Pointed Hopf actions on central simple division algebras
aa r X i v : . [ m a t h . R A ] J u l POINTED HOPF ACTIONS ON CENTRAL SIMPLE DIVISIONALGEBRAS
PAVEL ETINGOF AND CRIS NEGRON
Abstract.
We examine actions of finite-dimensional pointed Hopf algebras oncentral simple division algebras in characteristic 0. (By a Hopf action we meana Hopf module algebra structure.) In all examples considered, we show thatthe given Hopf algebra does admit a faithful action on a central simple divisionalgebra, and we construct such a division algebra. This is in contrast to earlierwork of Etingof and Walton, in which it was shown that most pointed Hopfalgebras do not admit faithful actions on fields. We consider all bosonizationsof Nichols algebras of finite Cartan type, small quantum groups, generalizedTaft algebras with non-nilpotent skew primitive generators, and an exampleof non-Cartan type. Introduction
This work is concerned with pointed Hopf actions on central simple division alge-bras, in characteristic 0. It is an open question [9, Question 1.1] whether or not anarbitrary finite-dimensional Hopf algebra can act inner faithfully on such a divisionalgebra. A conjecture of Artamonov also proposes that any finite-dimensional Hopfalgebra should act inner faithfully on the ring of fractions of a quantum torus [6,Conjecture 0.1], and it is known that the parameters appearing in such a quantumtorus cannot (all) be generic [13, Theorem 1.8].We focus here on examples, and consider exclusively pointed Hopf algebras withabelian group of grouplikes. Such algebras are well-understood via the extensivework of many authors, e.g. [16, 2, 3].
Theorem 1.1.
The following Hopf algebras admit an inner faithful Hopf action ona central simple division algebra: • Any bosonization H = B ( V ) ⋊ G of a Nichols algebra of a finite Cartantype braided vector space via an abelian group G (as defined in [2] ). • The small quantum group u q ( g ) of a semisimple Lie algebra g . • Generalized small quantum groups u ( D ) such that the space of skew primi-tives in u ( D ) generate Rep( G ) (as a tensor category), where G is the groupof grouplikes in u ( D ) . • Generalized Taft algebras T ( n, m, α ) , where m | n and α ∈ C . • The -dimensional Hopf algebra H = B ( W ) ⋊ Z / Z , where W is the -dimensional braided vector space with braiding matrix (cid:20) − i − i (cid:21) . Date : July 19, 2019.The author was supported by NSF Postdoctoral Research Fellowship DMS-1503147.
In each of the examples appearing in Theorem 1.1, an explicit central simpledivision algebra with an inner faithful action is constructed. We also consider ineach case whether the action we construct is Hopf-Galois.As mentioned in the abstract, our results contrast with those of Etingof-Walton[12, 14]. In [12] the authors show that any generalized Taft algebra T ( n, m, α ) whichadmits an inner faithful action on a field is a standard Taft algebra T ( m, m, B ( V ) ⋊ G are not directly consideredin [12, 14], this restriction on Taft actions already obstructs actions of generalbosonizations B ( V ) ⋊ G , as each pair ( g, v ) of a grouplike g ∈ G and ( g, v ∈ V generates a generalized Taft algebra in B ( V ) ⋊ G . Similarly, smallquantum groups outside of type A were shown to not act inner faithfully on fieldsin [12, 14].Our methods are based on the observation that, for H a pointed Hopf algebrawith abelian group of grouplikes G , and Q a central simple division algebra withan H -action, the skew primitives in H must act as inner skew derivations on Q (see Theorem 3.1 and Lemma 6.3 below). Hence actions of H on a given Q areparametrized by a choice of a grading by the character group of G , and a cor-responding choice of a collection of elements in Q which solve certain universalequations for (the skew primitives in) H .The universal approach to Hopf actions we have just described is discussed inmore detail, at least in the case of coradically graded H , in Section 7. Acknowledgements.
We thank Iv´an Angiono, Juan Cuadra, and Chelsea Wal-ton for helpful conversations. The first author was partially supported by the USNational Science Foundation grant DMS-1502244. The second author was sup-ported by the US National Science Foundation Postdoctoral Research FellowshipDMS-1503147.
Contents
1. Introduction 12. Preliminaries 23. Actions of generalized Taft algebras 54. Actions of graded finite Cartan type algebras 85. Actions for (generalized) quantum groups 126. Proof of Theorem 3.1 157. Coradically graded algebras and universal actions 18References 222.
Preliminaries
Conventions.
All algebras, vector spaces, etc. are over C . For a Hopf algebra H we let G ( H ) denote the group of grouplike elements. Given a Hopf algebra H and a grouplike g ∈ G ( H ) we let Prim g ( H ) denote the C -subspace of ( g, H ) = ⊕ g ∈ G Prim g ( H ) . Given a finite-dimensional Hopf algebra H and H -module algebra A , we say that A is H -Galois over its invariants A H if, under the corresponding H ∗ -coaction, A isan H ∗ -Galois extension of its coinvariants A H = A co H ∗ .2.2. The category
YD( G ) . We recall some standard notions, which can be foundin [2] for example. The category of Yetter-Drinfeld modules over a group G is thecategory of simultaneous left G -representations and left kG -comodules V whichsatisfy the compatibility ρ ( g · v ) = ( gv − g − ) ⊗ gv , where g ∈ G , v ∈ V , and ρ ( v ) = v − ⊗ v denotes the kG -coaction. This categoryis braided, with braiding c V,W : V ⊗ W → W ⊗ V, v ⊗ w ( v − w ) ⊗ v . We will focus mainly on Yetter-Drinfeld modules over abelian G , in which case theaction and coaction simply commute.For algebras A and B in YD( G ), we define the braided tensor product A ⊗ B asthe vector space A ⊗ B with product( a ⊗ b ) · ( a ′ ⊗ b ′ ) = (cid:0) a ( b − a ′ ) (cid:1) ⊗ (cid:0) b b ′ (cid:1) . The object A ⊗ B is another algebra in YD( G ) under the diagonal action and coac-tion. We can also define the braided opposite algebra A op , which is the vector space A with multiplication a · op b = ( a − b ) a .A Hopf algebra in YD( G ) is an algebra R in YD( G ) equipped with a coalgebrastructure such that the comultiplication ∆ R : R → R ⊗ R is a map of algebras inYD( G ). Such an R should also come equipped with an antipode S R : R → R whichis a braided anti-algebra and anti-coalgebra map satisfying S R ( r ) r = r S R ( r ) = ǫ ( r ), for each r ∈ R . Definition 2.1.
Given a Hopf algebra R in YD( G ), the bosonization of R is thesmash product algebra R ⋊ G .Any bosonization R ⋊ G is well-known to be a Hopf algebra with unique Hopfstructure (∆ , ǫ, S ) such that k [ G ] is a Hopf subalgebra, and on R ⊂ R ⋊ G we have∆( r ) = r ( r ) − ⊗ ( r ) , ǫ ( r ) = ǫ R ( r ) , S ( r ) = S k [ G ] ( r − ) S R ( r ) . The bosonization operation is also referred to as the Radford biproduct in theliterature.
Lemma 2.2.
Let A be an algebra in YD( G ) . Suppose R acts on A in such a waythat the action map R ⊗ A → A is a morphism in YD( G ) and r · ( ab ) = (cid:0) r ( r ) − a (cid:1)(cid:0) ( r ) b (cid:1) for r ∈ R , a, b ∈ A . Then A is a module algebra over the bosonization R ⋊ G , where G acts on A via the Yetter-Drinfeld structure and the R -action is unchanged.Proof. This is immediate from the definition of the comultiplication on the bosoniza-tion. (cid:3)
PAVEL ETINGOF AND CRIS NEGRON
Hopf actions on division algebras.
Recall that for a domain A which isfinite over its center, we have the division algebra Frac( A ), which one can constructas the localization via the center Frac( A ) = Frac( Z ( A )) ⊗ Z ( A ) A . Theorem 2.3 ([24, Theorem 2.2]) . Suppose a Hopf algebra H acts on a domain A which is finite over its center. Then there is a unique extension of this H -actionto an action on the fraction division algebra Frac( A ) . Remark 2.4.
The result from [24] is significantly more general than what we havewritten here. They show that an H -action extends to Frac( A ), essentially, whenevera reasonable algebra of fractions exists for A (with no reference to the center).When considering actions on division algebras, one can assess the Hopf-Galoisproperty for the extension Q H → Q via a rank calculation. Theorem 2.5 ([8, Theorem 3.3]) . Suppose a finite-dimensional Hopf algebra H actson a division algebra Q . Then Q is H -Galois over Q H if and only if rank Q H Q =dim H . Faithfulness of pointed Hopf actions.
Recall that Prim g ( H ) denotes thesubspace of ( g, H , for g an arbitrary grouplike.Take Prim g ( H ) ′ to be the sum of all the nontrivial eigenspaces for Prim g ( H ) underthe adjoint action of g .For finite-dimensional pointed H , we have that the nilpotence order of any g -eigenvector x in the degree 1 portion Prim g (gr H ) is less than or equal to the orderof the associated eigenvalue. So we see that the mapPrim g ( H ) ′ → Prim g ( H ) / C (1 − g ) = Prim g (gr H ) is an isomorphism. Now by the Taft-Wilson decomposition of the first portion ofthe coradical filtration F H [25], we have F H = C [ G ] ⊕ M g,h ∈ G h · Prim g ( H ) ′ , (1)where G = G ( H ). Lemma 2.6.
Let H be a finite-dimensional pointed Hopf algebra, and A be an H -module algebra. Suppose that the G ( H ) action on A is faithful, and that for each g ∈ G ( H ) the map Prim g ( H ) ′ → End k ( A ) is injective. Then the H -action on A isinner faithful.Proof. Take G = G ( H ). Suppose we have a factorization H → K → End k ( A ),where π : H → K is a Hopf projection. By considering the dual inclusion K ∗ → H ∗ we find that K is pointed as well. By faithfulness of the G -action we have that π | G is injective. Furthermore, each π | Prim g ( H ) ′ is injective by hypothesis, and eachPrim g ( H ) ′ maps to Prim g ( K ) ′ . By the decomposition (1), where we replace H with K , we find that the restriction F H → F K is injective. It follows that π isinjective [17, Prop. 2.4.2], and therefore an isomorphism. (cid:3) In the case in which the group of grouplikes G = G ( H ) is abelian, the entiregroup G acts on each Prim g ( H ), and we can decompose the sum of the primitivespaces Prim( H ) as C H ⊕ Prim( H ) = C [ G ] ⊕ Prim( H ) ′ , where Prim( H ) ′ is the sum of the nontrivial eigenspaces. Corollary 2.7.
Suppose H is finite-dimensional and pointed, with abelian group ofgrouplikes. Then an action of H on an algebra A is inner faithful provided G ( H ) acts faithfully on A and the restriction of the representation H → End k ( A ) to Prim( H ) ′ is injective.Proof. We have Prim( H ) ′ = ⊕ g Prim g ( H ) ′ in this case. (cid:3) Actions of generalized Taft algebras
We consider for positive integers m ≤ n , with m | n , the Hopf algebra T ( n, m, α ) = C h x, g i ( x m − α (1 − g m ) , g n − , gxg − − qx ) , where q is a primitive m -th root of 1. In the algebra T ( n, m, α ) the element g isgrouplike and x is ( g, α = 0, the division algebra we produce is the ring offractions of a quantum plane, while the division algebra we produce for T ( n, m, Generic actions of pointed Hopf algebras and Taft algebras.
Let ustake a moment to consider actions of pointed Hopf algebras in general, beforereturning to the specific case of generalized Taft algebras.We note that for a pointed Hopf algebra H each skew primitive x i determinesa Hopf embedding T ( n i , m i , α i ) → H . An action of H on an algebra A is thendetermined by an action of the group G ( H ) and compatible actions of the Hopfsubalgebras T ( n i , m i , α i ) → H . Whence we study actions of the generalized Taftalgebras T ( n, m, α ) in order to understand actions of pointed Hopf algebras moregenerally.The following result motivates most of our constructions, even when it is notexplicitly referenced. The proof is non-trivial and is given in Section 6. Theorem 3.1.
Suppose T ( n, m, α ) acts on a central simple algebra A , and fix ζ a primitive n -th root of with ζ nm = q . Let A = ⊕ ni =0 A i be the correspondingdecomposition of A into eigenspaces, so that g acts as ζ i on A i . Then there exists c ∈ A n/m such that x · a = ca − ζ | a | ac for each (homogeneous) a ∈ A . Furthermore,this element c satisfies the commutativity relation c m a − ζ m | a | ac m = α (1 − ζ m | a | ) a (2) for each homogeneous a ∈ A .Conversely, if A = ⊕ ni =0 A i is a Z /n Z -graded central simple division algebra, and c ∈ A n/m is such that c m a − ζ m | a | ac m = α ( ζ m | a | − a for each homogeneous a ∈ A ,then there is a (unique) action of the generalized Taft algebra T ( n, m, α ) on A givenby g · a = ζ | a | a and x · a = ca − ζ | a | ac which gives A the structure of a T ( n, m, α ) -module algebra. Now, for general H with abelian group of grouplikes, if H acts on a central simplealgebra A then we decompose A into character spaces A = ⊕ µ A µ for the actionof G . For each skew homogeneous ( g i , x i ∈ H , with associated PAVEL ETINGOF AND CRIS NEGRON character χ i , we have the generalized Taft subalgebra T ( n i , m i , α i ) → H . Byrestricting the action, and considering Theorem 3.1, we see that each x i acts on A as an operator x i · a = c i a − µ ( g i ) ac i , for a ∈ A µ , for an element c i ∈ A χ i . Whence the action of H is determined by a choice of a G ∨ -grading on A and a choice of elements c i ∈ A χ i satisfying relations (2) (as wellas all other relations for H ). We return to this topic in Sections 6 and 7.3.2. A Hopf-Galois action for generalized Taft algebras at α = 0 . Consider T ( n, m,
0) as above, with q a primitive m -th root of 1. It was shown in [12] thatthis algebra admits no inner faithful action on a field when n > m .Take K = C ( u, v ) and consider the cyclic algebra Q ( n, m ) = Q ζ ( n, m ) := K h c, w i / ( c n − u, w n − v, cw − ζwc ) , where ζ is a chosen primitive n -th root of 1 with ζ nm = q . The algebra Q ( n, m ) isa cyclic division algebra of degree n over K . Proposition 3.2.
The central simple division algebra Q ( n, m ) admits an innerfaithful T ( n, m, -action which is uniquely specified by the values g · c = qc, g · w = ζw, x · c = (1 − q ) c , x · w = 0 . Furthermore, Q ( n, m ) is T ( n, m, -Galois over its invariants Q ( n, m ) T ( n,m, .Proof. Take s = nm . The existence of the proposed inner faithful action followsby Theorem 3.1. So we need only address the Hopf-Galois property. Take T = T ( n, m,
0) and define [ c, a ] sk := ca − ( g · a ) c for arbitrary a ∈ Q ( n, m ).As for the Hopf-Galois property, we consider the basis of monomials { c i w j } n − i,j =0 for Q ( n, m ), considered as a vector space over the field K = C ( u, v ) = C ( c n , w n ).The elements c m and w n are both g -invariant andad sk ( c )( c m ) = [ c, c m ] = 0 , ad sk ( c )( w n ) = [ c, w n ] = 0 . So the degree m field extension K ( c m ) ⊂ Q ( n, m ) lies in the T -invariants. Thealgebra Q ( n, m ) is free over K ( c m ) on the left with basis { c i w j : 0 ≤ i < m, ≤ j < n } . Now, for a generic element f = X ≤ i L/K ) = h g i : 1 ≤ i ≤ s i ⋊ h σ i ∼ = ( Z /m Z ) s ⋊ Z /s Z . We consider the Ore extension L [ t ; σ ]. This algebra is a domain which is finiteover its center, and we take Q = Frac( L [ t ; σ ]) . We produce below an action of T ( n, m, 1) on Q .We first extend the automorphism g | L = Q si =1 g i : L → L , c i qc i , to anautomorphism g : Q → Q such that g ( t ) = ζt . We note that such an extension iswell-defined since ( g | L ) σ = σ ( g | L ). The automorphism g is order n , and we obtainan action of Z /n Z = G ( T ( n, m, Q . Lemma 3.3. Take Q as above, with the given Z /n Z -action. Then, at arbitrary a ∈ Q , each element c i ∈ Q satisfies c mi a − ( g m · a ) c mi = (1 − g m ) · a. Proof. Take ζ an s -th root of q as above. It suffices to provide the relation on L [ t ; σ ]. Any homogeneous element of L [ t ; σ ] may be written in the form bt r , with b ∈ L . Note that c mi − τ w /s for each i , where τ is a root of unity, and σ ( w /s ) = ζ m w /s . Note also that g m | L = id L . We therefore have τ − (( c mi − bt r − ( g m · bt r )( c mi − w /s bt r − b ( g m · t r ) w /s = w /s bt r − ζ mr bt r w /s = bt r σ r ( w /s ) − ζ mr bt r w /s = 0 . Thus ( c mi − y − ( g m · y )( c mi − 1) = 0 for all y ∈ L [ t ; σ ]. The fact that ( c mi − yy − implies that ( c mi − 1) satisfies the same relation for all a in the ring of fractions Q . We rearrange to arrive at the desired equation. (cid:3) Proposition 3.4. For any non-zero α ∈ C , there is an inner faithful T ( n, m, α ) -action on the central simple division algebra Q = Frac( L [ t ; σ ]) . This action is notHopf-Galois.Proof. We may assume α = 1. Take s = n/m , G = G ( T ( n, m, h g i , and let ζ be the give primitive n -th root of unity with ζ s = q . We provide a G -action on Q by letting g act as the above automorphism g ( c i ) = qc i , g ( t ) = ζt . If we grade Q PAVEL ETINGOF AND CRIS NEGRON as Q = ⊕ n − i =0 Q i , with g | Q i = ζ i · − , then c i ∈ Q s , and any choice c = c i providesan element which satisfies the equation c m a − ( g m · a ) c m = (1 − g m ) · a at each a ∈ Q . We therefore apply Theorem 3.1 to arrive at an explicit action of T ( n, m, 1) on Q .As for inner faithfulness, the fact that G acts faithfully on Q is clear, and thefact that ad sk ( c ) = 0 follows from the fact that ad sk ( c )( c ) = (1 − q ) c = 0. Thusthe action of T ( n, m, 1) is inner faithful by Corollary 2.7.As for the Hopf-Galois property, we consider the invariants L [ t ; σ ] G and de-compose L = ⊕ m − k =0 L ks , with g | L ks = q k · − . Then L = L [ α ], for arbitrarynonzero α ∈ L − s , and one calculates that the invariants is a polynomial ring L [ t ; σ ] G = L [ αt s ]. Now we have L [ t ; σ ] = L [ αt s ] · ( ⊕ s − j =0 Lt j ) = L [ αt s ] · { α a t b : 0 ≤ a < m, ≤ b < s } , from which one can conclude rank L [ t ; σ ] G L [ t ; σ ] = sm. Since σ is order s , we have L [ t ; σ ] G = L [ αt s ] ⊂ Z ( L [ t ; σ ]) , and ad sk ( c ) | L [ t ; σ ] G = 0. Hence the G -invariants in L [ t ; σ ] is the entire T ( n, m, Q = Frac( L [ t ; σ ]) = Frac( L [ t ; σ ] G ) ⊗ L [ t ; σ ] G L [ t ; σ ]to find that Q T = Q G = Frac( L [ t ; σ ] G ) anddim Q T Q = dim Q G Q = sm < nm = dim T ( n, m, . Hence the action is not Hopf-Galois, by Theorem 2.5. (cid:3) Actions of graded finite Cartan type algebras We consider a class of pointed Hopf algebras which generalize small quantumBorel algebras. These are pointed, coradically graded, Hopf algebras of finite Cartantype. We first recall the construction of these algebras, then provide correspondingcentral simple division algebras on which these Cartan type algebras act innerfaithfully.4.1. Cartan type algebras (following [2] ). Let V = C { x , . . . , x θ } be a braidedvector space of diagonal type, with braiding matrix [ q ij ]. Rather, the coefficients q ij are such that c V,V ( x i ⊗ x j ) = q ij x j ⊗ x i , where c V,V is the braiding on V . Weassume that the q ij are roots of unity so that V ∈ YD( G ) for a finite abelian group G . Following Andruskiewitsch and Schneider, we say V is of Cartan type if there isan integer matrix [ a ij ] such that the coefficient q ij satisfy q ij q ji = q a ij ii . (3)We always suppose a ii = 2 and 0 ≤ − a ij < ord( q ii ) for distinct indices i, j . Wesay V is of finite Cartan type if the associated Nichols algebra B ( V ) is finite-dimensional. We have the following fundamental result of Heckenberger. Theorem 4.1 ([16, Theorem 1]) . Suppose V is of Cartan type. Then the Nicholsalgebra B ( V ) is finite-dimensional if and only if the associated matrix [ a ij ] is offinite type, i.e. if and only if [ a ij ] is the Cartan matrix associated to a semisimpleLie algebra over C up to permutation of the indices. Consider V of finite Cartan type, we have the associated root system Φ, withbasis { α i } i indexed by a homogeneous basis for V . Let Γ be the associated unionof Dynkin diagrams. We decompose Φ into irreducible componentsΦ = a I ∈ π (Γ) Φ I . Throughout we assume the following two additional restrictions: • q ii is of odd order. • q ii is of order coprime to 3 when the associated component Φ I , with α i ∈ I ,is of type G .By [2, Lemma 2.3] we have that N i = ord( q ii ) is constant for all i with associatedsimple roots α i in a given component of the Dynkin diagram. For γ ∈ Φ + I we take N γ = N i for any i in component I .For finite Cartan type V and γ ∈ Φ + one has associated root vectors x α , whichare constructed via iterated braided commutators as in [1, 19]. Theorem 4.2 ([2, Theorem 5.1]) . Suppose R = B ( V ) is of Cartan type, and take N i = ord( q ii ) . Then R admits a presentation R = T V /I , where I is generated bythe relations • (Nilpotence relations) x N α γ for γ ∈ Φ + ; • ( q -Serre relations) ad sk ( x i ) − a ij ( x j ) ; Actions of finite Cartan type algebras. We call a Hopf algebra H of (finite) Cartan type if H = B ( V ) ⋊ G for V of (finite) Cartan type and G a finiteabelian group. For a G × G ∨ -homogeneous basis vectors x i ∈ V we write g i for thegroup element associated to x i , ∆ H ( x i ) = x i ⊗ g ⊗ x i , and χ i for the associatedcharacter Ad g ( x i ) = χ i ( g ) x i . Theorem 4.3. Take H = B ( V ) ⋊ G of finite Cartan type, and let [ q ij ] be thebraiding matrix for V = C { x , . . . , x θ } . Let [ a ij ] be the matrix encoding the rela-tions (3) , and suppose that the x i are ordered so that [ a ij ] is block diagonal witheach block a standard Cartan matrix associated to a Dynkin diagram. Then for anysubset Y = { µ , . . . , µ t } ⊂ G ∨ there is an H -action on the algebra A ( Y ) = C h c , . . . , c θ , w , . . . , w t i ( c i c j − q ij c j c i , c k w m − µ m ( g k ) w m c k : i < j ) and on the central simple division algebra Q ( Y ) = Frac( A ( Y )) . This action isuniquely specified by the values on the generators g · c i = χ i ( g ) c i , x j · c i = c j c i − q ji c j c i , g · w k = µ k ( g ) w k , x l · w k = 0 , and is inner faithful if and only if the subset { χ i } θi =1 ∪ Y generates G ∨ . The proof of Theorem 4.3 is given in Section 4.5. The main difficulty in producingsuch an action is showing that the proposed action does in fact satisfy the relationsof H .We note that the algebra Q ( Y ) is not H -Galois outside of type A . This followsby a rank calculation which we do not repeat here. In type A we have produceda Hopf-Galois action already in Proposition 3.2. The pre-Nichols algebra. Let G be a finite abelian group. Take V in YD( G )of finite Cartan type, and fix R = B ( V ). Consider a basis { x , . . . , x θ } for V , witheach x i homogeneous with respect to the G × G ∨ -grading. We take g i = deg G ( x i )and χ i = deg G ∨ ( x i ).Let [ q ij ] be the braiding matrix for V . We assume the orders ord( q ii ) are odd,and additionally that ord( q ii ) is coprime to 3 in type G . We recall here some workof Andruskiewitsch and Schneider. Theorem 4.4 ([2]) . For R = B ( V ) of finite Cartan type, the algebra R pre := T V / ( q - Serre relations) is a Hopf algebra in YD( G ) , with Hopf structure induced by the quotient T V → R pre . We refer to R pre as the distinguished pre-Nichols algebra associated to R , fol-lowing Angiono [4]. For H = R ⋊ G we call H pre := R pre ⋊ G the ADK form of H ,in reference to Angiono, de Concini, and Kac.As with the usual de Concini-Kac algebra, there is an action of the braid groupof R pre which gives us elements x γ = T σ ( x i ) as in [1, 19]. Theorem 4.5 ([2, Theorem 2.6]) . Let Z be the subalgebra of R pre generated bythe powers x N γ γ . The subalgebra Z is a Hopf subalgebra in R pre . For an algebra B in YD( G ) the total center Z tot ( B ) of B is the maximal subal-gebra for which the two diagrams Z ⊗ B c / / mult ❋❋❋❋❋❋❋❋❋ B ⊗ Z mult { { ①①①①①①①①① B B ⊗ Z c / / mult ❋❋❋❋❋❋❋❋❋ Z ⊗ B mult { { ①①①①①①①①① B commute. Proposition 4.6 ([2, Theorem 3.3]) . Consider Z in R pre , and take c = c R pre ,R pre . (i) The restriction of the braiding c to Z ⊗ R pre is an involution, i.e. c | Z ⊗ R pre =( c | R pre ⊗ Z ) − . (ii) The subalgebra Z is contained in the total center of R pre , Z ⊂ Z tot ( R pre ) . We note that in the case of the (classical) quantum De Concini-Kac-style Borel U DKq ( b ), the elements E N γ γ are actually central. However, in general this will notbe the case. One can view the centrality in the classical de Concini-Kac setting asa consequence of the fact that c | C E Nαα ⊗ U DKq ( b ) happens to be the trivial swap.4.4. Some technical lemmas.Lemma 4.7. The adjoint action of R pre on itself factors through the quotient R .Proof. It suffices to show that the adjoint action restricted to Z ⊂ R pre is trivial,since the kernel of the projection R pre → R is generated by the augmentation idealfor Z . For any (homogeneous) X ∈ Z and a ∈ R pre we havead sk ( X )( a ) = P i χ a ( g i ) X i aS ( X i )= P i χ a ( g i ) χ i (deg( a )) X i S ( X i ) a (Prop. 4.6 (ii))= P i χ a ( g i ) χ a ( g i ) − X i S ( X i ) a (Prop. 4.6 (i))= ( P i X i S ( X i )) a = ǫ ( X ) a, where in the above calculation g i is the G -degree of X i and χ i is the G ∨ -degree.Hence ad sk | Z factors through the counit, and the restriction of the adjoint actionto Z is trivial, as desired. (cid:3) Let us order the basis of primitives P ord = { x i } i so that the matrix [ a ij ] is blockdiagonal with each block a Cartan matrix of type A , D , E , etc. We take S ord := T V / (ad sk ( x i )( x j ) : i < j ) , s ord := T V / (ad sk ( x i )( x j ) , x N i i : i < j ) . These are both algebras in YD( G ). We let c i denote the images of the x i in S ord and/or s ord . Lemma 4.8. The projections T V → S ord and T V → s ord factor to give projections R pre → S ord and R → s ord respectively.Proof. In S ord we have ad sk ( c j )( c mj c i ) = (1 − q m + a ji jj ) c m +1 j c i for i < j , which impliesby induction ad sk ( c j ) − a ji ( c i ) = c − a ji j c i − a ji Y m =0 (1 − q m + a ji jj ) = 0 . When R has no exceptional relations the above relation is sufficient to produce theproposed surjection R pre → S ord . In the case of exceptional relations, one checksdirectly from the presentations of [1, Eq. 4.6, 4.13, 4.22, 4.27, 4.34, 4.41, 4.49] thatthe relations ad sk ( c i )( c j ), for i < j , imply all additional relations for R pre as well.If we consider the projection S ord → s ord , the addition of the relations c N i i to S ord imply the relations c N γ γ . So we also get the projection R → s ord . (cid:3) Proof of Theorem 4.3. Proof of Theorem 4.3. Take S = S ord . We have the adjoint action of R pre on itself,which induces an action of R pre on the braided symmetric algebra S . Since theaction of R pre on itself factors through R , the induced action on S also factors togive a well-defined action of R on S . The generators x i in this case act as theadjoint operators ad sk ( c i ). We integrate the natural action of G as well to get awell-defined action of H = R ⋊ G , which gives S a well-defined H -module algebrastructure (see Lemma 2.2).We note that the restriction of the action H → End k ( S ) produces an embedding V → End k ( S ), where V = R is the space of primitives in R . To see this clearly,note that for any linear combination v = P i κ i x i , and i v maximal in the orderedbasis P ord such that κ i v = 0, we have v · c i v = κ i v ad sk ( c i v )( c i v ) = (1 − q i v i v ) κ i v c i v = 0 . The action of H will however not be inner faithful in general, as G may not actfaithfully on S .We have the additional action of H on C [ w µ : µ ∈ Y ] given simply by the Hopfprojection H → C [ G ] and the prescribed G -action on C [ w µ : µ ∈ Y ], g · w µ = µ ( g ) w µ . We can therefore let H act diagonally on the tensor product C [ w µ : µ ∈ Y ] ⊗ S. Via the vector space equality C [ w µ : µ ∈ Y ] ⊗ S = C [ w µ : µ ∈ Y ] ⊗ S = A we get an H -action on A , which we claim gives it the structure of an H -modulealgebra. To show this it suffices to show that the multiplication is G -linear and R -linear independently.The fact that the multiplication on A is a map of G -representations follows fromthe fact that A is an algebra object in YD( G ). For R -linearity it suffices to showthat the braiding c : S ⊗ C [ w µ : µ ∈ Y ] → C [ w µ : µ ∈ Y ] ⊗ S is a map of R -modules,since S and C [ w µ : µ ∈ Y ] are both R -module algebras independently. However,this is clear as C [ w µ : µ ∈ Y ] is a trivial R -module. Whence we find that A isan H -module algebra, as proposed. We then get an induced action of H on thefraction field Q = Frac( A ) by Theorem 2.3.The fact that the H -action on Q is inner faithful when Y generates G ∨ followsby Corollary 2.7, since the restrictions G → End k ( A ) and V → End k ( A ) are bothinjective. (cid:3) Actions for (generalized) quantum groups We consider cocycle deformations of the Cartan type algebras considered in theprevious section. The primary example of such an algebra is the small quantumgroup u q ( g ) associated to a simple Lie algebra and root of unity q . However,more generally, one has the pointed Hopf algebras u ( D ) of Andruskiewitsch andSchneider. These algebras are determined by a combinatorial data D consisting ofa collection of Dynkin diagrams and a so-called linking data for these diagrams.We produce actions of the Hopf algebras u ( D ) on central simple division algebraswhich are constructed from their Angiono-de Concini-Kac form U ( D ). This actionis inner faithful if and only if the skew primitives in U ( D ), considered as a represen-tation of the grouplikes under the adjoint action, tensor generate Rep( G ( u ( D ))).In the case of a classical quantum group u q ( g ) we construct a faithful action on acentral simple algebra via quantum function algebras, without imposing restrictionson the interactions of grouplikes and skew primitives.5.1. Actions for u ( D ) . Let R = B ( V ) be of finite Cartan type. Take V in YD( G )for some abelian G and consider the bosonization H = R ⋊ G . Take a basis { x , . . . , x θ } for V consisting of G × G ∨ -homogeneous elements. Let g i be the G -degree of x i .We can consider V as object in YD( Z θ ) and take H pre := R pre ⋊ Z θ . Specifically, Z θ has generators t i , we have the group map Z θ → G , t i g i , and welet Z θ act on V via this group map. We take each x i ∈ V to be homogeneous of Z θ -degree t i . Lemma 5.1. For R = B ( V ) , and V of Cartan type as above, the algebra H pre isa domain which is finite over its center.Proof. Recall that R pre is finite over the subalgebra Z , which is generated by the x N γ γ and lies in the total braided center by Proposition 4.6. Hence R pre is finiteover the central subalgebra Z ′ generated by the powers x exp( G ) α . If we take K tobe the kernel of the projection K → Z θ → G , it follows that H pre is finite over Z ′ ⊗ C [ K ].We show that H pre is a domain. We first show that R pre is a domain. Just asin [10, § R pre via a normal ordering on the positive roots for the root system associated to V to get that gr R pre is a skew polynomial ring generated by the x α . In particular, gr R pre is a domain,and hence R pre is a domain. By considering the Z θ -grading on H pre given directlyby the Z θ factor, we see that H pre is a domain as well. (cid:3) We note that any Hopf 2-cocycle σ : H ⊗ H → C restricts to a Hopf 2-cocycleon H pre , via the projection H pre → H . Hence we can consider for any such σ thetwist H pre σ and Hopf projection H pre σ → H σ . Lemma 5.2. Consider any -cocycle σ : H ⊗ H → C with trivial restriction σ | G × G = 1 . Then the following holds: (i) The cocycle deformation H pre σ is (still) a domain. (ii) H pre σ is finite over its center. (iii) The adjoint action of H pre σ on itself factors through H σ .Proof. (i) By considering the associated graded algebra gr H pre σ with respect to thecoradical filtration, and Lemma 5.1, we see that H pre σ is a domain. In particular,gr H pre σ = H pre , which is a domain by Lemma 5.1.(ii) Let Π be the kernel of the projection Z θ → G , and take Z = Z ⋊ Π. Thenwe have an exact sequence of Hopf algebras Z → H pre → H . Therefore σ | Z ⊗ H pre = σ | H pre ⊗ Z = ǫ and H pre σ = H pre as a Z -bimodule. In particular H pre σ is a finite module over Z . Since Z is finite over the central subalgebra generated by the kernel Π of theprojection Z θ → G and the exp( G )-th powers of the generators for R pre , we seethat H pre σ is finite over its center.(iii) We note that the subalgebra Z = Z ⋊ Π in H pre σ is a Hopf subalgebra.Since H pre σ = H pre as a Z -bimodule, it follows that the adjoint action of Z on H pre σ is still trivial, by Proposition 4.7. Whence the adjoint action of H pre σ on H pre σ restricts trivially to Z , and from the exact sequence Z → H pre σ → H σ we see thatthe adjoint action factors through H σ . (cid:3) Theorem 5.3. Suppose that V ∈ YD( G ) is of finite Cartan type, and that V (tensor) generates Rep( G ) . Then for any -cocycle σ of H = B ( V ) ⋊ G with σ | G × G = 1 , the adjoint action of H σ on H pre σ is inner faithful. Consequently, theinduced action of H σ on the central simple division algebra Frac( H pre σ ) is innerfaithful.Proof. The fact that V generates Rep( G ) implies that all characters for G appear inthe decomposition of H pre into simples, under the adjoint action. So G acts faith-fully on H pre . Triviality of the restriction σ | G × G implies that the grading gr H pre σ with respect to the coradical filtration is the bosonization H pre . Semisimplicity of C [ G ] then implies an isomorphism of G -representations H pre σ ∼ = H pre . So we seethat G acts faithfully on H pre σ .All that is left is to verify that the restriction of the adjoint action H σ → End C ( H pre σ ) to the space of nontrivial ( g, g ( H pre σ ) ′ is in-jective. Note that H pre σ is a G -graded vector space (not algebra) with gradinginduced by comultiplication and projection H pre σ → H pre σ ⊗ H pre σ → H pre σ ⊗ C [ G ].Choose any such primitive v and a of trivial G -degree, i.e. a ∈ B ( V ) ⊂ H pre σ . Notethat v ∈ V , and hence v has a canonical lift to H σ . We have v · ad a = σ ( v, a ) a + σ ( g, a ) va + σ ( g, a )˜ ga σ − ( v, a )+elements in degree G −{ e } . So we see that it suffices to show that the e -degree term is nonvanishing.Take i minimal with a ∈ F i H σ , where we filter with respect to the coradicalfiltration. Then, since gr H σ = H , σ ( v, a ) a + σ ( g, a ) va + σ ( g, a ) a σ − ( v, a ) = va mod F i H σ . Since H is a domain, va is nonzero, and we conclude v · ad a is nonzero. It followsthat the restriction of the adjoint action to each Prim g ( H σ ) ′ is injective, and theadjoint action of H σ on H pre σ is inner faithful by Lemma 2.6. (cid:3) We are particularly interested in the generalized quantum groups u ( D ) = u ( D , λ, µ )of Andruskiewitsch and Schneider [2]. These algebras are determined by a collec-tion of Dynkin diagrams and a “linking data” D = ( D , λ, µ ) between the Dynkindiagrams. As far as the above presentation is concerned, we have u ( D ) = ( B ( V ) ⋊ G ) σ = H σ for a finite Cartan type V and a cocycle σ which restricts trivially to the group-likes [2, Section 5.2, 5.3], [5, Corollary 1.2]. A direct application of Theorem 5.3yields Corollary 5.4. Suppose V ∈ YD( G ) is of finite Cartan type, and that V generates Rep( G ) . Then the generalized quantum group u ( D ) associated to any linking data D admits an inner faithful action on a central simple division algebra. Remark 5.5. The supposition that V generates Rep( G ) is a serious restriction. Forclassical quantum groups u q ( g ), for example, the space of skew primitives generatesRep( G ) if and only if q is relatively prime to the determinant of the Cartan matrixfor g . For generalized Taft algebras T ( n, m, α ), we have such generation if and onlyif m = n .5.2. More refined actions for standard quantum groups. Let q be an oddroot of 1, g be a simple Lie algebra, and u q ( g ) be the corresponding small quantumgroup. We assume additionally that the order of q is coprime to 3 when g is of type G . Proposition 5.6. There is an inner faithful action of u q ( g ) on Frac( O q ( G )) , where G is the simply-connected, semisimple, algebraic group with Lie algebra g . Further-more, this action is Hopf-Galois. In particular, u q ( g ) acts inner faithfully on acentral simple division algebra.Proof. By definition, O q ( G ) is the finite dual of the Lusztig, divided powers, quan-tum group U q ( g ). We have the action of u q ( g ) on O q ( G ) by left translation x · f := ( a f ( ax )) for x ∈ u q ( g ) , f ∈ O q ( G ) . This action is faithful as it reduces to a faithful action of u q ( g ) on the quotient u q ( g ) ∗ .The exact sequence C → u q ( g ) → U q ( g ) → U ( g ) → C [18] gives an exactsequence C → O ( G ) → O q ( G ) → u q ( g ) ∗ → C . (By an exact sequence C → A → B → C → C we mean that A → B is a faithfullyflat extension with B ⊗ A C ∼ = C , and that A is the C -coinvariants in B .) Thesubalgebra O ( G ) is central in O q ( G ), and O q ( G ) is finite over O ( G ). Furthermore, O q ( G ) is a domain [7, III.7.4]. So we take the algebra of fractions Frac( O q ( G )) toarrive at a central simple division algebra on which u q ( g ) acts inner faithfully.As for the Hopf-Galois property, faithful flatness of O q ( G ) over O ( G ) implies that O q ( G ) is a locally free O ( G )-module, and also O ( G ) = O q ( G ) u q ( g ) [21, Theorem2.1]. From the equality Frac( O q ( G )) = Frac( O ( G )) ⊗ O ( G ) O q ( G ) one calculatesrank Frac( O ( G )) Frac( O q ( G )) = rank O ( G ) O q ( G ) = dim( u q ( g ))and Frac( O ( G )) = Frac( O q ( G )) u q ( g ) . It follows that the given extension is Hopf-Galois by Theorem 2.5. (cid:3) Proof of Theorem 3.1 We first establish some general information regarding skew derivations of centralsimple algebras, then provide the proof of Theorem 3.1.6.1. Bimodules in Yetter-Drinfeld categories and skew derivations. Givena field K we write YD K ( G ) for the category of Yetter-Drinfeld modules over thegroup algebra KG . We always assume K is of characteristic 0. Lemma 6.1. Let A be an algebra in YD K ( G ) . There is an equivalence of categoriesbetween the subcategory of A -bimodules in YD K ( G ) and right A op ⊗ K A -modules in YD K ( G ) . This equivalence takes a bimodule M to the Yetter-Drinfeld module M along with the right A op ⊗ K A -action m · ( a ⊗ b ) := ( m − a ) m b .Proof. Straightforward direct check. (cid:3) Recall that in characteristic 0, a finite-dimensional semisimple K -algebra A isseparable over K . Lemma 6.2. Let G be an abelian group and A be an algebra in YD( G ) , which issemisimple as a C -algebra. Let K be a central invariant subfield in A over which A is finite. Then the algebra A is projective as an A op ⊗ K A -module.Proof. Since G is abelian, the Yetter-Drinfeld structure on A is equivalent to a G × G ∨ -grading on A . Take G ′ = G × G ∨ . We claim that A ⊗ K A → A admitsa homogeneous degree 0 section, as a map of bimodules. To see this one simplytakes an arbitrary separability idempotent e and expands e = P g,h ∈ G ′ e g ⊗ e h witheach e g ⊗ e h ∈ A g ⊗ K A h . Take e ′ = P g e g ⊗ e g − . Since the multiplication on A is homogeneous we see that m ( e ′ ) = 1. Furthermore, since the multiplication onthe right and left of A ⊗ A preserves the grading, we see that ae ′ = e ′ a for eachhomogeneous a ∈ A , and hence each a ∈ A . So the map A → A ⊗ K A , 1 e ′ ,provides a degree 0 splitting of the multiplication map. By Lemma 6.1 we see thatthe projection A op ⊗ K A → A, a ⊗ b ab is split as well, and hence that A is projective over A op ⊗ K A . (cid:3) Lemma 6.3. Take G abelian, and let A be a G -module central semisimple algebra.Let K be a central invariant subfield over which A is finite, and let M be a K -central A -bimodule in Rep( G ) . Then every K -linear, homogeneous, ( g, -skew derivation f : A → M , for g ∈ G , is inner. By homogeneous we mean the following: if we decompose A and M into characterspaces A = ⊕ µ A µ , M = ⊕ µ M µ , then f ( A µ ) ⊂ M µσ for some fixed σ ∈ G ∨ . So f is homogeneous of degree σ here. By an inner skew derivation we mean there is c ∈ M σ so that f = [ c, − ] sk : a ( ca − ( g · a ) c ). Proof. Take σ = deg G ∨ ( f ). We choose a non-degenerate form b : G × G → C × and let G ∨ act on A and M via the isomorphism f b : G ∨ → G provided bythe form. Then we decompose A and M into character spaces A = ⊕ µ A µ and M = ⊕ µ M µ , and the corresponding G -gradings A = ⊕ g A g and M = ⊕ g M g aresuch that A g = A µ and M g = M µ for µ with g = f b ( µ ). There is a unique shift M [ h ] of the G -grading on M so that M σ = ( M [ h ]) g . In this way A and M [ h ] areobjects in YD K ( G ), and M [ h ] is an A -bimodule in YD K ( G ).Consider M [ h ] as an A op ⊗ K A -module. As in [22, Proposition 3.3(1)], one canshow thatExt A op ⊗ K A ( A, M [ h ]) = { Skew derivations } / { Inner derivations } . Since A is separable, this cohomology group vanishes. Whence we conclude thateach skew derivation of M is inner. (cid:3) Proof of Theorem 3.1. We consider again the algebra T ( n, m, α ). We willneed the following result. Proposition 6.4 ([11, Proposition 3.9]) . Suppose H is a finite-dimensional Hopfalgebra acting on an algebra A which is finite over its center. Then A is finite overthe invariant part of its center Z ( A ) H = Z ( A ) ∩ A H . From a G -module algebra A , an element c ∈ A i , and fixed g ∈ G , we let [ c, − ] sk : A → A denote the endomorphism [ c, a ] sk := ca − ( g · a ) c . We now prove Theorem 3.1. Proof of Theorem 3.1. Take G = G ( T ( n, m, α )) = h g i , and ζ a primitive n -th rootof 1 with ζ n/m = q . We fix A a G -module central simple algebra, which we decom-pose as A = ⊕ ni =1 A i so that g | A i = ζ i · − . We claim that, for an arbitrary element c ∈ A n/m , we have [ c, − ] m sk ( a ) = c m a − ζ m | a | ac m . (4)The skew commutator here employs the action of the generator g . The equality (4)will imply the desired result, as for any T ( n, m, α )-action on A , which extends thegiven action of G , we will have x · − = [ c, − ] sk for some c ∈ A n/m by Lemma 6.3.In our application of Lemma 6.3 here we take K = Z ( A ) T . So we seek to prove (4).We note that q m ( m − / = (cid:26) q m/ = − m is even1 when m is odd = ( − m +1 . So the desired relation (4) can be rewritten as[ c, − ] m sk ( a ) = c m a + ( − m ζ m | a | q m ( m − / ac m . (5)We have directly [ c, − ] m sk ( a ) = c m a + m X l =1 ( − l ζ l | a | ω l c m − l ac l , (6) for coefficients ω i ∈ Q ( ζ ). The coefficient ω l can be deduced as follows: Each c appearing on the right of c m − l ac l indicates an integer i so that at the i -th iterationof [ c, [ c, − ] i − ( a )] sk = c [ c, − ] i − ( a ) − ζ | a | +( i − m [ c, − ] i − ( a ) c = c [ c, − ] i − ( a ) − ζ | a | q ( i − [ c, − ] i − ( a ) c we take the summand q ( i − [ c, − ] i − ( a ) c . Each choice of l such distinct positions { k , . . . , k l } ⊂ { , . . . , m } contributes a summand with q -coefficient ( Q lj =1 q ( k j − ).Take [ m − 1] = { , . . . , m − } . Considering all possible choices for the subset { k , . . . , k l } gives ω l = X ≤ k < ··· 1] which do not contain thegiven j ∈ [ m − j . Then ω l = ω l ( j ) + ω ′ l ( j ) . (9)Note that ω ′ l ( j ) = q j ω l − ( j ), where ω ( j ) is formally taken to be 1. Then theexpression (8) gives ω l = 1 l ! m − X j =0 q j ( l − ω l − ( j ) = 1 l m − X j =0 q j ω l − ( j ) . (10)We have already seen that ω = 0. We take l < m and suppose that ω k = 0 forall k < l . Then the decomposition ω k = ω k ( j ) + ω ′ k ( j ) for all j ∈ [ m − 1] implies ω k ( j ) = − ω ′ k ( j ) = − q j ω k − ( j − for all k < l and j . Hence, from (10), ω l = l − P m − j =0 q j ω l − ( j )= − l − P m − j =0 q j ω ′ l − ( j )= − l − P m − j =0 q j ω l − ( j )= ( − l − P m − j =0 q j ω ′ l − ( j )...= ( − l − l − P m − j =0 q lj ω ( j ) = ( − l − l − (1 − q lm ) / (1 − q l ) = 0 . Hence ω l = 0 for all l < m . One recalls our initial expression (6) to arrive finallyat the desired equality [ c, − ] m sk ( a ) = c m a − ζ m | a | ac m . (cid:3) Coradically graded algebras and universal actions Let us fix now a coradically graded, pointed Hopf algebra H with abelian groupof grouplikes. We may write H = B ( V ) ⋊ G , with G abelian and V in YD( G ). Fixalso a homogeneous basis { x i } i for V with respect to the G × G ∨ -grading providedby the Yetter-Drinfeld structure.7.1. The universal algebra. We consider the (Hopf) free algebra T V in YD( G )as a module algebra over itself under the adjoint action a · adj b := a (cid:0) ( a ) − b ) S (cid:0) ( a ) (cid:1) . Consider a presentation B ( V ) = T V / ( r , . . . , r l ) with each r i homogeneous withrespect to the G × G ∨ -grading, as well as the grading on T V by degree.Define A univ as the quotient A univ = A univ ( V ) := T V / ( r i · adj a : 1 ≤ i ≤ l, a ∈ T V ) . We note that A univ is a connected graded algebra in YD( G ), as all relations canbe taken to be homogeneous with respect to all gradings. Furthermore, the adjointaction of the free algebra on itself induces an action of T V on A univ . We let c i denote the image of x i ∈ V in A univ . Lemma 7.1. The adjoint action of T V on A univ induces an action of B ( V ) on A univ . This action is specified on the generators by x i · a = [ c i , a ] sk := c i a − ( g i · a ) c i .Proof. Evident by construction. (cid:3) Since each relation for B ( V ) in T V must act trivially on A univ we have immedi-ately Corollary 7.2. For any r in the kernel of the projection T V → B ( V ) , and arbitrary a ∈ T V , A univ has the relation r · adj a = 0 . In particular, the B ( V ) -module algebra A univ is independent of the choice of relations for B ( V ) . Definition 7.3. For given V in YD( G ), with G abelian, we call A univ ( V ) the universal algebra for V .We would like to construct from A univ central simple H -division algebras, andtherefore would like to develop means of understanding when A univ itself is finiteover its center. Lemma 7.4. Suppose the kernel I of the projection T V → B ( V ) contains a rightcoideal subalgebra R ⊂ I such that (a) R is a graded subalgebra in YD( G ) , (b) R is finitely generated and (b) the quotient T V / ( R + ) is finite-dimensional.Then the algebra A univ ( V ) is finitely presented and finite over its center.Proof. Enumerate a homogeneous generating set { r , . . . , r d } for R . By homoge-neous we mean homogeneous with respect to the G × G ∨ -grading as well as the Z -grading. Define B = T V / ( R + ) = T V / ( r , . . . , r d ) and A = T V / ( r i · adj a ) i , where a runs over homogeneous elements in T V . Note that B is a finite-dimensional Hopfalgebra in YD( G ), by hypothesis, and surjects onto B ( V ). Note also that A surjectsonto A univ .Take I k to be the ideal in T V generated by the relations r i · adj a for r i withdeg( r i ) ≤ k , and homogeneous a ∈ T V . Let J k be the ideal generated by the[ r i , a ] sk = r i a − ( ga ) r i for r i with deg( r i ) ≤ k and a homogeneous, where g =deg G ( r i ). Since each [ r i , − ] sk is a skew derivation, J k is alternatively generated bythe relations [ r i , x j ] sk for varying i and j . We would like to show I k = J k for all k .We have I = J = 0.We have for each relation∆( r i ) = r i ⊗ ⊗ r i + X m f m ⊗ h m , where the f m ∈ R and the h m ∈ T V , and deg( f m ) , deg( h m ) < deg( r i ), since R iscoideal subalgebra. Suppose we have I k − = J k − for some k . Then I k = ( r i · adj a : deg( r i ) = k ) a ∈ T V + I k − = ( r i · adj a : deg( r i ) = k ) a ∈ T V + J k − , and one also computes for r i of degree k , r i · adj a = [ r i , a ] sk + P m χ a (deg G ( h m )) f m aS ( h m )= r i a + χ a ( g ) aS ( r i ) + P m χ a ( g ) af m S ( h m ) mod J deg( r i ) − = r i a + χ a ( g ) a (cid:0) ( r i ) S (( r i ) ) − r i )= r i a − χ a ( g ) ar i = [ r i , a ] sk , where in the above computation deg G ( r i ) = g and deg G ∨ ( a ) = χ a . Hence I k = J k and, by induction, we have( r i · adj a ) i,a = ∪ k> I k = ∪ k> J k = ([ r i , x j ] sk ) i,j . The above identification provides a presentation A = T V / ([ r i , a ] sk ) i,a = T V / ([ r i , x j ] sk ) i,j . (11)Let R ′ be the image of R in A . Via the relations (11) we see that R ′ is thequotient of a skew polynomial ring which is finite over its center, and also that R ′ is normal in A , in the sense that ( R ′ ) + A = A ( R ′ ) + . Note that a bounded below Z -graded module M over a Z ≥ -graded algebra T with T = C is finitely generatedif and only if the reduction C ⊗ T M is finite-dimensional. So we see that A isfinite over R ′ , and hence finite over its center, as the reduction C ⊗ R ′ A = B isfinite-dimensional by hypothesis.The center of R ′ is finite over C [ r exp( G ) i : 1 ≤ i ≤ d ] and hence finitely generated.In particular, the center of R ′ is Noetherian. As A is finite over Z ( R ′ ) it followsthat any ideal in A is finitely generated as well. Whence the kernel of the surjection A → A univ is finitely generated, and we see that A univ is finitely presented. (cid:3) Remark 7.5. In the notation of Lemma 7.4, one can produce coideal subalgebrasin I ⊂ T V by considering, for example, subalgebras generated by coideals in T V which are contained in I .The most immediate way for the hypotheses of Lemma 7.4 to be satisfied is ifa generating set of relations for B ( V ) can, in its entirety, be chosen to generate acoideal subalgebra in T V . Lemma 7.6. Suppose there is a choice of homogeneous relations { r , . . . , r d } for B ( V ) so that the subalgebra R generated by the r i in T V forms a coideal sub-algebra. (For example, this occurs when the relations for B ( V ) can be chosento be primitive.) Then A univ is finite over its center, and has a presentation A univ = T V / ([ r i , x j ] sk ) i,j .Proof. The fact that A univ is finite over its center follows by Lemma 7.4. Thepresentation by skew commutators was already provided in the proof of Lemma 7.4. (cid:3) In non-Cartan, diagonal, type the stronger hypotheses of Lemma 7.6 are notalways met. (There are certainly examples in which they are met, however. SeeSection 7.3.) Indeed, one can show for some simple super-type algebras that A univ does not have the desired commutator relations. In some more regular settings,however, we expect that the conditions of Lemma 7.6 will be met. One can prove,for example, that this occurs for the quantum Borel in small quantum sl at q a3-rd root of 1.7.2. Central simple division algebras via the universal algebra. Take A univ = A univ ( V ), as above, and H = B ( V ) ⋊ G . Consider any field K with a G -action,which we consider as an algebra in YD( G ) by taking the trivial G -grading, andalso as a trivial B ( V )-module algebra. We may take the tensor product K ⊗ A univ to get a well-defined B ( V )-module algebra in YD( G ) (cf. proof of Theorem 4.3).Consider now any quotient A ( K, I ) := K ⊗ A univ /I via a prime G -ideal I such that A ( K, I ) is (a domain which is) finite over its center.Since B ( V ) acts by skew commutators on K ⊗ A univ , any such ideal will additionallybe an H = B ( V ) ⋊ G -ideal. In this case the ring of fractions Q ( K, I ) := Frac( K ⊗ A univ /I )is a central simple division algebra on which B ( V ) acts faithfully, by [24, Theorem2.2]. Definition 7.7. A pair ( K, I ) of an field K with a G -action and a prime G -ideal I in K ⊗ A univ is called a pre-faithful pair if the quotient A ( K, I ) is finite over itscenter. A pre-faithful pair is called faithful if the H -action on A ( K, I ) is innerfaithful.Note that when A univ is finite over its center, A ( K, I ) is finite over its center forany choice of K and I (see Lemmas 7.4 and 7.6). Also, there are practical conditionson K and I which ensure that H acts inner faithfully on A ( K, I ). For example, ifthe sum K ⊕ V generates Rep( G ) and the composition V → A univ → A ( K, I ) isinjective then the H -action on A ( K, I ) is inner faithful. In what follows we consider H -module structures on a given algebra Q which areinduced by a B ( V )-module structure in YD( G ). An additional YD( G )-structureon an H -module algebra Q consists only of a choice of an additional action of thecharacter group G ∨ on Q , which is compatible with the given H -action. Proposition 7.8. Suppose H = B ( V ) ⋊ G acts inner faithfully on a central simpledivision algebra Q . Then (1) Q admits an H -module algebra map f : A univ → Q so that x i · a = [ f ( c i ) , a ] sk for each x i ∈ Prim( H ) ′ and a ∈ Q . (2) Q contains an H -division subalgebra of the form Q ( K ′ , I ′ ) for some pre-faithful pair ( K ′ , I ′ ) . (3) If the H -action on Q is induced by a B ( V ) -module algebra structure in YD( G ) , then Q contains an H -division subalgebra Q ′ over which Q is afinite module, and which admits an embedding Q ′ → Q ( K, I ) into a divisionalgebra associated to a faithful pair. In particular, the existence of such Q impies the existence of a faithful pair for H .Proof. (1) By Lemma 6.3 the x i act on Q as skew derivations x i · a = [ c ′ i , a ] sk = c ′ i a − ( g i · a ) c ′ i for some c ′ i ∈ Q of G ∨ -degree χ i . (Here ( g i , χ i ) denotes the G × G ∨ -degree of x i in B ( V ).) We claim that the assignment f ( c i ) = c ′ i provides the necessary mapof (1). Indeed, the corresponding map F : T V → Q , F ( x i ) = c ′ i is a well-defined T V ⋊ G -module map, and factors through A univ as any relation r for B ( V ) issuch that F ( r · a ) = r · F ( a ) = 0. Whence there is a well-defined G -algebra map f : A univ → Q , f ( c i ) = c ′ i , which commutes with the skew derivations x i · − , and istherefore a map of H -module algebras.(2) Take K ′ to be a G -subfield in Q which is contained in the B ( V )-invariants,and which contains Z ( Q ) H . By Proposition 6.4 Q is finite over K ′ . The B ( V )-invariance of K ′ tells us that all the c ′ i ∈ Q , from (1), skew commute with K ′ .Hence the map f of (1) extends to f ′ : K ′ ⊗ A univ → Q . Take I ′ = ker( f ′ ) to obtainthe desired pre-faithful pair.(3) Via the Yetter-Drinfeld structure on Q , we may take each c ′ i ∈ Q of theappropriate G × G ∨ -degree ( g i , χ i ). The map A univ → Q is then a map in YD( G ),and inner faithfulness ensures that the composite V → A univ → Q is injective.(Otherwise homogeneous elements in the kernel would act trivially on Q .)Take Q ′ = Q ( K ′ , I ′ ) with K ′ and I ′ as in (2), and let S = Sym( W ) where W is a (finite-dimensional) G -representation such that W ⊕ Q ′ generates Rep( G ) asa tensor category. If we take S as a trivial G -comodule, the diagonal H -action onthe tensor product S ⊗ Q ′ gives it an H -module algebra structure. This algebra isa domain which is finite over its center, and so we take the ring of fractions to geta central simple algebra Q ′′ = Frac( S ⊗ Q ′ ) on which H -acts inner faithfully. If wetake K to be the image of the G -algebra Frac( S ⊗ K ′ ) in Q ′′ , and I the kernel ofthe map K ⊗ A univ → Q ′′ , then we see Q ′′ = Q ( K, I ). (cid:3) Remark 7.9. We have a faithful braided functor YD( G ) → YD( G × G ∨ ) so thatHopf algebras in YD( G ) are sent to Hopf algebras in YD( G × G ∨ ), and an extensionof an H -action on Q to a B ( V )-action in YD( G ) is equivalent to an action of thepointed algebra B ( V ) ⋊ ( G × G ∨ ) on Q . So, in terms of the general question of(non-)existence of actions of pointed, coradically graded, Hopf algebras on central division algebras, one may deal only with actions of Nichols algebras in Yetter-Drinfeld categories.In particular, the non-existence of a faithful pair ( K, I ) for a particularly patho-logical braided vector space V in some YD( G ) would provide a negative resolutionto [9, Question 1.1]. One could also attempt to approach actions on quantum tori [6,Conjecture 0.1] via A univ .Proposition 7.8 is, of course, why we refer to A univ as the universal algebra for H .7.3. A non-Cartan example. We provide a small example to illustrate the man-ner in which A univ can be employed to obtain results outside of Cartan type. Con-sider V = C { x , x } the 2-dimensional braided vector space with braiding matrix[ q ij ] = (cid:20) − √− − √− (cid:21) . We take V as an object in YD( Z / Z ) with each of the x i homogeneous of degree g , where g generates Z / Z , and g · x = − x , g · x = √− x .Note that V is a faithful Z / Z -representation, and that V is not of Cartan type,as q q = −√− q = − R = B ( V ) has relations x = 0 , x = 0 , ad sk ( x ) ( x ) = 0 , ad sk ( x ) ( x ) = 0 . (12)One can check directly, or use the fact that x is primitive, to see that the rela-tion x = 0 implies the relation ad sk ( x ) ( x ) = 0. Hence we have the minimalpresentation B ( V ) = C h x , x i / ( x , x , ad sk ( x ) ( x )) . One sees that each of the minimal relations for B ( V ) is primitive in the tensoralgebra T V (see [1]). 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