aa r X i v : . [ m a t h . QA ] O c t Universal quantum (semi)groups and Hopf envelopes
Marco Andr´es Farinati * Abstract
We prove that, in case A ( c ) = the FRT construction of a braided vector space ( V, c ) ad-mits a weakly Frobenius algebra B (e.g. if the braiding is rigid and its Nichols algebra isfinite dimensional), then the Hopf envelope of A ( c ) is simply the localization of A ( c ) by asingle element called the quantum determinant associated with the weakly Frobenius alge-bra. This generalizes a result of the author together with Gast ´on A. Garc´ıa in [FG], wherethe same statement was proved, but with extra hypotheses that we now know were un-necessary. On the way, we describe a universal way of constructing a universal bialgebraattached to a finite dimensional vector space together with some algebraic structure givenby a family of maps { f i : V ⊗ n i → V ⊗ m i } . The Dubois-Violette and Launer Hopf algebraand the co-quasi triangular property of the FRT construction play a fundamental role onthe proof. Introduction
Given c : V ⊗ V → V ⊗ V a solution of the braid equation, or equivalently, R : V ⊗ V → V ⊗ V a solution of the Yang-Baxter equation, the FRT construction (Faddeev-Reshetikhin-Takhtajan) produces a coquasitraingular bialgebra A ( c ) , so that its comodule category is natu-rally braided, V is a comodule over A ( c ) , and the map c is recovered as the categorical braiding.The FRT construction gives a standard way of constructing quantum semigroups, it is a bial-gebra that is never a Hopf algebra (unless the trivial case V = 0 ), and the problem of getting aHopf algebra by inverting a quantum determinant is a classical one, for instance, this problemis present in Manin’s work [M]. In [FG] we give a partial answer, motivated by the theoryof finite dimensional Nichols algebras, we could exhibit very explicit examples generalizingquantum grassmannian algebras and other similar approaches to quantum determinants. Theadjective “explicit” in [FG] is double: we give explicit formulas for the quantum determinantand explicit formulas for the antipode. However, the main results in [FG] has hypothesis of twokind: the first main result has a theoretical assumption, that we know is not always satisfied,but that lead to a general statement, and the second main result has an ad hoc hypothesis, thatis easy to check -from a computational point of view- in concrete examples, but we couldn’tgive a general statement where that hypothesis holds. This situation is solved in this paper.We emphasis that in [FG] we give a framework that generalizes various previous situations,such as quantum grassmanian algebras (qga) or Frobenius quantum spaces (Fqs) introducedby Manin [M, M2], the quantum determinants constructed by Hayashi [H] for multiparametricquantum deformations of O (SL n ) , O (GL n ) , O (SO n ) , O ( O n ) and O (Sp n ) , the quantum exterioralgebras (qea) in the work of Fiore [F] for SO q ( N ) , Oq ( N ) , and S pq ( N ) . Also, qea’s appear inthework of Etingof, Schedler and Soloviev [ESS]. All these qga’s Fqs and qea’s, defined and * Dpto de Matem´atica FCEyN UBA - IMAS (Conicet). e-mail: [email protected]. Partially supported byUBACyT 2018-2021 “K-teor´ıa y bi´algebras en ´algebra, geometr´ıa y topolog´ıa” and PICT 2018-00858 “Aspectosalgebraicos y anal´ıticos de grupos cu´anticos”.
Section 1 we define the universal bialgebra associ-ated with a map f : V ⊗ n → V ⊗ n where V is a finite dimensional vector space. Obviousgeneralization for families of maps is also presented.In Section 2 we recall the Hopf envelope of a bialgebra and study in detail the case of theHopf algebra associated with a non-degenerate bilinear form.In
Section 3 we recall the main construction in [FG]: the definition of Weakly GradedFrobenius Algebra (WGFA) and the corresponding candidate for the explicit formula of theantipode. After that, using the coquasitriangular property of the FRT construction and theuniversal construction developed in Section 1 for a modification of the the Dubois-Violete andLauner’s Hopf algebra, the main result (Theorem 3.11) is proved:
Theorem:
If the FRT construction admits a Weakly Graded Frobenius Algebra (WGFA, see Def-inition 3.1), then its Hopf envelope is the localization with respect to a single element, the quantumdeterminant associated with the WGFA. In Section 4 we briefly comment other applications of the universal idea of the first section,and comparaison with other works.
Acknowledgements:
I wish to thank Gast ´on A. Garc´ıa for fruitful discussions on the de-velopments of this work. Research Partially supported by the projects UBACyT 2018-2021“K-teor´ıa y bi´algebras en ´algebra, geometr´ıa y topolog´ıa” and PICT 2018-00858 “Aspectos al-gebraicos y anal´ıticos de grupos cu´anticos”.
Let k be a field, V a finite dimensional vector space of dimension n , let n , n ∈ N , and f : V ⊗ n → V ⊗ n a linear map. Fix { x i } ni =1 a basis of V and consider C the coalgebra with basis { t ji } ni,j =1 andcomultiplication ∆( t ji ) = n X k =1 t ki ⊗ t jk We consider V as C -comodule via ρ ( x i ) = X j t ji ⊗ x j T C the tensor algebra on C , with bialgebra structure extending the comultiplication of C . Since V is a C -comodule, then it is a T C -comodule, but because
T C is a bialgebra, it followsthat V ⊗ ℓ is a T C -comodule for any ℓ ∈ N . The structure map is as follows:For multi-indices I, J ∈ { , , . . . , n } ℓ (i.e. I = ( i , i , · · · , i ℓ ) ) denote x I := x i ⊗ x i ⊗ · · · ⊗ x i ℓ ∈ V ⊗ ℓ t JI := t j i t j i · · · t j ℓ i ℓ ∈ T C
In this notation, the
T C -comodule structure of V ⊗ ℓ is given by ρ ( x I ) = X J ∈{ ,...,n } ℓ t JI ⊗ x J If f ( x I ) = P J f JI x J , it need not be T C -colinear, the condition ρ ( f ( x I )) = (id ⊗ f ) ρ ( x I ) isprecisely the commutativity of the following diagram x I ❴ (cid:15) (cid:15) ✫ , , V ⊗ n f / / ρ (cid:15) (cid:15) V ⊗ n ρ (cid:15) (cid:15) P J f JI x J ❴ (cid:15) (cid:15) P J t JI ⊗ x J ✓ / / T C ⊗ V ⊗ n id ⊗ f / / T C ⊗ V ⊗ n P J,K f JI t KJ ⊗ x K P J,K t JI f KJ ⊗ x K That is, f is colinear if and only if X J,K t JI f KJ ⊗ x K = X J,K f JI t KJ ⊗ x K ( ∀ I, K ) So, we define the two-sided ideal I f := (cid:10) P J ( t JI f KJ − f JI t KJ ) : ∀ I, K (cid:11) and the algebra A ( f ) := T C/ I f Theorem 1.1. A ( f ) is a bialgebra. More precisely, I f ⊆ Ker( ǫ ) , where ǫ : T C → k is the algebra map determined by δ ( t ji ) = δ ji ,and ∆ I f ⊆ I f ⊗ T C + T C ⊗ I f . In other words, I f is a bi-ideal. Proof.
Recall, for t JI = t j i · · · t j a i a , the comultiplication is given by ∆( t JI ) = ∆( t j i ) · · · ∆( t j a i a )= ( X ℓ t ℓ i ⊗ t j ℓ ) · · · ( X ℓ a t ℓ a i a ⊗ t j a ℓ a ) = X ℓ ,...,ℓ a t ℓ i · · · t ℓ a i a ⊗ t j ℓ · · · t j a ℓ a = X L ∈{ ,...,n } a t LI ⊗ t JL So, ∆ X J ( t JI f KJ − f JI t KJ ) ! = X J,L ( t LI ⊗ t JL f KJ − f JI t LJ ⊗ t KL )= X J,L ( t LI ⊗ ( t JL f KJ − f JL t KJ ) + t LI f JL ⊗ t KJ − f JI t LJ ⊗ t KL ) X J,L ( t LI ⊗ ( t JL f KJ − f JL t KJ ) + t LI f JL ⊗ t KJ − f LI t JL ⊗ t KJ )= X L t LI ⊗ ( X J t JL f KJ − f JL t KJ ) + X J ( X L t LI f JL − f LI t JL ) ⊗ t KJ Also ǫ X J t JL f KJ − f JL t KJ ! = X J δ JL f KJ − f JL δ KJ = f KL − f KL = 0 We conclude that the ideal generated by P J t JL f KJ − f JL t KJ is a coideal, contained in Ker ǫ . Remark 1.2. Variation: families of maps.
The above construction generalizes easily for fami-lies of maps. If F := { f i : V n i → V m i } i ∈ I is a family of maps, then I F := P i ∈ I I f i is a sum ofbi-ideals, so, it is a bi-ideal and A ( F ) := T C/ I F is a bialgebra Proposition 1.3.
The construction A ( F ) satisfies the following universal property: If V is a comoduleover a bialgebra A , with structure map ρ A : V → A ⊗ V , and F = { f i : V ⊗ n i → V ⊗ m i } i ı I is afamily of A -colinear maps, then there exists a unique bialgebra morphism π : A ( F ) → A such thatthe A -comodule structure on V is the one comming from A ( F ) via π , that is, the following diagram iscommutative V ρ A / / ρ $ $ ■■■■■■■■■■ A ⊗ VA ( F ) ⊗ V π ⊗ id V O O Proof.
Fix a basis x , . . . x n of V then ρ A ( x i ) = P j a ji ⊗ x j for a unique choice of elements { a ji } i,j in A . Define the map π : A ( F ) → A via t ji a ji , since all f i are A -colinear it follows that π iswell defined. Example 1.4.
Let V = k [ x ] /x considered as unital algebra. That is, we have two maps, themultiplication and the unit: m : V ⊗ V → Vu : k = V ⊗ → V Consider { , x } as a basis of k [ x ] /x = k ⊕ kx . If ρ (1) = a ⊗ b ⊗ xρ ( x ) = c ⊗ d ⊗ x the condition of u being co-linear, since ρ (1 k ) = 1 ⊗ k forces a = 1 and b = 0 . Because x = 0 , (id ⊗ m )( ρ ( x ⊗ x )) = c ⊗ m (1 ⊗
1) + cd ⊗ m (1 ⊗ x ) + dc ⊗ m ( x ⊗
1) + d ⊗ m ( x ⊗ x )= c ⊗ cd + dc ) ⊗ x we have in A ( m ) the relations c = 0 and dc = − cd . Notice the (co)matrix comultiplication ∆ a = a ⊗ a + b ⊗ c, ∆ b = a ⊗ b + b ⊗ d ∆ c = c ⊗ a + d ⊗ c, ∆ d = c ⊗ b + d ⊗ d a = 1 and b = 0 gives ∆ c = c ⊗ d ⊗ c, ∆ d = d ⊗ d That is, d is group-like and c is a skew primitive. We conclude A = A ( { m, u } ) = k { c, d } / h c , cd + dc i = k [ N ] k [ c ] /c A comodule structure over k [ N ] k [ c ] /c is precisely a d.g. structure, in this case, this con-struction gives the natural grading | | = 0 , | x | = 1 , together with the differential ∂x = 1 , ∂ . Notice that -at least if ∈ k -, the abelianization A ab := A/ ([ A, A ]) = k [ d ] = k [ N ] . Inthe “classical” setting one gets only a natural grading on k [ x ] / ( x ) (or a Torus action), but inthis non-commutative or “quantum semigroup” action, one gets the differential structure dueto the element c . Remark 1.5.
Usual classical objects that are invariant under the group of automorphisms donot need to be automatically “quantum invariant”. For example, if A is a finite dimensionalalgebra, one can consider the trace map tr : A → k given by tr( a ) = tr (cid:16) a ′ a · a ′ (cid:17) In the above example, tr(1) = 2 and tr( x ) = 0 , but (id ⊗ tr) ρ ( x ) = (id ⊗ tr)( c ⊗ d ⊗ x ) = 2 c = 0 = ρ (tr x ) That is, tr is not A ( m, u ) -colinear. Similarly, the Killing form of a finite dimensional Lie algebra ( g , [ , ]) is not necessarily colinear with respect to the algebra A ([ , ]) . Nevertheless, one canalways add the relation associated with that operation, form instance, tr is always colinearwith respect to A ( m, u, tr) , and the Killing form will be colinear using A ([ , ] , κ ) . In [DV-L], the authors define a Hopf algebra associated with a non-degenerate bilinear formin the following way:Let b : V ⊗ V → k be a non degenerate bilinear map, write b ( x i ⊗ x j ) = b ij ∈ k and denote b ij the matrix coefficients of the inverse of the matrix ( B ) ij = b ij , and as in theprevious section, { x i : i = 1 . . . , n } is a basis of V . Dubois-Violette and Launer define the k -algebra with generators { t ji : i, j = 1 , . . . , n } and relations (sum over repeated indices). b µν t µλ t νρ = b λρ (1) b µν t λµ t ρν = b λρ (2)In our setting, V is a comodule over the free bialgebra with generators t ji and V ⊗ is acomodule via (sum over repeated indices) ρ ( x i ⊗ x j ) = t ki t lj ⊗ x k ⊗ x l , Considering k as trivial comodule, the colinearity of b requires ρ ( x i ⊗ x j ) = 1 ⊗ b ( x i ⊗ x j ) = 1 ⊗ b ij ? = (id ⊗ b )( ρ ( x i ⊗ x j )) = t ki t lj ⊗ b kl A ( b ) are t ki t lj b kl = b ij (3)This is the same as equation (1).In terms of matrices, consider B ∈ M n ( k ) ⊂ M n ( A ( b )( c )) and t ∈ M n ( A ( b )) given by ( B ) ij = b ij and ( t ) ij = t ji ; the relation (3) is: t · B · t = B or equivalently B − · t · B · t = Id because B is invertible in M n ( k ) , and so it is invertible in M n ( A ( b )) . We see that t has a left inverse B − · t · B , we will show that t also have B − · t · B as right inverse. For that denote U := t · ( B − · t · B ) We want to show that U = id . We compute B − t BU and get B − t BU = B − t B t B − t B and using t · B · t = B we get B − ( t B t ) B − t B = B − BB − t B = B − t B But from B − t BU = B − t B it follows that t BU = t B and so B − t B t BU = B − t B t B Recall t B t = B , so B − t B t = id and we conclude from the above equation that BU = B which clearly implies U = Id .Notice that U = Id means t · B − · t · B = id or equivalently t · B − · t = B − and the components of this equation is precisely equation (2), so, equation (2) is redundantand Dubois-Violette and Launer Hopd algebra coincides with the bialgebra A ( b ) . In particular, A ( b ) is a Hopf algebra, the antipode is given by S ( t ji ) = b jk t lk b li We mention that the equations S ( h ) h = ǫ ( h ) and h S ( h ) = ǫ ( h ) h = t ji (and for all i, j ) mean, respectively, ( B − t B ) t = id and t ( B − t B ) = id The fact that Dubois-Violette and Launer construction gives a Hopf algebra is well-known,but for completeness we include the following:
Proof that the antipode is well-defined:
Recall the relation t ki t lj b kl = b ij ; we want to see that theopposite relation is valid for S ( t ji ) , that is S ( t lj ) S ( t ki ) b kl ? = b ij Let us denote e B ji := S ( t lj ) S ( t ki ) b kl = b la t ca b cj b kd t ed b ei b kl = δ ak t ca b cj b kd t ed b ei = t ck b cj b kd t ed b ei We have e B ji = t ck b cj b kd t ed b ei So e B ji t iu = t ck b cj b kd t ed b ei t iu = t ck b cj b kd ( t ed b ei t iu )= t ck b cj b kd b du = t ck b cj δ ku = t cu b cj = b ij t iu In matrix notation, e B t = B t t . Since t is invertible, it follows that e B ji = b ij as desired. Remark 2.1.
Other situations where the universal bialgebra is already Hopf, outside non-degenerate bilinear forms, are possible; see for instance [BD] or [CWW]. However, the Dubois-Violette and Launer’s Hopf algebra will be enough for our purpose.
It is well-known that the forgetful functor from Hopf algebras to bialgebras has a left adjoint,the general construction is due to Takeuchi [T]. In other words, if B is a bialgebra, then thereexists a Hopf algebra H ( B ) together with a bialgebra map ι B : B → H ( B ) such that every map f : B → H from B into a Hopf algebra H factors in a unique way through H ( B ) : B ι B " " ❊❊❊❊❊❊❊❊ ∀ f / / HH ( B ) ∃ ! e f < < The general construction can be complicated: if B is given by generators b , . . . , b m and rela-tions, one should add extra generators b ′ , · · · , b ′ m so that S ( b i ) = b ′ i , and relations in order toget the Antipode axiom, but also one should add S ( b i ) = S ( b ′ i ) as generator, say b ′′ i , and so on.It is not clear in general if the Hopf envelope of a finitely generated bialgebra is finitely gen-erated. In some cases, very few elements are really necessary in order to get a Hopf algebra.For example, if one knows a priori that S = id (e.g. if B is commutative or cocommutative),then the double of the original generators will be enough. A particularly simple example is B = O ( M n ( k )) , whose Hopf envelope is O ( GL n ( k )) = O ( M n ( k ))[det − ] . That is, we only adda single commuting generator D − , with the relation D − · det = det · D − = 1 .7n the framework of universal biagebra A ( F ) associated with a family of maps F , onecan also consider H ( F ) := H ( A ( F )) , that is, the Hopf envelope of A ( F ) . It will have theanalogous universal property as A ( F ) but within the Hopf algebras. It is not clear how manygenerators are really necessary to add, but very different things can happen. From this point ofview, the FRT construction (see subsection 3.1) is an opposite example of the Dubois-Violetteand Launer’s Hopf algebra: The universal bialgebra associated with a non degenerate bilinearform is already a Hopf algebra, while the FRT construction is never Hopf (unless the trivial case V = 0 ). However, the main result of this paper is to show a general circumstance where theHopf envelope of the FRT construction is the (in general non-commutative - though normal)localization with respect to a single element, that we call the quantum determinant.Notice that if H is a Hopf algebra and I a bi-ideal such that S ( I ) ⊆ I , then clearly H/I is aHopf algebra. This can be applied to the following situation:
Theorem 2.2.
Let V be a finite dimensional vector space and let us fix a linear isomorphism Φ : V → V ∗∗ (not necessary the canonical one). Consider the maps ev l : V ∗ ⊗ V → kϕ ⊗ x ϕ ( x )ev r : V ⊗ V ∗ → kx ⊗ ϕ Φ( x )( ϕ ) If W := V ⊕ V ∗ , then the universal bialgebra on W such that the decomposition W = V ⊕ V ∗ and thebilinear maps ev l y ev r are colinear, is already a Hopf algebra. We will denote it by H (ev l , ev r ) .Proof. If b : W ⊗ W → k is the bilinear map determined by b ( v, w ) = 0 = b ( φ, ψ ) , b ( φ, v ) =ev l ( φ, v ) , b ( v, φ ) = ev r ( φ, v ) , then b is clearly non-degenerate; its universal bialgebra is Hopfbecause it coincides with Dubois-Violette and Launer’s one. Let us call it H ( b ) . Let x , . . . , x n be a basis of V , x , . . . , x n its dual basis, that for covinience we will denote x n +1 , . . . , x n . Recall H ( b ) has generators t ji : i, j = 1 , . . . , n and the antipode is given by S ( t ji ) = b jk t lk b li Since b ( V, V ) = b ( V ∗ , V ∗ ) = 0 , the matrix B ∈ k n × n of b has a structure of × blocks of size n × n of the form B = (cid:18) ∗∗ (cid:19) Similar block structure for its inverse. One may write, for i, j = 1 , . . . , nS ( t n + ji ) = n X k,l =1 b n + j,k t lk b li = n X k =1 n X l = n +1 b n + j,k t lk b li = n X k,l =1 b n + j,k t n + lk b n + l,i and similarly S ( t jn + i ) = n X k,l =1 b j,k t lk b l,n + i = n X k = n +1 n X l =1 b j,k t lk b l,n + i = n X k,l =1 b j,n + k t ln + k b l,n + i If we add the condition “the decomposition W = V ⊕ V ∗ is colinear”, this is the same as requirethat the projector π V : V ⊕ V ∗ → V ⊕ V ∗ i x i , ( i = 1 , . . . , n ) x n + i i = 1 , . . . , n ) should be colinear. (Notice π V ∗ = id W − π V , so we don’t need to add the other projector to thefamily of maps). We have, for i = 1 , · · · , n : ρ ( x i ) = n X j =1 t ji ⊗ x j = n X j =1 t ji ⊗ x j + n X j =1 t n + ji ⊗ x n + j ⇒ (id ⊗ π V )( ρ ( x i )) = n X j =1 t ji ⊗ x j ρ ( x n + i ) = n X j =1 t jn + i ⊗ x j = n X j =1 t jn + i ⊗ x j + n X j =1 t n + jn + i ⊗ x n + j ⇒ (id ⊗ π V )( ρ ( x n + i )) = n X j =1 t jn + i ⊗ x j If one ask π V to be colinear then the relations needed are t n + ji = 0 = t jn + i ∀ i, j = 1 , . . . , n We see from the previous computation that the ideal generated by { t n + ji , t jn + i , i, j = 1 , . . . , n } is stable by the antipode and so, the quotient bialgebra A ( b, π V ) = H ( b ) / h t n + ji = 0 = t jn + i , i, j = 1 , . . . , n i is a Hopf algebra. Since π V : W → W is A ( b, π V ) colinear, then π V ∗ = id W − π V is colinear too,hence, V ⊂ W and V ∗ ⊂ W are subcomodules. Since b is colinear, we conclude that ev l and ev r are colinear maps too, because they can be computed using b and restrictions from W intosubcomodules, that is, using compositions with colinear inclusions.As a corollary, we can give a proof of Theorem 3.11, that is the same statement as in [FG]but almost without hypothesis. After recalling the main objects of interest here: the FRT construction and Weakly GradedFrobenius Algebras, we prove our main result. A ( c ) A braided vector space is a pair ( V, c ) , where V is k -vector space and c ∈ End( V ⊗ V ) is a solutionof the braid equation: ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ) in End( V ⊗ V ⊗ V ) , (4)In [FRT], the authors define a bialgebra associated with an R -matrix, but to have an R -matrixis equivalent to have a braiding considering c := τ ◦ R , where τ : V ⊗ V → V ⊗ V is the usual9ip. In terms of the matrix coefficients of c , the F RT construction is the k -algebra generatedby t ji with relations X k,ℓ c kℓij t rk t sℓ = X k,ℓ t ki t ℓj c rskℓ ∀ ≤ i, j, r, s ≤ n. (5)It turns out that the FRT construction is exactly A ( c ) .The fact that c is a solution of the braid equation implies the very important fact that A ( c ) is a co-quasi-triangular bialgebra. 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. (Notice the indices ij and ji in the definition of r .) In particular, the category of A ( c ) -comodulesis braided. In this subsection and the following we recall the main definitions and results of [FG]. We beginwith the definition of weakly graded Frobenius algebra, that extends the notion of Frobeniusquantum space introduced by Manin in [M2, §8.1]. The motivation is to produce quantumdeterminants together with quantum Cramer-Lagrange identities, hence a formula for the an-tipode. The paradigmatic examples are finite dimensional Nichols algebras associated withrigid solutions of the braid equation.
Definition 3.1. [FG, 2.1] Let A be a bialgebra and V ∈ A M . An algebra B is called a weaklygraded-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 , ρ ( B n ) ⊆ A ⊗ B n , where ρ : 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 . We notice that 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]. Definition 3.2.
Let B be a WGF algebra for A and write B top = k b for some = b ∈ B . Wecall such an element a volume element for B . Since B top is an A -subcomodule, ρ ( b ) = D ⊗ b forsome group-like element D ∈ A . We call this element D the quantum determinant in A associatedwith B . 10 otation 3.3. 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 . For ≤ 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 3.4. If V = k n , c = − τ on V ⊗ V , A ( c ) = O ( M n ( k )) , B = Λ V , then b = x ∧ · · · x n isthe usual volume form, the elements w j = ( − i +1 x ∧ · · · b x j · · · ∧ x n give a ”dual basis” withrespect to { x , . . . , x n } . The elements T ji are the minors of the generic matrix. The bialgebra O ( M n ( k )) is not Hopf, but its localization O ( GL ( n, k )) = O ( M n ( k ))[det − ] is a Hopf algebra.One of the main goals in [FG] was to generalize the Lagrange formula for expanding thedeterminant by rows, and hence to have a natural candidate for the antipode on the localiza-tion by quantum determinants. The general statement is: Proposition 3.5. [FG, 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. (6)We recall a result of Hayashi. Lemma 3.6. [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. In particular, D ∈ A ( c ) is a group-like element in a cqt bialgebra, so we have a bialgebraisomorphism J : A ( c ) → A ( c ) such that Da = J ( a ) D for all a ∈ A ( c ) Definition 3.7.
Let A ( c )[ D − ] be the k -algebra generated by A ( c ) and a new element D − sat-isfying the relations DD − = 1 = D − D. (7)It easy to see that A ( c )[ D − ] is indeed a (non commutative) localization of A ( c ) in D . We denoteby ι : A ( c ) → A ( c )[ D − ] the canonical map. Notice that, in virtue of Hayashi’s result, D is anormal element, so, the general non-commutative localization can be computed in terms ofleft (or right) fractions: a general element in A ( c )[ D − ] is of the form D − n a for some n ∈ N and a ∈ A ( c ) . The next result follows from [H, Theorem 3.1], see also [FG, Lemma 2.13]. Lemma 3.8. A ( c )[ D − ] is a coquasitriangular bialgebra. .3 Main result In this section• ( V, c ) is a finite dimensional braided vector space,• A ( c ) is the FRT construction,• we assume that A ( c ) admits a WGF algebra B (see Definition 3.1), denote D its associatedquantum determinant.Since D ∈ A ( c ) is a group-like element, it must be invertible in the Hopf envelope of A ( c ) .In [FG] we studied the problem of deciding if inverting D is enough in order to get a Hopfalgebra. In other words, to decide if A ( c )[ D − ] is a Hopf algebra, and moreover, to give theformula for the antipode. The first main result (see 3.3 for the notation of the T ji ’s) in [FG] is: Theorem 3.9. [FG, Theorem 2.19]) If the canonical map ι : A ( c ) → A ( c )[ D − ] is injective thenthe category of A ( c )[ D − ] -comodules is rigid, tensorially generated by V and k D − . As a conse-quence, A ( c )[ D − ] is a Hopf algebra. Moreover, the formula for the antipode is given on generatorsby S ( D − ) = D , and S ( t ji ) := T ji D − ( ∀ ≤ i, j ≤ n ) . With the notation of the automorphism J = J D given by Hiyashi’s theorem, the secondmain results in [FG] is: Theorem 3.10. [FG, 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. (8) Then A ( c )[ D − ] is a Hopf algebra and the formula for the antipode on generators is given by S ( D − ) = D , and S ( t ji ) := T ji D − for all ≤ i, j ≤ n . Now, without assuming ι : A ( c ) → A ( c )[ D − ] (there are examples with generic braidingsof diagonal type where ι is not injective, see last example in [FG]), nor asuming (8), we canprove the main result of this paper: Theorem 3.11. A ( c )[ D − ] is always a Hopf algebra.Proof. First notice that equations (6) (Proposition 3.5) and (8) are equations in A ( c ) , but in A ( c )[ D − ] one can write respectively as n X k =1 t ki T jk D − = δ ji (9)and δ ji = n X k =1 D − J ( T ki ) t jk = n X k =1 T ki D − t jk (10)We know equation (9) is true because of (6), and it means that defining S ( t ji ) = T ji D − , in case S is well-defined and antimultiplicative, it will satisfy the antipode axiom on the right. In [FG]we show that (10) implies that S is actually well-defined as antialgebra map, and that also S satisfies the antipode axiom on the left. So, it will be enough to show that equation (6) impliesequation (10). 12sing that A ( c )[ D − ] is coquasitriangular, the category of comodules is braided, and so,the following two comodules are isomorphic: V ∗ := B top − ⊗ kD − ∗ V := kD − ⊗ B top − Fix an isomorphism. Define W := ∗ V ⊕ V and b : W × W → k the bilinear map as follows: b ( v, v ′ ) = 0 = b ( φ, φ ′ ) ∀ v, v ′ ∈ V, φ, φ ′ ∈ ∗ V Define b ( φ, v ) through the multiplication m : B top − ⊗ V → B top : ∗ V ⊗ V = kD − ⊗ B top − ⊗ V id ⊗ m −→ kD − ⊗ B top = kD − ⊗ kD ∼ = k and finally b ( v, φ ) through the fixed isomorphism ∗ V ∼ = V ∗ : V ⊗ ∗ V = V ⊗ ( kD − ⊗ B top − ) ∼ = V ⊗ V ∗ = V ⊗ ( B top − ⊗ D − ) m ⊗ id −→ kD ⊗ kD − ∼ = k It is clear that b is non degenerate, so H ( b ) is a Hopf algebra. But also, because the multiplica-tion in B is A ( c ) colinear, and the isomorphism kD − ⊗ kD ∼ = k is A ( c )[ D − ] colinear, we getthat b is also A ( c )[ D − ] colinear. Using the universal property for H ( b ) we get a map H ( b ) → A ( c )[ D − ] Moreover, V ∗ and V are A ( c )[ D − ] comodules, hence the projection π V : V ⊕ V ∗ → V iscolinear, and the above epimorphism factor through H ( b ) → H (ev l , ev r ) → A ( c )[ D − ] Recall the set of generators { t ′ ji , i, j = 1 , . . . , n } of H ( b ) and H (ev l , ev r ) . Notice the subset { t ′ ji , i, j = 1 , . . . , n } maps into the generators { t ji , i, j = 1 , . . . , n } of A ( c ) .We also know from Proposition 3.5 that the following equation holds: n X k =1 t ki T jk = δ ji D for all ≤ i, j ≤ n. so, in A ( c )[ D − ] we have n X k =1 t ki T jk D − = δ ji (11)or in matrix notation t · T = id n × n where ( t ) ij = t ji and ( T ) ij = T ji D − are elements of M n ( A ( c )[ D − ]) .Now one can compute in M n ( H ( b )) the equality given by the antipode property, for i, j =1 , . . . , n : δ ji = ǫ ( t ′ ji ) = ( t ′ ji ) S (( t ′ ji ) ) = n X k =1 t ′ ki S ( t ′ jk ) = n X k,l,r =1 t ′ ki b kl t ′ rl b rj H (ev l , ev r ) , using t n + ba = 0 = t bn + a , the above sum gives = n X k =1 2 n X l,r =1 t ′ ki b kl t ′ rl b rj And because the only possible nonzero coefficients of the bilinear form are b k,n + l , or b n + k,l ( k, l = 1 , . . . , n ) this is the same as = n X k =1 2 n X l =1 n X r =1 t ′ ki b kl t ′ n + rl b n + r,j But also, in H (ev l , ev r ) we have t ′ n + rl = 0 for l, r = 1 , . . . , n , so = n X k,l,r =1 t ′ ki b k,n + l t ′ n + rn + l b n + r,j Similar equation of the left-axiom of the antipode, shows that the matrix t ′ ∈ M n ( H (ev l , ev r )) is invertible (and not only the ( t ′ ji ) ni,j =1 -matrix in M n ( H (ev l , ev r )) .Equation (11) says that the matrix t ∈ M n ( A ( c )[ D − ]) has right inverse, but because t is theimage of t ′ ∈ M n ( H (ev l , ev r )) and t ′ is invertible we conclude that t is also invertible, hence, ithas right inverse and it is equal to the left inverse. This is precisely condition (10) Z -Graded vector spaces and graded coactions Assume W = L p ∈ Z W p is a graded vector space such that W p is finite dimensional for every p ∈ Z . Let C p := End( W p ) ∗ and C gr := M p ∈ Z C p For every p , fix { x ( p )1 , . . . , x ( p )dim W p } a basis of W p , denote { t ( p ) ji } dim W p i,j =1 the corresponding basisof C p , then W is a C p -comodule by defining ρ (cid:0) x ( p ) i (cid:1) := dim W p X j =1 t ( p ) ji ⊗ x j It verifies ρ ( W p ) ⊆ C p ⊗ W p ⊆ C gr ⊗ W p , that is, it is a graded C gr -comodule. Define T ( C gr ) the tensor algebra with comultiplication extending the comultiplication of C gr . W is also a(graded) T ( C gr ) -comodule, and hence W ⊗ n is a T ( C gr ) -comodule for any n .If F = { f i : V ⊗ n i → V ⊗ m i } i ı I is a family of family of linear maps, the ideal I F can bedefined exactly in the same way, A gr ( F ) = T C gr / I F will be a bialgebra and W will be agraded A gr ( F ) -comodule. Remark 4.1. If dim V = P p ∈ Z dim V p < ∞ then C gr then one can also consider C = End k ( V ) ∗ ,and E = { e p : V → V } the family of projectors corresponding the direct summands V p (i.e. e p e q = δ p,q e q , Im ( e p ) = V p ). We have T ( C gr ) = T C/ I E . If F = { f i : V ⊗ n i → V ⊗ m i } is a familyof graded maps, then A gr ( F ) is a quotient of A ( F ) : A ( F ) ։ A ( E ∪ F ) = A gr ( F )
14n particular, if B a graded algebra, with unit u and multiplication m : B ⊗ B → B , onemay consider graded comodule structures on B , and they will be governed by A gr ( u, m ) . Example 4.2.
Let V = k [ x ] /x be considered as a unital algebra with grading | | = 0 and | x | = 1 . Every graded component is of dimension 1, so, the graded comodule structures arenecessarily of the form ρ (1) = a ⊗ ρ ( x ) = d ⊗ with a and d group-like elements. If the unit map u : k → k [ x ] /x is colinear then a = 1 ,and A gr ( u, m ) = k [ d ] ∼ = k [ N ] , one lose the differential structure (compare with Example 1.4).However, we will see that in some cases the graded comodule structures may still be veryinteresting, since they will correspond, in the quadratic case, to Manin bialgebras.An easy lemma is the following: Lemma 4.3.
Let B be a connected associative graded algebra, namely B − n = 0 for n > and B = k .Denote B = V and assume B is generated by V . That is, B = T V / ( R ) where R is homogeneous(but not necessarily concentrated in some specific degree). Then A gr ( B ) := A gr ( u, m ) ( u is the unitand m the multiplication) is generated by C := End( V ) ∗ . That is, the map T ( C ) ֒ → T ( C gr ) inducesa surjective map T ( C ) → A gr ( B ) .Proof. If { x , . . . , x n } is basis of V = B , then the elements of the form x i . . . x i p generates B p .Denote ρ ( x i ) = P j t (1) ji ⊗ x j . Since ρ : B → A gr ( B ) ⊗ B is an algebra map, ρ ( x i . . . x i p ) = X j ...,j p t (1) j i i · · · t (1) j p i p ⊗ x i · · · x i p and we see that C generates A gr ( B ) . Recall a quadratic algebra is a k -algebra of the form B = T V / ( R ) with R ⊆ V ⊗ . We willassume further that V is finite dimensional. In the seminal work [M], Manin define operations • , ◦ and ( − ) ! on quadratic algebras. He proves that given a quadratic algebra B = T V / ( R ) ,then end( B ) := B ! • B is a bialgebra, B is an end( B ) comodule-algebra, the structure map ρ : B → end( B ) ⊗ B satisfies ρ ( B p ) ⊆ end( B ) ⊗ B p ( ∀ p ) , and moreover, end( B ) is universalwith respect to those properties (see [M2, Section 6.6]). As a corollary we have: Proposition 4.4.
Let B be a finitely generated quadratic algebra: B = T V / ( R ) . If u B : k → B denotes the unit map and m B : B ⊗ B → B its multiplication, then A gr ( B ) := A gr ( u B , m B ) = B ! • B Proof.
Both algebras are generated by V ∗ ⊗ V = End k ( V ) ∗ and they share the same universalproperty. The unique isomorphism determined by the universal properties is the identity ongenerators. Remark 4.5.
In case of a quadratic algebras B = T V / ( R ) , the subspace of defining relations of A gr ( B ) = B ! • B is clear. It would be interesting to expand the class of algebras B where thedefining relations of A ( B ) (or A gr ( B ) if B is graded) can be made explicit.15 .3 N -homogeneous algebras In [Po], the author follows Manin construction for N -homogeneous algebras. That is, if A = T V /R where R ⊆ R ⊗ N for some N ≥ . Define R ⊥ ⊆ ( V ∗ ) ⊗ N ∼ = ( V ⊗ N ) ∗ the annihilator of R and A ! := T ( V ∗ ) /R ⊥ . Notice ( A ! ) ! ∼ = A . Similarly he defines the operation • as follows. Fortwo N -homogeneous algebras A = T V / ( R ) and B = T W/ ( S ) (where R ⊆ V ⊗ N and S ⊆ W ⊗ N for the same N ), he define A • B := T ( V ⊗ W ) / ( τ ( R ⊗ S )) where τ : ( V ⊗ N ) ⊗ ( W ⊗ N ) → ( V ⊗ W ) ⊗ N is defined by τ ( v ⊗ · · · ⊗ v N ⊗ w ⊗ · · · ⊗ w N ) = ( v ⊗ w ) ⊗ ( v ⊗ w ) ⊗ · · · ⊗ ( v N ⊗ w N ) ∈ ( V ⊗ W ) ⊗ N Denoting end ( A ) := A ! • A , it is a bialgebra and A is a left comodule algebra over it. Ana-logus considerations for the algebra A ! . Finally, he defines e ( A ) as the quotient of end ( A ) bythe relations of end ( A ! ) (or vice versa). We notice that in our setting, for any finitey gener-ated graded algebra A , the universal bialgebra A gr ( A ) is defined, independently of the degreeof the relations, and also works for homogeneous relations of eventualy different degrees,and the same for a pair of graded algebras ( A, A ′ ) with the same set of generators. For N -homogeneous graded algebras, we don’t say that our approach is easier or better than the onein [Po], but we say that our approach is sufficiently general and flexible to addapt perfectly tothe N -homogeneous case, and also for multigraded case, or even to the non-graded case (if thealgebras are finite dimensional). Let Q = ( Q , Q ) be a finite quiver. If Q has no oriented cycles, then kQ is a finite dimensional k -algebra and one may consider the multiplication map m : kQ ⊗ kQ → kQ and unit u : k → kQ , and consequently the universal bialgebra A ( Q ) := A ( m, u ) . But also, the path algebra kQ is naturally graded by length of paths; that is, | x i | = 0 ∀ i ∈ Q and | x α | = 1 ∀ α ∈ Q . So, even if Q happens to have cycles, if Q is finite, maybe kQ is not finite dimensional, but kQ is a locallyfinite graded vector space, generated as algebra in degree 0 and 1. Hence, the graded version A gr ( Q ) is defined, and generated by End( V ) ∗ ⊕ End( V ) ∗ where V = k [ Q ] and V = k [ Q ] are the vector spaces spanned by Q and Q respectively. Since Q and Q are sets, the vectorspaces V and V have cannonical bases { x i : i ∈ Q } , { x α : α ∈ Q } , and End( V ) ∗ ⊕ End( V ) ∗ is the coalgebra with basis { t ji : i, j ∈ Q } ∪ { t βα : α, β ∈ Q } and comultplication ∆ t ji = X k ∈ Q t ki ⊗ t jk ∆ t βα = X γ ∈ Q t γα ⊗ t βγ Remark 4.6.
If one consider the universal bialgebra associated to the graded object kQ and themultiplication map m : kQ → kQ → kQ (i.e. one ignores the unit of this algebra) then A gr ( m ) is the bialgebra generated by { t ji : i, j ∈ Q } ∪ { t βα : α, β ∈ Q } and relations t ki t kj = δ ik t ki t t ( β ) i t βα = δ i,t ( α ) t βα βα t s ( β ) j = δ j,s ( α ) t βα ( s and t are the source and target maps s, t : Q → Q ) with comultiplication induced by ∆ t ji = X k ∈ Q t ki ⊗ t jk ∆ t βα = X γ ∈ Q t γα ⊗ t βγ Proof.
It is straightforward form the relations defining kQ : x i x j = δ ij x i ∀ i, j ∈ Q x i x α = δ i,t ( α ) x α ∀ i ∈ Q , α ∈ Q x α x j = δ j,s ( α ) x α ∀ j ∈ Q , α ∈ Q and the multiplicativity of the structure map. As an illustration we show the second set ofrelations. From x i x α = δ i,t ( α ) x α applying ρ we get X j ∈ Q X β ∈ Q t ji t βα ⊗ x j x β = δ i,t ( α ) X β ∈ Q t βα ⊗ x β But x j x β = x β if j = t ( β ) and zero otherwise, so we get X β ∈ Q t t ( β ) i t βα ⊗ x β = δ i,t ( α ) X β ∈ Q t βα ⊗ x β Since { x β } β ∈ Q are l.i. the result follows.Changing notaton t ji ↔ y ij and t βα ↔ y pq , the universal bialgebra A gr ( m : kQ ⊗ → kQ ) is not the same (they consider weak bialgebras) but very similar to the ones considered in Lemma4.2 of [HWWW]. C ∗ -context In the C ∗ -algebra context, Wang define (see [W]) a C ∗ -algebra associated with a finite dimen-sional C ∗ -algebra. That definition includes that a natural state is colinear. The C ∗ -algebradefinition is more restrictive, for instance, for the C ∗ -algebra given by the group algebra C [ G ] of a finite group G , it is showed in [KSW] that the corresponding C ∗ -Hopf algebra is commu-tative, while it is a relatively easy exercise to show that the universal construction of Section1 for the algebra M ( k ) gives a non commutative nor cocommutative bialgebra. Using that C [ S ] ∼ = C × C × M ( C ) we see that our construction gives an object different from the onedefined by Wang. 17 .6 Lie/Leibniz algebras In [AM], the authors define a commutative bialgebra associated with a Lie or Leibniz algebraby studying the adjoint of the functor A h ⊗ A where A is a commutative k -algebra and h is a fixed Leibniz (or Lie) algebra. The bracket inthe current algebra is given by [ x ⊗ a, y ⊗ a ′ ] := [ x, y ] ⊗ aa ′ The Leibniz (resp. Lie) algebra A ⊗ h is called the current algebra. From the adjoint functorof the current algebra they define what they call the universal algebra of h . This (necessarycommutative) algebra turns out to be a quotient of a polynomial algebra in n variables ( n =dim h ) that in fact is a bialgebra, with a universal property among commutative bialgebrascoacting on h . Recall that a Leibniz algebra is a generalization of a Lie algebra in the sense thatthe operation is not required to be antisymmetric, but it satisfies a choice of the Jacobi identity: [ x, [ y, z ]] = [[ x, y ] , z ] − [[ x, z ] , y ] Notice that there is a permutation of the letters x, y, z , so, if A is not commutative, it is not clearhow to define a Leibniz structure on h ⊗ A , and the point of view in [AM] do not generalizesto noncommutative algebras. However, from the point of view of our universal construction(Section 1), the non-commutative universal bialgebra coacting on h is clear: just consider thebracket operation as a map [ − , − ] h : h ⊗ → h , and A ( h ) := A ([ − , − ] h ) will give the universal(in general non-commutative) bialgebra such that h is a comodule and [ − , − ] h is colinear. Theabelianization A ( h ) ab := A ( h ) / ([ A ( h ) , A ( h )]) will be of course commutative, and since the ideal generated by brackets is always a bi-ideal,this is also a bialgebra, and satisfies the same universal property of A ( h ) but among commu-tative bialgebras. We conclude that the universal commutative bialgebra constructed in [AM]is the abelianization of A ( h ) . Similar comments for the Hopf envelope H ( h ) and H ( h ) ab . How-ever, the advantage of having a noncommutative universal bialgebra/Hopf coacting on h isclear. For instance skew-derivations, (e.g. differential graded structures) are detected by non-commutative bialgebras. In a similar way to the example k [ x ] / ( x ) , the smallest non-abelianLie algebra already gives a nontrivial (non commutative, nor cocommutative) universal object: Example 4.7.
Let h be the non-commutative 2-dimensional Lie algebra h = kx ⊕ ky with anti-symmetric bracket [ x, y ] = x . Writing C as the 4-dimensional coalgebra with basis a, b, c, d , ρ ( x ) = a ⊗ x + b ⊗ y, ρ ( y ) = c ⊗ x + d ⊗ y, ∆( a ) = a ⊗ a + b ⊗ c, ∆( b ) = a ⊗ b + b ⊗ d ∆( c ) = c ⊗ a + d ⊗ c, ∆( d ) = c ⊗ b + d ⊗ d The structure map on x ⊗ y is computed using the standard diagonal structure: ρ ( x ⊗ y ) = ( a ⊗ x + b ⊗ y )( c ⊗ x + d ⊗ y ) = ac ⊗ ( x ⊗ x ) + ad ⊗ ( x ⊗ y ) + bc ⊗ ( y ⊗ x ) + bd ⊗ ( y ⊗ y ) The requirement (id ⊗ [ , ]) ρ ( x ⊗ y ) = ρ ([ x, y ]) gives ac ⊗ [ x, x ] + ad ⊗ [ x, y ] + bc ⊗ [ y, x ] + bd ⊗ [ y, y ] = ( ad − bc ) ⊗ x ρ ([ x, y ]) = ρ ( x ) = a ⊗ x + b ⊗ y that is, ( ad − bc ) = a, b or equivalently b = 0 and ad = a . Before checking the other conditions for colinearity using x ⊗ y , y ⊗ x and y ⊗ y , we see that b = 0 implies that a is group-like. In the Hopf envelope H ( h ) , a must be invertible, and ad = a forces d = 1 . It is an easy exercise that these conditionsare enough to get the bracket colinear: the bialgebra freely generated by a ± and c with ∆( a ) = a ⊗ a, ∆( c ) = c ⊗ a + 1 ⊗ c is the universal Hopf algebra, coacting as ρ ( x ) = a ⊗ xρ ( y ) = c ⊗ x + 1 ⊗ y If we don’t require d to be invertible, it i easy to check that the other conditions on the colin-earity of the bracket are ad = da = a , cd = dc . So the universal bialgebra is A ( h ) = k h a, c, d i / ( ad = da = a, cd = dc ) with comultiplication ∆( a ) = a ⊗ a, ∆( d ) = d ⊗ d, ∆( c ) = c ⊗ a + d ⊗ c and coaction ρ ( x ) = a ⊗ x, ρ ( y ) = c ⊗ x + d ⊗ y Remark 4.8. If G is a group and h is a G -graded Lie algebra, because of the antisymmetry ofthe bracket, one may assume that G is abelian. But for Leibniz algebras, grading over non-commutative groups makes perfect sense, and grading over a non-commutative group G isthe same as a coaction over the non-commutative Hopf algebra k [ G ] . References [AGV] A. A
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