and ev r : V ⊗ V ∗ → k by the followingcompositions. For the right evaluation: V ⊗ V ∗ ev r , , V ⊗ ( B top − ⊗ k D − ) / / k b ⊗ k D − ∼ = / / k x ⊗ ( w ⊗ D − ) ✤ / / xw ⊗ D − ev r ( x, w ⊗ D − ) b ⊗ D − ✤ / / ev r ( x, w ⊗ D − ) and similarly on the left: ∗ V ⊗ V ev ℓ , , ( k D − ⊗ B top − ) ⊗ V / / k D − ⊗ k b ∼ = / / k ( D − ⊗ w ) ⊗ x ✤ / / D − ⊗ wx ev ℓ ( D − ⊗ w, x ) b ⊗ D − ✤ / / ev ℓ ( D − ⊗ w, x ) Observe that everything depends on the choice of the “volume element” b .11 emark 2.16. By (WGF4) one may define two (possibly different) bases for B top − which areright and left dual to a given basis { x , . . . , x n } of V , say { w ℓ , . . . , w nℓ } and { w r , . . . , w nr } , satis-fying for all ≤ i, j ≤ n that x i w jℓ = δ ji b , w jr x i = δ ji b . For V ∗ = B top − ⊗ k D − and ∗ V = k D − ⊗ B top − as above, define left and right coevalu-ation maps coev ℓ : k → V ⊗ ∗ V and coev r : k → ∗ V ⊗ V by coev ℓ (1) := X i x i ⊗ ( D − ⊗ w iℓ ) , coev r (1) := X i ( D − ⊗ w ir ) ⊗ x i . By a direct computation we obtain the following:
Lemma 2.17. V ∗ and ∗ V are, respectively, right and left duals of V in H ( c ) M . Remark 2.18.
Using similar arguments as before, one has that ( V ∗ ) ∗ = k b ⊗ V ⊗ k D − and ∗ ( ∗ V ) = k D − ⊗ V ⊗ k b . In particular, V ∗∗ ≃ V ≃ ∗∗ V .The next theorem is our first main result. It states that H ( c ) is indeed a Hopf algebra,provided the canonical map ι : A ( c ) → H ( c ) is injective. This is the case when D is not a zerodivisor in A . Theorem 2.19.
If the canonical map ι : A ( c ) → H ( c ) is injective then H ( c ) M is rigid, tensoriallygenerated by V and k D − . As a consequence, H ( c ) is a coquasitriangular Hopf algebra. Moreover, theformula for the antipode is given on generators by S ( D − ) = D , and for all ≤ i, j ≤ n : S ( t ji ) := T ji D − . Proof.
Identify the elements of A with their image in H ( c ) under the canonical map. Let M ∈ H ( c ) M and fix a basis { m , . . . , m k } of M as vector space. Denote by h ki ∈ H ( c ) the elementssuch that the coaction on m i is given by λ ( m i ) = P j h ji ⊗ m j . If all h ji belong to the image of A under the canonical map, then clearly M is an A -comodule. Since D is normal, each h ji can bewritten as a polynomial in D − with coefficients in (the image of) A , say: h ji = N ij X k =0 a ijk D − k with a ij ∈ A for all ≤ i, j ≤ n. Thus, for all N ≥ N ij we have h ji D N ∈ A . Taking N = max i,j { N ij } , we have that f M = M ⊗ k b ⊗ N = M ⊗ N − times z }| { k b ⊗ · · · ⊗ k b is an A -comodule: a basis is { m i ⊗ b ⊗ b ⊗ · · · ⊗ b } ≤ i ≤ n , and the A -comodule structure is λ ( m i ⊗ b ⊗ b ⊗ · · · ⊗ b ) = X j h ji D N ⊗ m i ⊗ b ⊗ b ⊗ · · · ⊗ b ∈ A ⊗ f M Now using that k b ⊗ k D − ∼ = k we have that f M ⊗ N − times z }| { k D − · · · ⊗ k D − ∼ = M . That is, M isisomorphic to a tensor product of an A -comodule and the H ( c ) -comodule k D − . Thus H ( c ) M is tensorially generated by A ( c ) -comodules and k D − . Since A ( c ) M is tensorially generated by V , the first assertion of the statement follows. 12inally, we prove the formula for the antipode. Since D and D − are group-like, we have S ( D ) = D − and S ( D − ) = D . For the generators t ji we proceed as follows: from Proposition2.6 we know that P k t ki T jk = δ ji D . So, in H ( c ) it holds that X k t ki T jk D − = δ ji for all ≤ i, j ≤ n. Now, since H ( c ) is a Hopf algebra, we must have that P ℓ S ( t ℓi ) t kℓ = ε ( t ki ) = δ ki . From one sideone gets ( ∗ ) := X k X ℓ S ( t ℓi ) t kℓ T jk D − = X k δ ki T jk D − = T ji D − , but, changing the order of the double sum, we obtain ( ∗ ) = X ℓ X k S ( t ℓi ) t kℓ T jk D − = X ℓ S ( t ℓi ) δ jℓ = S ( t ji ) . Consequently, S ( t ji ) = T ji D − for all ≤ i, j ≤ n . Remark 2.20.
The antipode verifies both
S ∗ id = uε and id ∗S = uε , so we also have X k T ki D − t jk = δ ji . In particular, if D is central, in addition to P k t ki T jk = δ ji D , one must have P k T ki t jk = δ ji D. In thegeneral case, it holds that X k J ( T ki ) t jk = δ ji D, where J is as in Lemma 1.4. It would be interesting to have a direct proof of this fact in A ( c ) .Usually, one does not know a priori if ι : A → H ( c ) is injective, and there are exampleswhere this map actually have non-zero kernel. Also, it is difficult to check by computer if theelement D is a zero divisor or not. We give below a ”computer adapted version” of Theorem2.19, without the assumption of the canonical map ι : A → H ( c ) being injective. Theorem 2.21.
Assume the following equality holds in A ( c ) for all ≤ i, j ≤ n : n X k =1 J ( T ki ) t jk = δ ji D. (7) Then H ( c ) is a coquasitriangular Hopf algebra and the formula for the antipode on generators is givenby S ( D − ) = D , and S ( t ji ) := T ji D − for all ≤ i, j ≤ n .Proof. Define an algebra map ϕ : H ( c ) → H ( c ) by ϕ ( t ji ) = t ji and ϕ ( u ji ) = T ji D − for all ≤ i, j ≤ n . It is clear that the FRT relations (4) and (2) are the same, also ϕ (cid:0) X k t ki u jk (cid:1) = X k t ki T jk D − = δ ji DD − = δ ji . For the remaining relations, notice that for a ∈ H ( c ) , we have D − J ( a ) = aD − and so ϕ (cid:0) X k u ki t jk (cid:1) = X k T ji D − t jk = X k D − J ( T ji ) t jk = D − X k J ( T ji ) t jk = D − δ ji D = δ ji ϕ is a well-defined algebra map. Moreover, by 2.4 it follows that ϕ is indeeda bialgebra map. In particular, B is also a weakly graded-Frobenius algebra for H ( c ) and thereexists a group-like element D on H ( c ) which is mapped to D . Since H ( c ) is a Hopf algebra, D − is a group-like element whose image D − is contained in the image of ϕ . Consequently, f is surjective and H ( c ) is a Hopf algebra. By the universal property of H ( c ) , it follows that H ( c ) is also coquasitriangular.The formula for the antipode follows the same lines as in the proof of Theorem 2.19.We end this section with two corollaries. The first one states that actually both Hopf alge-bra H ( c ) and H ( c ) coincide. The second one is a resume that stresses the results for Nicholsalgebras. Corollary 2.22.
Assume that (7) holds in A ( c ) for all ≤ i, j ≤ n . Then H ( c ) and H ( c ) are isomorphicas Hopf algebras.Proof. By the proof of Theorem 2.21, we know that there is a surjective Hopf algebra map ϕ : H ( c ) → H ( c ) such that ϕ ( D ) = D , ϕ ( t ji ) = t ji and ϕ ( u ji ) = T ji D − for all ≤ i, j ≤ n ,where D is the quantum determinant of H ( c ) associated with B . To prove that both algebrascoincide, we show that ϕ is bijective. Define the algebra map f : H ( c ) → H ( c ) by f ( t ji ) = t ji and f ( D − ) = D − . Clearly, it is a well-defined bialgebra map, which satisfies that u ji = f ( T ji D − ) .Indeed, define the matrices t , T, J ( T ) , t and u by ( t ) ij = t ji , ( T ) ij = T ji , ( J ( T )) ij = J ( T ji ) , ( t ) ij = t ji and ( u ) ij = u ji . In particular, by Proposition 2.6 and our assumptions, we know that f ( t ) = t , u · t = id = t · u , t · T = DI and J ( T ) · t = DI.
Thus, f ( T ) = ( u · t ) · f ( T ) = u · ( f ( t ) · f ( T )) = u · ( f ( t · T )) = u D. Namely, f ( T ji ) = u ji D and so u ji = f ( T ji D − ) . Hence, we conclude that the algebra map f : H ( c ) → H ( c ) is surjective. Since by definition we have that f ◦ ϕ = id and ϕ ◦ f = id , theclaim is proved. Corollary 2.23.
Let ( V, c ) be a rigid finite-dimensional braided vector space such that the associatedNichols algebra B ( V, c ) is finite-dimensional. If the canonical map ι : A ( c ) → H ( c ) is injective orequation (7) is satisfied, then B ( V, c ) is a braided Hopf algebra in H ( c ) M .Proof. Follows from Proposition 1.7, the fact that ι : A ( c ) → H ( c ) is a bialgebra map and that B ( V, c ) is a braided Hopf algebra in A ( c ) M . Question 2.24.
If the canonical map ι : A → H ( c ) is injective, one can easily see that thehypothesis of Theorem 2.21 is superfluous. We do not know if it is actually superfluous ingeneral, or at least superfluous in some situation, e.g. noetherianity. In this subsection, we give an explicit formula for the quantum determinant in the case wherethe braided vector space is given by a linearization of a set-theoretical solution of the braidequation. Using the coquasitriangular structure, we also give the commuting relations be-tween the quantum determinant and the generators of the bialgebra.Let ( X, c ) be a set-theoretical solution of the braid equation and write c ( i, j ) = ( g i ( j ) , f j ( i )) for all i, j ∈ X, f, g : X → Fun(
X, X ) . We say that the solution ( X, c ) is non-degenerate if the images of f and g are bijections. Let V = k X be the vector space linearly spanned by X and consider themap on V , also written as c , obtained by linearizing c . Then ( V, c ) is a braided vector space.Set { x i } i ∈ X for the linear basis of V . A map q : X × X → k is called a cocycle if the map c q : V ⊗ V → V ⊗ V given by c q ( x i ⊗ x j ) = q ij x g i ( j ) ⊗ x f j ( i ) for all i, j ∈ X, where q ij = q ( i, j ) , is a solution of the braid equation. It turns out that the braiding c q is rigidif and only if c is non-degenerate, see [AG2, Lemma 5.7].Assume | X | = n and let ( c kℓij ) i,j,k,ℓ ∈ X be the ( n × n ) -matrix given by c q ( x i ⊗ x j ) = P k,ℓ c kℓij x k ⊗ x ℓ . By the formula above, it follows that c kℓij = q ij δ k,g i ( j ) δ ℓ,f j ( i ) . Let B ( V, c q ) be the Nichols algebra associated with the rigid braided vector space ( V, c q ) . If dim B ( V, c q ) is finite, then B ( V, c q ) = B = L Ni =0 B i with dim B N = 1 . Assume further that B N = k b ; with b a ”volume element”. Then for any element x i · · · x i N ∈ B N with x i s ∈ V ,there exists α i ,...,i N ∈ k such that x i · · · x i N = α i ,...,i N b . Proposition 2.25.
Assume b = x j · · · x j N with x j s ∈ V for all ≤ s ≤ N . Then D = X ≤ i ,...,i N ≤ n α i ,...,i N t i j t i j · · · t i N j N . Proof.
Since λ ( x i ) = P ni =1 t ji ⊗ x j for all ≤ j ≤ n , we have that λ ( b ) = λ ( x j · · · x j N ) = X ≤ i ,...,i N ≤ n t i j t i j · · · t i N j N ⊗ x i x i · · · x i N = X ≤ i ,...,i N ≤ n α i ,...,i N t i j t i j · · · t i N j N ⊗ b Since λ ( b ) = D ⊗ b , the assertion follows.Next we give some formulas concerning the quantum determinant and the coquasitrian-gular structure. Proposition 2.26.
Let ≤ a, b ≤ n and denote recursively a = f j ( a ) and a k = f j k ( a k − ) . Then r ( D, t ba ) = δ b,a N α g a ( j ) ,g a ( j )) ...,g aN − ( j N ) q j ,a q j ,f a ( j ) · · · q j N ,a N − . Proof.
From Proposition 2.25 we get r ( D, t ba ) = X ≤ i k ≤ n α i ,...,i N r ( t i j t i j · · · t i N j N , t ba )= X ≤ i k ≤ n X ≤ ℓ k ≤ n α i ,...,i N r ( t i j , t ℓ a ) r ( t i j , t ℓ ℓ ) · · · r ( t i N j N , t bℓ N − )= X ≤ i k ≤ n X ≤ ℓ k ≤ n α i ,...,i N c i ,ℓ a,j c i ,ℓ ℓ j · · · c i N bℓ N − j N = X ≤ i k ≤ n α i ,...,i N X ≤ ℓ k ≤ n q a,j δ i ,g a ( j ) δ ℓ ,f j ( a ) q ℓ ,j δ i ,g ℓ ( j ) δ ℓ ,f j ( ℓ ) · · ·· · · q ℓ N − ,j N δ i N ,g ℓN − ( j N ) δ b,f jN ( ℓ N − ) = α g a ( j ) ,g a ( j )) ...,g aN − ( j N ) q j ,a q j ,f a ( j ) · · · q j N ,a N − δ b,a N , Remark 2.27.
Assume q is a constant cocycle with q ij = q ∈ k × for all ≤ i, j ≤ n . Then r ( D, t ba ) = δ b,a N α g a ( j ) ,g a ( j )) ...,g aN − ( j N ) q N . Corollary 2.28.
Let ≤ a, b ≤ n . As before, denote a k = f j k ( a k − ) with a (0) = a , and set b k = f j k ( b k − ) with b N = b . Assume q is a constant cocycle with q ij = q ∈ k × for all ≤ i, j ≤ n . Then α g a ( j ) ,g a ( j )) ...,g aN − ( j N ) Dt ba N = α g b ( j ) ,g b ( j )) ...,g bN − ( j N ) t b a D. In particular, if α g a ( j ) ,g a ( j )) ...,g aN − ( j N ) = 0 , we have that J ( t aa N ) = α − g a ( j ) ,g a ( j )) ...,g aN − ( j N ) α g b ( j ) ,g b ( j )) ...,g bN − ( j N ) t b a . Proof.
Follows directly from Remark 1.3 (4).
Remark 2.29.
Let ( X, ⊲ ) be a rack and consider the associated set-theoretical solution of thebraid equation c ( x, y ) = ( x ⊲ y, x ) for all x, y ∈ X . Set V = k X and consider the covering group G := G ( X, c ) = h g x : x ∈ X i / ( g x g y = g x⊲y g x ) . Let q be a -cocycle and write c q ∈ Aut( V ⊗ V ) for the braiding on V . Then V is a left k G -comodule with structure map δ V : V → k G ⊗ V given by δ V ( x ) = g x ⊗ x for all x ∈ X . Moreover, c q is a k G -colinear map and, by the universal property of A ( c q ) , there exists a bialgebra map f : A ( c q ) → k G such that ( f ⊗ id) λ V = δ V . In particular, we have that g x ⊗ x = δ V ( x ) = ( f ⊗ id) λ V ( x ) = ( f ⊗ id) X y ∈ X t yx ⊗ y = X y ∈ X f ( t yx ) ⊗ y, which implies that f ( t yx ) = δ x,y g x for all x, y ∈ X . Clearly, f is a well-defined bialgebra map.As the FRT relations of A ( c q ) are given by q ij t ℓ ⊲ ki ⊲ j t ℓi = q ℓ,k t ℓi t kj for all i, j, k, ℓ ∈ X , one mightview k G , as the universal Hopf algebra of the algebra generated by the elements { g x = t xx } x ∈ X satisfying the relations t x⊲yx⊲y t xx = t xx t yy for all x, y ∈ X. It is a bialgebra with the coalgebra structure determined by t xx being group-like for all x ∈ X .Suppose that B ( V, c q ) is finite-dimensional and denote by D ∈ A ( c q ) the associated quan-tum determinant. As D satisfies (5), we must have that δ i,j f ( D ) = n X k =1 f ( t ki ) f ( T jk ) = f ( t ii ) f ( T ji ) = g i f ( T ji ) for all i, j ∈ X. Since f ( D ) , g i are group-like elements in k G , we have that f ( T ji ) = 0 if i = j and f ( T ii ) = g − i f ( D ) ∈ G for all i ∈ X . From the (quantum) geometrical point of view, one may considerthe universal map ˜ f : H ( c q ) → k G as the inclusion of the (non-commutative) subgroup ofdiagonal matrices into the quantum group associated with H ( c q ) . Compare with examples inSubsection 3.6. 16 Examples
In this section we provide examples and formulas given by our main results. Recall that givena set-theoretical solution of the braid equation s : X × X → X × X , s ( x, y ) = ( g x ( y ) , f y ( x )) for all x, y ∈ X , there is a so-called derived solution that is of rack type τ s : X × X → X × X ,that is, it is of the form τ s ( x, y ) = ( y, x ⊳ s y ) for all x, y ∈ X (see for instance [AG2, Proposition5.4] and references therein). Now for every n , one can let act the braided group B n using s or τ s . The derived solution has the remarkable property that there exists a bijection in X n intertwining these two possible actions. As a consequence, for any non-zero constant cocycle q , the dimensions of the homogeneous components Nichols algebras attached to ( X, qs ) and ( X, qτ s ) are the same. A very particular case is when s = id ; we also have τ s = id and sonecesarily τ s is the flip. Hence, for involutive set theoretical solutions ( X, s ) , one always havethat the dimension of B ( X, − s ) is finite, and equal to the dimension of the exterior algebra.Nevertheless, the FRT-construction on the braiding s is far from being trivial: Example 3.1 isthe smallest example of a non-trivial set-theoretical involutive solution, and Example 3.2 isalso comming from an involutive set-theoretical solution whose quantum determinant is notcentral. × example Let X = { , } , and consider the set-theoretical solution of the braid equation given by s : X × X → X × X given by s (1 ,
2) = (1 , , s (2 ,
1) = (2 , , s (1 ,
1) = (2 , , s (2 ,
2) = (1 , . Write ( t ji ) i,j = (cid:0) a bc d (cid:1) . Then, the FRT relations (4) are a = d , ab = cd, ba = dc, ac = bd, ca = db, b = c . The Nichols algebra B = B ( V, c ) associated with V = k x ⊕ k y and the linearization of c = − s is the k -algebra generated by x , y with relations x + y = 0 , xy = 0 = 2 yx. If char ( k ) = 2 , then dim B is finite and B has a basis { , x, y, x } . The volume element b can betaken to be b = x . Since λ ( x ) = a ⊗ x + b ⊗ y , we have that λ ( b ) = λ ( x ) = ( a ⊗ x + b ⊗ y ) = a ⊗ x + ab ⊗ xy + ba ⊗ yx + b ⊗ y = ( a − b ) ⊗ x . Thus, we obtain that D := a − b is a group-like element. One can check by hand that it iscentral (hence J = id ) and { ω = x, ω = − y } is a ”dual basis” with respect to the volumeelement b . We compute the coaction to get the values of the T ji : λ ( ω ) = λ ( x ) = a ⊗ x + b ⊗ y = a ⊗ ω − b ⊗ ω ,λ ( ω ) = λ ( − y ) = − c ⊗ x − d ⊗ y = − c ⊗ ω + d ⊗ ω . Actually, it is quite difficult to check whether D is a zero divisor or not. However, to check thecondition of Theorem 2.21 is a very easy task (one can check it directly by hand or use GAP)and conclude that H ( s ) = A [ D − ] =: GL( X, − s ) is a Hopf algebra. The antipode is given by S ( a ) = aD − , S ( b ) = − cD − , S ( c ) = − bD − , S ( d ) = dD − . Since the quantum determinant D is central, we may also consider the Hopf algebra SL( X, − s ) given by A ( s ) / ( D − . It is the algebra presented by SL( X, − s ) = k h a, b, c, d : a = d , ab = cd, ba = dc, ac = bd, ca = db, b = c , a − b = 1 i . .2 Involutive and non-central example For X = { , , } , consider the set-theoretical solution of the braid equation given by s ( i, i ) =( i, i ) for i = 1 , , , s ( i, j ) = ( j, i ) for i, j = 2 , , and s (1 ,
2) = (3 , , s (1 ,
3) = (2 , , s (2 ,
1) = (1 , , s (3 ,
1) = (1 , . Clearly, this solution is involutive. For ( t ji ) i,j = (cid:18) a b cd e fg h i (cid:19) , the FRT relations reads c = b , g = d , h = f , i = e ,ba = ac, ca = ab, da = ag, db = cg, dc = bg, ea = ai,eb = ci, ec = bi, eg = di, f a = ah, f b = ch, f c = bh,f g = dh, f i = eh, ga = ad, gb = cd, gc = bd, gh = f d, gi = ed,ha = af, hb = cf, hc = bf, hd = gf, hg = df, hi = ef, ia = ae,ib = ce, ic = be, id = ge, if = he, ig = de, ih = f e. Let V = k X and write s also for the braiding given by the linearization of s . As the set-theoretical solution is involutive, the Nichols algebra B ( V, − s ) is finite-dimensional, its maxi-mal degree is . Our construction gives the quantum determinant D = ae − af + bdf − bed − cde + cf d It is group-like, normal but not central : D commutes with a but bD = − Dc , cD = − Db , dD = − Dg , gD = − Dd , De = iD , Di = eD , Df = hD , f D = Dh . On the other hand, thesenon-commutation relations give us the formula for the automorphism J : J ( a ) = a, J ( b ) = − c, J ( d ) = − g, J ( e ) = i, J ( f ) = h. Also, it holds that J = id . One can check directly that the hypothesis of Theorem 2.21 holdsand conclude that H ( s ) =: GL( X, − s ) is a Hopf algebra. We also have the explicit formula forthe antipode: (cid:0) S ( t ji ) (cid:1) ij = − f h + ei − ce + bf ch − bi − f g + dh − cd + ae cg − aheg − di bd − af − bg + ai D − . Remark 3.1.
The relations defining the FRT-construction in this example are not very enlight-ning, however, we exhibit them the following reasons: first, to stress the fact that our con-structions are very explicit; second, to show that our methods apply to every braiding com-ming from a set theoretical involutive solution, as it has a finite-dimensional Nichols algebraattached to it. Also, even for very elementary solutions (e.g. a braiding coming from a settheoretical involutive solution on a set with 3 elements!), the Hopf algebras that arise in thisway are non-trivial, since the quantum determinants in these cases are not necessarily central.And third, the number of set-theoretical involutive solutions on a finite set X grows really fastwith respect to the cardinal of X , so, one has a big number of exotic examples.18 .3 Fomin-Kirillov algebras Before introducing quantum determinants for Fomin-Kirillov algebras, we first apply our con-struction to solutions of the braid equation given by a rack and a cocycle.A rack is a pair ( X, ⊲ ) where X is a non-empty set and ⊲ : X × X → X is a map such that x ⊲ ( y ⊲ z ) = ( x ⊲ y ) ⊲ ( x ⊲ z ) and x ⊲ is bijective for every x, y, z ∈ X . Every rack gives aset-theoretical solution of the braid equation by setting c ( i, j ) = ( i ⊲ j, i ) for all i, j ∈ X. A rack 2-cocycle q : X × X → k × , ( i, i ) q i,j is a function such that q i,j ⊲ k q j,k = q i ⊲ j,i ⊲ k q i,k for all i, j, k ∈ X. Let ( X, ⊲ ) be a rack with | X | = n and let q : X × X → k × be a cocycle. Then, one may definea braiding on the vector space V = k X by c q ( x i ⊗ x j ) = q ij x i ⊲ j ⊗ x i for all i, j ∈ X. If we write c ( x i ⊗ x j ) = P nk,ℓ =1 c k,ℓi,j x k ⊗ x ℓ , then we have that c k,ℓi,j = q ij δ i ⊲ j,k δ i,ℓ for all i, j ∈ X. In particular, the FRT-relations defining A ( c q ) have the following form q ij t ki ⊲ j t ℓi = q ℓ,ℓ ⊲ − k t ℓi t ℓ ⊲ − kj Moreover, if we replace ℓ ⊲ − k by k we get q ij t ℓ ⊲ ki ⊲ j t ℓi = q ℓ,k t ℓi t kj Let n ∈ { , , } . The Fomin-Kirillov algebras E n arise as Nichols algebras when one con-sider the solution of the braid equation associated with the racks given by the conjugacy classesof transpositions in S n and a constant cocycle, see [MS], [AG2], [GGI] for more details. We de-scribe explicitly the case when n = 3 and q ij = − for all i, j ∈ X .Let X = O S be the rack of transpositions in S and consider the constant cocycle q ij = − . Let V = k O S be the braided vector space associated with them and take the basis x = x (12) , x = x (13) and x = x (23) on V . Then c ( x i ⊗ x j ) = − x i ⊲ j ⊗ x i for all ≤ i, j ≤ . In thiscase, A ( c q ) is generated by the elements { t ji } ≤ i,j ≤ satisfying the relations t ki ⊲ j t ℓi = t ℓi t ℓ ⊲ kj forall ≤ i, j, k, ℓ ≤ . Because the cocycle is constant we may write t ℓ ⊲ ki ⊲ j t ℓi = t ℓi t kj (8)The Nichols algebra B ( O S , − associated with this rack and cocycle is finite-dimensional andit is generated by the elements x , x , x satisfying the relations x i = 0 , x x + x x + x x = 0 , x x + x x + x x = 0 , (9)for all ≤ i ≤ . It has dimension 12 and its volume element is in degree N = 4 . In our case,we may take B = k x x x x and the volume element b = x x x x . In particular, by Remark19.27, we have that r ( D, t ji ) = δ i,j for all ≤ i, j ≤ . Using relations (9) one may prove that x x x x = 0 , x x x x = − b , x x x x = 0 , x x x x = − b ,x x x x = 0 , x x x x = 0 , x x x x = b , x x x x = 0 ,x x x x = − b , x x x x = b , x x x x = 0 , x x x x = 0 ,x x x x = b , x x x x = − b , x x x x = 0 , x x x x = b ,x x x x = 0 , x x x x = 0 , x x x x = − b , x x x x = b ,x x x x = 0 , x x x x = − b , x x x x = 0 . For example, since x x + x x + x x = 0 and x i = 0 , it follows that x x x x = 0 , x x x x = 0 , x x x x = 0 , x x x x = 0 . On the other hand, x x x x + x x x x = 0 , which implies that x x x x = − x x x x = − b .Moreover, for all ≤ r ≤ we have that x r ⊲ x r ⊲ x r ⊲ x r ⊲ = x x x x . Writing ( t ji ) i,j =1 , , = (cid:18) a b cd e fg h i (cid:19) , the relations in A ( c ) read ba = ac = cb, bc = ab = ca, da = ag = gd, db = bi = id,dc = ch = hd, dg = ad = ga, dh = cd = hc, di = bd = ib,ea = ai = ie, eb = bh = he, ec = cg = ge, ed = df = f e,ef = de = f d, eg = ce = gc, eh = be = hb, ei = ae = ia,f a = ah = hf, f b = bg = gf, f c = ci = if, f g = bf = gb,f h = af = ha, f i = cf = ic, hg = gi = ih, hi = gh = ig. Thus, the quantum determinant is given by D = a e − abdf + a f − abgi + b d − abgi − abdf + b f − abdf + c d − abgi + c e = c e + c d + b f + b d − abgi − abdf + a f + a e . One can check explicitly by hand using the relations above (or using GAP and non-commutativeGr ¨obner basis) that D is a central element; in particular, the hypothesis of Theorem 2.21 holds.Thus, H ( c ) =: GL( O S , − is a Hopf algebra. The formula for the antipode follows fromconsidering dual bases in the Nichols algebra and finding the elements T ji ; this can be doneexplicitly. For example, S ( a ) = ( − f bi + f ah − ech + eai ) D − . As D is a central group-like element, one may also define the Hopf algebra SL( O S , − givenby A ( c ) / ( D − .We end this example with a question suggested by the referee. Question 3.2.
The example above relies heavily on computations. Following our construction,it is possible to describe the quantum function algebras H ( c ) = GL( O S n , − for n = 4 or n = 5 .However, our methods need computer assistance since the dimension of the Nichols algebrafor n = 4 is 576 and its top degree is 12, while for n = 5 the dimension of B is 8294400 and itstop degree is 40. It would be interesting to present H ( c ) = GL( O S n , − in a more conceptualway, since this algebra would give some insight on the Fomin-Kirillov algebras.20 .4 Quantum determinants for quantum planes In [AJG], the authors consider all solution of the QYBE in dimension and give several ex-amples of finite-dimensional Nichols algebras, arranged in families R , , R ,i ( i = 1 , , , ),and R ,i ( i = 1 , , ). They remark that, up to now, the only case known where one can finda quantum determinant and localize A ( c ) to obtain a Hopf algebra is R , , due to a result ofTakeuchi [T]. As our method only has as hypothesis dim B < ∞ , we can apply it in the othercases (and for certain parameters) to obtain quantum determinants.For example, let us consider the case R , (with k = − and pq = 1 , according to thenotation in [AJG]). The braiding associated with this two-dimensional vector space V k,p,q is (cid:0) c ( x i ⊗ x j ) (cid:1) ≤ i,j ≤ = (cid:18) − x ⊗ x kq x ⊗ x − x ⊗ x kp x ⊗ x − x ⊗ x (cid:19) . The corresponding Nichols algebra is presented as follows B ( V k,p,q ) = T ( V k,p,q ) / ( x x − kq x x , x , x ) . A PBW-basis is given by { , x , x , x x } , dim B ( V k,p,q ) = 4 and one may take the volumeelement b = x x . Write ( t ji ) i,j =1 , = (cid:0) a bc d (cid:1) . Then the FRT relations read ab = kp ba, ac = kq ca, bc = q cb, ad − da = kp bc, cd = kp dc, bd = kq db. The quantum determinant is given by D = ad − kp bc . This element is not central, it verifies aD = Da , dD = Dd , but Db = p bD and Dc = q cD . Hence J ( a ) = a, J ( d ) = d, J ( b ) = p b, J ( c ) = q c. The matrices T and J ( T ) are T = ( T ji ) = (cid:18) d kqb − kpc a (cid:19) , J ( T ) = ( J ( T ji )) = (cid:18) d kpb − kqc a (cid:19) One can easily check that t · T = D · id = J ( T ) · t , where t = ( t ji ) = (cid:0) a bc d (cid:1) . Another feature of [AJG] is the presentation by generators and relations of families of Nicholsalgebras having quadratic relations. In [X], the author presents another example over a quan-tum plane considered in loc. cit. , but with no quadratic relations. It is a finite-dimensionalNichols algebra with all relations of order bigger than . As example, we compute the quan-tum determinant for the Nichols algebra associated with the quantum plane V , .Let ξ be a primitive 6-root of unity and write ξ = − ω , with ω a primitive 3-root of 1. Let B be the k -algebra generated by x and y with relations x = 0 , y − x y − yx + xyx = 0 , y x + xy − yxy = 0 , ξx y + ξ yx + xyx = 0 . The volume element is b = x yxy , and the quantum determinant for ( t ji ) i,j = (cid:18) a bc d (cid:19) , is D = ( − ω + ω ) b dbdc + ( − ω − ω ) b dbcd + ( − ω − ω ) b dad + ( ω − ω ) b cbd + ωb cbc + ω b cadc + (2 ω + ω ) badbd + ωbadbc − badacd − ωbacbdc − ωbacbcd + ( ω + 2 ω ) abdbd − ω abdadc − abdacd − ω abcbdc + ω abcad + a dbcd + a dad . .6 Two examples of the non-injectiveness of the canonical map ι : A ( c ) → H ( c ) We thank Peter Schauenburg for providing us this example, together with an argument. Nev-ertheless, we exhibit a different argument to show that the canonical map is not injective.Consider the vector space V = k x ⊕ k y with the following braiding c ( x ⊗ x ) = x ⊗ x, c ( x ⊗ y ) = y ⊗ x, c ( y ⊗ x ) = x ⊗ y, c ( y ⊗ y ) = 2 y ⊗ y. The FRT relations of A ( c ) are ba − ab = 0 , b = 0 , bd = 0 , ca − ac = 0 , cb − bc = 0 ,c = 0 , cd = 0 , da − ad = 0 , db − bd = 0 , dc − cd = 0 , which are equivalent to ab = ba, ac = ca, ad = da, bc = cb, bd = db = cd = dc = b = c . Thus, A ( c ) = A = k [ a, b, c, d ] / ( b , c , bd, cd ) is a commutative bialgebra. Assume k is alge-braically closed of characteristic zero. Then, it is clear that A cannot inject into a Hopf algebrabecause it has nilpotent elements. Also, a direct proof (valid on any characteristic) in this par-ticular case can be given as follows: the bialgebra A is a quotient of k [ a, b, c, d ] = O (M ) , thus D := ad − bc is a group-like element in A . Notice that Λ V is not the Nichols algebra of ( V, qc ) for any q ∈ k × , because the former is finite-dimensional and the latter infinite-dimensional,but nevertheless it is a weakly graded-Frobenius algebra for A . Since b = 0 = bd , it followsthat bD = 0 and cD = 0 . Thus, D is a zero divisor. If there is a bialgebra map f : A → K with K a Hopf algebra, then f ( D ) is invertible and so f ( b ) = 0 .Also, being A commutative, we have that J = id , and the left inverse of a matrix withcommuting entries is the same as a right inverse, so hypothesis of Theorem 2.21 are fullfilled.Hence, H ( c ) is a Hopf algebra. In this concrete example, we see clearly that b and c are killedwhen inverting D , and we get the isomorphism H ( c ) ∼ = A ( b = c = 0) [( ad − bc ) − ] = k [ a, d ][( ad ) − ] = k [ a ± , d ± ] , with ∆( a ) = a ⊗ a and ∆( d ) = d ⊗ d , so H ( c ) ∼ = k [ Z × Z ] . We end the paper with another example that the canonical map is not necessarily injective.From our calculations one obtains another proof, under certain hypothesis on the braiding,of the very well-known fact that a Nichols algebra associated with a quantum linear space ofdimension n is realizable as a braided Hopf algebra in the category of k [ Z n ] -comodules.Let V be a finite-dimensional vector space with basis { x i } ≤ i ≤ n and consider the following diagonal braiding on it: c ( x i ⊗ x j ) = q ij x j ⊗ x i for all ≤ i, j ≤ n, where q ij ∈ k × satisfy that q ij q ji = 1 for i = j and q ii are primitive N i -roots of 1, with N i ∈ N and N i > . Then the Nichols algebra B ( V, c ) has generators x , . . . , x n and relations x i x j = q ij x j x i , x N i i = 0 .
22n particular, dim B ( V, c ) = Q ni =1 N i , and a volume element is given by b = x N − · · · x N n − n .Notice that all N i may be different, and not necessarily equal to 2. That is, this is a family ofnon-quadratic, non-homogeneous algebras.Recall that, for a braiding c of diagonal type, the FRT relations of A ( c ) are given by q kℓ t ki t ℓj = q ij t ℓj t ki for all ≤ i, j ≤ n. This implies in particular that A ( c ) is non-commutative if q kℓ = q ij . Besides, it holds that q ℓk t ℓj t ki = q ji t ki t ℓj and consequently t ki t ℓj = q − kℓ q ij t ℓj t ki = q − kℓ q ij q − ℓk q ji t ki t ℓj . By our assumptions on the braiding, this is nothing else that t ki t ℓj = t ki t ℓj if i = j and k = ℓ . Onthe other hand, if i = j , we get t ki t ℓi = q − kℓ q ii q − ℓk q ii t ki t ℓi . Thus, for k = ℓ we obtain that t ki t ℓi = q ii t ki t ℓi . If moreover N i = 2 , it holds that t ki t ℓi = 0 . Similarly,we have that t ki t kj = q − kk t ki t kj for i = j and k = ℓ, ( t ki ) = q − kk q ii ( t ki ) for i = j and k = ℓ. From the considerations above we get the following lemma:
Lemma 3.3.
Under the assumptions above, for i = j we have: ( a ) If q − jj q ii = 1 then ( t ji ) = 0 . ( b ) If N k = 2 , then t ki t kj = 0 = t ik t jk . Corollary 3.4.
Assume that for i = j it holds that q − jj q ii = 1 and N k = 2 for all ≤ k ≤ n . Then ( a ) The quantum determinant is D = ( t ) N − ( t ) N − · · · ( t nn ) N n − . ( b ) t ji = 0 for all i = j , and t kk is group-like for all ≤ k ≤ n as elements in H ( c ) . ( c ) H ( c ) ∼ = k [( t ) ± , ( t ) ± , . . . , ( t nn ) ± ] ≃ k [ Z n ] . In particular, the canonical map is not injectiveif q kℓ = q ij for some ≤ i, j, k, ℓ ≤ n .Proof. ( a ) The claim follows by a direct computation. Indeed, by Lemma 3.3, for ≤ ℓ ≤ N k − one has that λ ( x ℓk ) = X i ,...,i ℓ ( t i k t i k · · · t i ℓ k ) ⊗ x i x i · · · x i ℓ = ( t kk ) ℓ ⊗ x ℓk . In particular, this implies that ( t kk ) ℓ is a group-like element for all ≤ k ≤ n and ≤ ℓ ≤ N k − .Hence, λ ( x N − · · · x N n − n ) = ( t ) N − ( t ) N − ⊗ x N − · · · x N n − n and the assertion is proved. ( b ) Since q kℓ t ki t ℓj = q ij t ℓj t ki , it follows that q ij t ii t jj = q ij t jj t ii , which implies that t ii t jj = t jj t ii forall ≤ i, j ≤ n . Thus, for all ≤ k ≤ n we may write D = ( t kk ) N k − D ′ for some D ′ ∈ A ( c ) .Since D is invertible in H ( c ) and N k > , we have that t kk is a unit in H ( c ) . On the other hand,as t ki t kj = 0 for i = j , it follows that t ki t kk = 0 for i = k , from which follows that t ki = 0 for i = k .The last claim follows from the very definition of the comultiplication in H ( c ) . ( c ) From the considerations above, we obtain H ( c ) = A ( t ji : i = j ) [ D − ] = k [ t , t , . . . , t nn ][ D − ] = k [( t ) ± , ( t ) ± , . . . , ( t nn ) ± ] ∼ = k [ Z n ] . eferences [A] N. A NDRUSKIEWITSCH , An Introduction to Nichols Algebras.
In Quantization, Ge-ometry and Noncommutative Structures in Mathematics and Physics . A. Cardona, P.Morales, H. Ocampo, S. Paycha, A. Reyes, eds., pp. 135–195, Springer (2017).[AC] N. A
NDRUSKIEWITSCH and J. C
UADRA , On the structure of (co-Frobenius) Hopf alge-bras , J. Noncommutative Geometry (2013), Issue 1, pp. 83–104.[AG1] N. A NDRUSKIEWITSCH and M. G RA ˜ NA , Braided Hopf algebras over non-abelian finitegroups , Colloquium on Operator Algebras and Quantum Groups (Vaquer´ıas, 1997),Bol. Acad. Nac. Cienc. (C ´ordoba) (1999), 45–78.[AG2] N. A NDRUSKIEWITSCH and M. G RA ˜ NA , From racks to pointed Hopf algebras , Adv. inMath. (2), 177–243 (2003).[AJG] N. A
NDRUSKIEWITSCH and J. M. J
URY G IRALDI , Nichols algebras that are quantumplanes , J. Linear and Multilinear Algebra.[CWW] A. C
HIRVASITU , C. W
ALTON and X. W
ANG , On quantum groups associated to apair of preregular forms, to appear in J. Noncommutative Geometry.[ER] P. E
TINGOF and V. R
ETAKH , Quantum determinants and quasideterminants , Asian J.Math. Vol. 3, No. 2 (1999), 345–352.[ESS] P. E
TINGOF , T. S
CHEDLER and A. S
OLOVIEV , Set-theoretical solutions to the quantumYang-Baxter equation , Duke Math. J. (1999), no. 2, 169–209.[FRT] L.D. F
ADDEEV , N.Y U . R ESHETIKHIN and L.A. T
AKHTAJAN , Quantization of Liegroups and Lie algebras , Leningrad Math. J. 1 (1990) 193.[F] G. F
IORE , Quantum groups SO q ( N ) , Sp q ( n ) have q -determinants, too , J. Phys. A (1994), no. 11, 3795–3802.[FK] S. F OMIN and A. N. K
IRILLOV , Quadratic algebras, Dunkl elements and Schubert cal-culus , Progress in Mathematics (1999), 146–182.[GJG] G. A. G
ARC ´ IA and J. M. J URY G IRALDI , On Hopf algebras over quantum subgroups ,J. Pure Appl. Algebra, Volume 223 (2019), Issue 2, 738–768.[GGI] G. A. G
ARC ´ IA and A. G ARC ´ IA I GLESIAS , Finite dimensional pointed Hopf algebrasover S . Israel J. Math. 183 (2011), 417– 444.[Gr] M. G RA ˜ NA , Zoo of finite dimensional Nichols algebras of non-abelian group type , avail-able at http://mate.dm.uba.ar/ ∼ lvendram/zoo/ .[Gu] D. G UREVICH , Algebraic aspects of the Quantum Yang-Baxter Equation , Leningrad J.Math. (1991), 801–828 .[H] T. H AYASHI , Quantum groups and quantum determinants , J. Alg. (1992), 146–165.[JL] N. J
ING and M. L IU , R -matrix realization of two-parameter quantum group U r,s ( gl n ) ,Commun. Math. Stat. (2014), no. 3-4, 211–230.[JoZ] P. J ØRGENSEN and J. J. Z
HANG , Gourmet’s Guide to Gorensteinness , Adv. Math. Vol-ume , Issue 2, (2000), 313–345. 24JZ] N. J
ING and J. Z
HANG , Multiparameter quantum Pfaffians . arXiv:1701.07458 .[KKZ] E. K IRKMAN , J. K
UZMANOVICHA and J.J. Z
HANG , Gorenstein subrings of invariantsunder Hopf algebra actions , J. Algebra (2009) 3640–3669.[KL] D. K
ROB and B. L
ECLERC , Minor identities for quasi-determinants and quantum deter-minants , Comm. Math. Phys. (1995), no. 1, 1–23.[LR] L. L
AMBE and D. R
ADFORD
Introduction to the Quantum Yang-Baxter Equationand Quantum Groups: An Algebraic Approach, Mathematics and Its ApplicationsVol. 423, Dordrecht: Kluwer Academic Publishers. xx, 293 p. (1997).[M] Y. M
ANIN , Some remarks on Koszul algebras and quantum groups , Annales de l’institutFourier, tome 37, n o 4 (1987), p. 191–205.[M2] Y. M
ANIN , Quantum groups and noncommutative geometry , Universit´e de Montr´eal,Centre de Recherches Math´ematiques, Montreal, QC, 1988.[MS] A. M
ILINSKI and H.-J. S
CHNEIDER , Pointed indecomposable Hopf algebras over Cox-eter groups.
In New trends in Hopf algebra theory (La Falda, 1999), volume 267 ofContemp. Math., pages 215236. Amer. Math. Soc., Providence, RI, 2000.[N] W. D. N
ICHOLS , Bialgebras of type one , Comm. Algebra (1978), 1521–1552[PW] B. P ARSHALL and J.P. W
ANG , Quantum linear groups , Mem. Amer. Math. Soc. (1991), no. 439, vi+157 pp.[Ra] D. E. R ADFORD , Hopf algebras , Series on Knots and Everything, 49. World ScientificPublishing Co. Pte. Ltd., Hackensack, NJ, 2012.[Sch] P. S
CHAUENBURG , On Coquasitriangular Hopf Algebras and the Quantum Yang-Baxter Equation, Algebra-Berichte. . M ¨unchen: R. Fischer. 76 p. (1992).[T] M. T AKEUCHI , A two-parameter quantization of GL(n) (summary), Proc. Japan Acad.Ser. A Math. Sci. (1990), 112–114.[X] R. X IONG , On Hopf algebras over the unique -dimensional Hopf algebra with-out the dual Chevalley property. Preprint: arXiv:1712.00826 . M. A. F
ARINATI
I.M.A.S. CONICET - D
EPARTAMENTO DE M ATEM ´ ATICA ,F.C.E. Y N., U
NIVERSIDAD DE B UENOS A IRES C IUDAD U NIVERSITARIA P ABELL ´ ON I(1428) C
IUDAD DE B UENOS A IRES , A
RGENTINA
E-mail address: [email protected]
G. A. G
ARC ´ IA CM A LP, D
EPARTAMENTO DE M ATEM ´ ATICA ,F ACULTAD DE C IENCIAS E XACTAS U NIVERSIDAD N ACIONAL DE L A P LATA — CONICETC. C. 172 — 1900 L A P LATA , A
RGENTINA
E-mail address: [email protected]@mate.unlp.edu.ar