Delta sets and polynomial identities in pointed Hopf algebras
aa r X i v : . [ m a t h . QA ] J a n DELTA SETS AND POLYNOMIAL IDENTITIES IN POINTEDHOPF ALGEBRAS
YURI BAHTURIN AND SARAH WITHERSPOON
Abstract.
We survey a vast array of known results and techniques in thearea of polynomial identities in pointed Hopf algebras. Some new results areproven in the setting of Hopf algebras that appeared in papers of D. Radfordand N. Andruskiewitsch - H.-J. Schneider. Introduction
In this paper our main concern is the determination of conditions under whicha pointed Hopf algebra H over a field F is a PI-algebra, that is H satisfies anontrivial polynomial identity f ( x , . . . , x n ) = 0. Here f ( x , . . . , x n ) is a nonzeroelement of the free associative algebra (i.e. algebra of noncommutative polynomials)in the variables x , . . . , x n . For an excellent modern source of information aboutPI-algebras we refer the reader to Giambruno - Zaicev’s monograph [9]. As acrash course, here we recall a few facts about PI-algebras that are relevant to ourmain problem, particularly some of the many results known about group algebras,universal enveloping algebras, and smash products.Our main new results are: (1) A generalization in Theorem 4.4, to color Liesuperalgebras, of Kochetov’s classification of Lie superalgebras and their smashproducts satisfying polynomial identities. (2) A classification in Theorem 8.1 of alarge class of pointed Hopf algebras, arising in work of Andruskiewitsch - Schneider,satisfying polynomial identities. Along the way we also discuss delta sets in pointedHopf algebras. They play a crucial role in PI theory and also prove useful in othersituations.We begin in Sections 2 and 3 by recalling some definitions and summarizing someof the general theory we will need as well as some known results about cocommu-tative Hopf algebras, particularly group algebras, universal enveloping algebras ofLie algebras, and their smash products. In Section 4 we state our generalizationof Kochetov’s work to color Lie superalgebras: In Theorem 4.4 we give necessaryand sufficient conditions for U ( L ) F G to be a PI-algebra, where U ( L ) is the uni-versal enveloping algebra of a color Lie superalgebra L , G is an acting group, and F has characteristic 0. In the case of positive characteristic, these conditions arenot known. The proof of Theorem 4.4 is deferred to Section 9; we show there thatKochetov’s proof generalizes with no obstacles.To prepare for the proof of Theorem 4.4 as well as to set the stage for understand-ing a further large class of pointed Hopf algebras introduced by Andruskiewitsch Keywords:
Hopf algebras, algebras with polynomial identities, delta sets.
Primary 16T05, Secondary 16W50, 17B37.The first author acknowledges a partial support by NSERC Discovery Grant 2019-05695. Thesecond author acknowledges partial support by NSF grants DMS-1665286 and DMS-2001163. and Schneider, in Section 5 we collect some more specialized known techniques forgroup algebras, Lie superalgebras, and smash products. We recall delta sets andknown results for groups, Lie algebras, and Hopf algebras, and adapt them to colorLie superalgebras.Radford’s Hopf algebras F ( q ) feature in Section 6; these are Hopf algebras gen-erated by one grouplike and one skew-primitive element depending on a scalar q .They play a key role as subalgebras in more general pointed Hopf algebras, and ourwork in Section 6 is accordingly called upon later. We show that F ( q ) is a PI-algebraif and only if q is a root of unity, and we give further results on the degree of thecorresponding polynomial identity as well as some results on delta sets for F ( q ) .In Section 7 we define Hopf algebras U ( D , λ ) depending on some data D , λ ;these are the pointed Hopf algebras arising in Andruskiewitsch and Schneider’sclassification of finite dimensional pointed Hopf algebras, but with more generalgroups of grouplikes allowed. In Section 8 we give in Theorem 8.1 necessary andsufficient conditions for U ( D , λ ) to be a PI-algebra and study delta sets.Throughout, F will be a field, of arbitrary characteristic unless stated otherwise.2. PI-algebras
Note that every finite-dimensional algebra R over a field F is PI: if dim F R = n − R satisfies the so called standard identity :(2.1) s n ( x , . . . , x n ) = X σ ∈ Sym( n ) sgn( σ ) x σ (1) · · · x σ ( n ) = 0 . At the other extreme, commutative algebras of any dimension satisfy s ( x, y ) = xy − yx = [ x, y ] = 0. Generally, for any PI-algebra R there exist numbers m and n such that s n ( x , . . . , x n ) m = 0 is an identity in R .Subalgebras, factor-algebras and extensions (including direct products) of PI-algebras are PI. An important theorem of A. Regev [26] says the following. Theorem 2.2. If A and B are PI-algebras over an arbitrary field F , then A ⊗ F B is a PI-algebra. In particular, if n is a natural number and S is a PI-algebra then M n ( S ) = M n ( F ) ⊗ F S is a PI-algebra. One of the frequent applications of this is to extensionsof the field of coefficients to other fields or even to commutative algebras S over F . In all these cases the algebra obtained by an extension of coefficients from aPI-algebra remains PI.Another corollary of the techniques of Regev’s Theorem, which is useful whendealing with the quantum commutator structure of an algebra, is the following (see[6, Proposition 4.1.11]). In what follows, [ a, b ] q = ab − qba . Proposition 2.3.
Suppose that an associative algebra A over a field F satisfiesa nontrivial polynomial identity of degree d . Then for any q ∈ F , A satisfies anontrivial identity of the form (2.4) f q ( x , . . . , x n , y , . . . , y n ) = X σ ∈ Sym( n ) λ σ [ x , y σ (1) ] q · · · [ x n , y σ ( n ) ] q = 0 , where n = 3 d and λ σ ∈ F . ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 3
The following theorem covers PI-algebras that are also a Hopf algebras. Recallthe adjoint action of H on itself: (ad x )( y ) = P x yS ( x ) where S is the antipodeand comultiplication takes x to P x ⊗ x . Theorem 2.5.
Suppose that a Hopf algebra H over a field F satisfies a nontrivialpolynomial identity of degree d . Then H satisfies a nontrivial identity of the form (2.6) f ( x , . . . , x n , y , . . . , y n ) = X σ ∈ Sym( n ) λ σ (ad x )( y σ (1) ) · · · (ad x n )( y σ ( n ) ) = 0 , where n = 3 d and λ σ ∈ F . In [14] the proof of this theorem (attributed by M. Kochetov to Y. Bahturin)is given for smash products of the form A H , where A is an H -module algebra.In that case x , . . . , x n ∈ H and y , . . . , y n ∈ A . However, it works in our moregeneral situation, without any changes. The estimate n = 3 d is borrowed from [7,Lemma 4.1].Being PI is “almost” equivalent to having an upper bound on the dimension ofsimple modules. On the one hand, a direct calculation shows that a matrix algebra M n ( K ) of order n over an F -algebra K does not satisfy a nontrivial identity ofdegree less than 2 n . A much sharper result is a theorem by Amitsur - Levitzkystating that s n ( x , . . . , x n ) = 0 holds in M n ( F ) and there are no identities ofdegree less than 2 n . Now using the Density Theorem, we can easily see that, forany n , if an algebra A has a simple module of dimension ≥ n over its centralizer D = End A V then A does not satisfy any identity of degree less than 2 n .On the other hand, if A is semiprimitive and all simple (left, right) modules are ofdimension at most n over F then A is a PI-algebra (satisfying s n ( x , . . . , x n ) = 0).Indeed such an algebra is a subCartesian product of primitive algebras End D V .If dim F D = ℓ and dim D V = m , with ℓm = n , then A is a subCartesian productof simple algebras of dimension ℓm = mn . If F is a splitting field for D then F ⊗ End D V ∼ = M √ mn , so s n ( x , . . . , x n ) = 0 must hold.We complete this introduction by citing two important theorems about the struc-ture of PI-algebras. The first one is an old theorem by I. Kaplansky [17], dealingwith primitive algebras. Theorem 2.7.
Let R be a primitive algebra, satisfying a polynomial identity ofdegree d . Then R is simple, the center of R is a field and dim Z ( R ) R ≤ [ d/ . A very useful theorem due to E. C. Posner [24] deals with prime algebras.
Theorem 2.8. (E. C. Posner [24])
Let R be a prime algebra over a field F , satisfyinga nontrivial polynomial identity of degree d . Let C be the center of R and Q thefield of quotients of C . Then the algebra R Q = Q ⊗ C R of central quotients of R is central simple of dimension n , n = [ d/ , over its center K . Moreover, theidentities of R are the same as the identities of R Q and the same as the identitiesof M n ( K ) . If R has no zero divisors, then R is a subalgebra of a division algebraof dimension n over its center. Cocommutative Hopf algebras
A well-known theorem on pointed cocommutative Hopf algebras is the following.
Theorem 3.1.
Let H be a pointed cocommutative Hopf algebra over a field F .Let G = G ( H ) be the group of group-like elements of H and H the connected BAHTURIN AND WITHERSPOON component of 1. Then G acts on H by conjugation and H is isomorphic to thesmash product H F G via h g hg for h ∈ H , g ∈ G . An algebra H over a field F is PI if and only if its extension to the algebraicclosure of F is PI. Since the condition of pointedness is automatically satisfiedif F is algebraically closed, this theorem essentially reduces cocommutative Hopfalgebras to the smash products of connected cocommutative Hopf algebras andgroup algebras, where the group acts by Hopf algebra automorphisms.Following Cartier (see [3]), one uses the term hyperalgebra for any connectedcocommutative bialgebra. The existence of the antipode is automatic for such bial-gebras (see [4, 2.2.8]), so they are in fact Hopf algebras. It is well known (see [18,5.6]) that in characteristic 0 any hyperalgebra H is isomorphic to the universalenveloping algebra U ( L ) of the Lie algebra L = P ( H ) of its primitive elements.So the question of polynomial identities of a cocommutative Hopf algebra of char-acteristic 0 reduces to the study of the smash product of a universal envelopingalgebra U ( L ) and a group algebra F G , where G acts on L by automorphisms.We next recall some known results about identities of cocommutative Hopf al-gebras.If F is of characteristic zero, then a group algebra F G is PI if and only if G has anabelian subgroup of finite index [11]. An enveloping algebra U ( L ) of a Lie algebra L is PI if and only if L is abelian [16, 5]. A smash product H = U ( L ) F G is PIif and only if L is abelian and G has an abelian normal subgroup A of finite indexwhich acts trivially on L [10, 12].These results were used in [15] to prove the following. Theorem 3.2.
Let H be a cocommutative Hopf algebra of characteristic 0. Thenthe following conditions are equivalent: (1) H is PI as an algebra, (2) There exists a normal commutative subHopfalgebra A ⊂ H such that H/HA + is finite-dimensional, (3) There exists a normal commutative subHopfalgebra B ⊂ H such that H isa finitely generated left B -module, (4) The Lie algebra L = P ( H ) of primitive elements is abelian and there ex-ists a normal subHopfalgebra C ⊂ corad( H ) of finite index such that C iscommutative and the adjoint action of C on L is trivial. In the case of fields of positive characteristic p >
0, a group algebra F G is PI [21]if and only if G contains a subgroup of finite index whose commutator subgroup isa finite p -group.The question of when an arbitrary hyperalgebra over a field of positive char-acteristic is PI, remains largely open. Some cases where the answer is known arelisted below.An enveloping algebra U ( L ) is PI [4] if and only if there is an abelian subalgebra M of finite codimension in L and all inner derivations are algebraic of finite boundeddegree. The last condition means that there is a natural number n such that eachinner derivation ad x , x ∈ L , is annihilated by a polynomial of degree n .A restricted enveloping algebra u ( L ) of a restricted Lie algebra L is PI [22, 23]if and only if L has restricted ideals N ⊂ M ⊂ L such that dim L/M, dim N ≤ ∞ ,[ M, M ] ⊂ N , [ N, N ] = 0 and there is natural n such that x [ p n ] = 0, for each x ∈ N . ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 5 If H is a reduced hyperalgebra over a perfect field F (that is, H ∗ has no nilpotentelements), then H is PI [14] if and only if H is commutative.In the case of a smash product U ( L ) F G , over a field F of characteristic p > G -invariant ideal H ⊂ L of finite codimension andall inner derivatives are algebraic of bounded degree,(2) there exists a normal subgroup A ⊂ G of finite index such that the com-mutator subgroup [ A, A ] is a finite abelian p -group,(3) A acts trivially on L .In the case of u ( L ) F G , char F = p >
0, this algebra is PI ([7]) if and only if(1) there are G -invariant restricted ideals N ⊂ M ⊂ L such that(a) dim L/M, dim
N < ∞ ,(b) M/N and N are abelian,(c) the p -map on N is nilpotent,(2) there is a subgroup A ⊂ G such that(a) | G : A | < ∞ ,(b) [ A, A ] is a finite p -group,(3) A acts trivially on M/N .In the general case of connected Hopf algebras, Mikhail Kochetov [14] conjecturedthat the following is true.
Conjecture 3.3.
Let H be a hyperalgebra over a perfect field F . If char F > , assume also that H is reduced. Let G be a group acting on H by bialgebraautomorphisms. Then the smash product H F G is PI if and only if (1) H is commutative, (2) there exist normal subgroups G ⊂ G ⊂ G such that G/G is finite, G /G is abelian, and G is a finite p -group if char F = p > and trivial if char F = 0 , (3) G acts trivially on H . In [14], the author proves that the conjecture is true if H = H gr where thelatter algebra is the associated graded algebra for H with respect to the coradicalfiltration on H . 4. Braided Hopf algebras
Let T be an abelian group and let β : T × T → F × be a skew-symmetricbicharacter, sometimes called “color”. A T -graded algebra L = L t ∈ T L t with acommutator product ( x, y ) → [ x, y ] is called a β -Lie superalgebra , or a color Liesuperalgebra , if the following hold for any x ∈ L t , y ∈ L u , z ∈ L v :[ x, y ] = β ( t, u )[ y, x ] ( color anticommutativity ) , (4.1) [[ x, y ] , z ] = [ x, [ y, z ]] − β ( t, u )[ y, [ x, z ]] ( color Jacobi identity ).(4.2)A T -graded associative algebra A = L t ∈ T A t becomes a color Lie superalgebra ifone sets(4.3) [ x, y ] β = xy − β ( t, u ) yx, for any homogeneous x ∈ A t , y ∈ A u . Given a color (or β -) Lie superalgebra L , the (universal) enveloping algebra U ( L ) for L is defined as follows. First of all, U ( L ) is a T -graded associative alge-bra, generated by L , which is a β -subsuperalgebra of U ( L ) (that is, closed under BAHTURIN AND WITHERSPOON β -commutator (4.3)). Second, given a T -graded associative algebra A and any ho-momorphism of β -superalgebras ϕ : L → A , there exists a (unique) homomorphismof associative algebras f : U ( L ) → A whose restriction to L is ϕ .Given an alternating bicharacter β : T × T → F × , for any t ∈ T , we must haveeither β ( t, t ) = 1 and then t is called even , or β ( t, t ) = −
1, and then t is called odd .The set of all even elements in T is a subgroup T + , the set of odd elements is acoset of T + , which we denote by T − . Respectively, if V = L t ∈ T V t is a T -gradedspace then we write V + = L t ∈ T + V t and V − = L t ∈ T − V t . If L is a β -superalgebrawith L = L + , then we call L a β -Lie algebra.A theorem from [6] says that in the case of characteristic zero, if L is a β -Liealgebra, then U ( L ) is PI if and only if L is abelian. If L is a general β -superalgebra L = L + ⊕ L − , then U ( L ) is PI if and only if there exists a homogeneous L + -submodule M ⊂ L − such that dim L − /M, dim[ L + , M ] < ∞ , and [ L + , L + ] =[ M, M ] = 0.If L is a β -Lie superalgebra , then L is a Yetter-Drinfeld module over F T , that is, L ∈ F T F T YD , where the action of T on L is given by t ∗ x u = β ( t, u ) x u . The same istrue for U ( L ). The smash product U ( L ) F T becomes a Hopf algebra, called the bosonization of U ( L ).A generalization of such algebras is the smash product of the form U ( L ) F G ,where G is a group (not necessarily abelian) acting on L (hence on U ( L )) by T -graded automorphisms: g ∗ L t = L t , for any g ∈ G and any t ∈ T .There are no papers where the polynomial identities of such bosonizations for β -Lie superalgebras have been examined. But there is a result of M. Kochetov[13], where char( F ) = 0, L is an ordinary Lie superalgebra, L = L + L , and G is an arbitrary group (not just G = T ). Kochetov shows that H = U ( L ) F G is aPI-algebra if and only if there exist L - and G -invariant subspaces N ⊂ M ⊂ L and an abelian subgroup of finite index A ⊂ G such that dim L /M, dim N < ∞ ,[ L , L ] = [ M, M ] = 0, [ L , M ] ⊂ N , and A acts trivially on L and M/N .The following result, although quite expected, is nevetherless new.
Theorem 4.4.
Let F be a field of characteristic 0. Let β : T × T → F × bean alternating bicharacter on a finite abelian group T , L a β -Lie superalgebra,with finite support, G a group acting on L by T -graded automorphisms, U ( L ) theenveloping algebra for L , F G the group algebra for G . Then U ( L ) F G is a PI-algebra if and only if the following hold. (1) L + is abelian; (2) There is an abelian subgroup A of finite index in G which acts trivially on L + ; (3) There is a G -invariant L + -submodule M in L − such that [ M, M ] = 0 and dim[ L + , M ] < ∞ ; (4) A acts trivially on M/ [ L + , M ] . We give a proof of this result below in Section 9.If F is a field of positive characteristic and L is a color Lie superalgebra or colorLie p -superalgebra over F , the conditions under which U ( L ) F G or u ( L ) F G is aPI-algebra are not known. ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 7 Techniques
Group-theoretical prelude.
If a group G has a subgroup A of finite index m whose commutator subgroup [ A, A ] is either or of order p k in case char F = p > R = F G satisfies a nontrivial polynomial identity. Thereason is that in this case R can be viewed as a free left S -module of rank m , where S = F A . The right regular action of R on itself imbeds R in the matrix algebra M m ( S ) ∼ = M m ( F ) ⊗ S . By Regev’s Theorem [26], stated here as Theorem 2.2, thetensor product of two PI-algebras is PI. If A is abelian, then S is commutative([ x, y ] = 0 holds in S ), hence R is PI. If A is p -abelian, that is, [ A, A ] = p k andchar F = p >
0, then one easily shows that [ x, y ] p k = 0 is satisfied in S . Again, R is PI.The above condition on G is also necessary for F G to satisfy a nontrivial poly-nomial identity but proving this needs much more sophistication.If G is a group then one defines the Delta set ∆( G ) to be the characteristicsubgroup consisting of the elements having only finitely many conjugates. In otherwords, ∆( G ) is the union of the set of all finite congugacy classes. A group G with G = ∆( G ) is called an FC-group . Given a natural number k , the Delta set ∆ k ( G )is the set of all elements in G whose conjugacy classes have at most k elements. Animportant theorem of B.H. Neumann and J. Wiegold [28] says that if k is a naturalnumber and G is a group whose commutator subgroup [ G, G ] is finite of order k then G = ∆ k ( G ); conversely, if G = ∆ k ( G ) then | [ G, G ] | < ( k ) k .If a group G has an abelian (normal) subgroup A of finite index then A ⊂ ∆( G )so that ∆( G ) has finite index in G . In [27] using Posner’s Theorem 2.8, M. Smith hasshown that if F G is a prime group ring satisfying a nontrivial polynomial identitythen G has an abelian subgroup of finite index. In [21] D. S. Passman has shown,without any conditions on the primeness of F G , that if F G satisfies a polynomialidentity of degree n and k = ( n !) then the set ∆ k ( G ) has index in G not exceeding( k + 1)!. So if F G is PI, then the existence of a subgroup of finite index with finitecommutator subgroup follows.Passman also produced an example in characteristic p > Lie algebras.
A special feature of Lie algebras, compared with groups, is that,in the case char F = 0, every nonabelian Lie algebra L has an infinite-dimensionalirreducible representation ρ : L → End V . This extends to an irreducible represen-tation of U ( L ). Using Kaplansky’s Theorem 2.7, we can conclude that U ( L ) is aPI-algebra in the case char F = 0 if and only if L is abelian.Another feature is that the enveloping algebra U ( L ) of a Lie algebra L overany field has no zero divisors. In particular, U ( L ) is always a prime algebra. SoPosner’s Theorem 2.8 applies. This theorem works not only in the case of envelopingalgebras of Lie algebras but also in other situations, such as enveloping algebrasof color Lie algebras or some smash products, which can be viewed as skew grouprings of groups with coefficients in an algebra. An important paper related to thisapproach is [10].In the case of enveloping algebras of Lie algebras over a field of characteristic p >
0, see [4], where Posner’s Theorem 2.8 was used in combination with thetechnique of Delta sets, as defined below.
BAHTURIN AND WITHERSPOON
Let L be a Lie algebra over the field F and let U ( L ) denote its universal envelopingalgebra. In case char F = p >
0, assume that L is restricted and u ( L ) is its restrictedenveloping algebra. Then U ( L ), u ( L ), and the group ring F [ G ], for a group G , areall Hopf algebras and hence are similar in many ways. In particular, since questionson group algebras have been solved using ∆-methods, it was therefore reasonableto try to find similar techniques in the Lie context. To this end one considers(5.1) ∆ = ∆( L ) = { x ∈ L | dim F [ x, L ] < ∞} , the (restricted) Lie ideal of L introduced in [4]. In the same way, as in the case ofgroups, we have Delta sets∆ n ( L ) = { x ∈ L | dim F [ x, L ] ≤ n } . Clearly, ∆( L ) is an ideal in L . The sets ∆ n ( L ) are not ideals. Still, one says that∆ n ( L ) has codimension m in L if there is an m -dimensional subspace V in L suchthat L = ∆ n ( L ) ⊕ V and m is the minimal number with this property.It was shown in [4] that if some ∆ n ( L ) is of codimension m in L , then L hasa subalgebra M such that L/M and [
M, M ] are finite-dimensional. Here in placeof the B. H. Neumann - Wiegold theorem in the case of groups one uses a generalresult about bilinear maps due to P. M. Neumann [19]:
Theorem 5.2. If f : U × V → W is a bilinear map such that for each u ∈ U one has dim F f ( u, V ) ≤ m and for each v ∈ V one has dim F f ( U, v ) ≤ n then dim F f ( U, V ) ≤ mn . Lie superalgebras and smash products.
We conclude that if a Delta set fora group or Lie algebra is of finite index or codimension then this strongly affectsthe structure of the group algebra of the group or (restricted) enveloping algebraof the Lie algebra in question. This remains true when one considers color Liesuperalgebras. However, in the case of a color Lie superalgebra L = L t ∈ T L t ,defined by an alternating bicharacter β : T × T → F × , one has to consider “graded”Delta sets as follows. Definition 5.3.
For any t, u ∈ T , m ∈ N one sets (1) ∆ mt,u ( L ) = { x ∈ L t | dim [ x, L u ] ≤ m } ; (2) ∆ mt ( L ) = T u ∈ T ∆ mt,u ( L ) ; (3) ∆ t ( L ) = S m ∈ N ∆ mt ( L ) ; (4) ∆( L ) = L t ∈ T ∆ t ( L ) . Let q ∈ F and consider a q -commutator [ a, b ] q = ab − qba . An important resultabout PI-algebras is the following (see [6, Theorem 4.2.3]). Theorem 5.4.
Let L = L t ∈ T L t be a color (restricted) Lie superalgebra over anarbitrary field F and fix t, u ∈ T . Suppose that in U ( L ) or u ( L ) some nontrivial q -polynomial f ( x , . . . , x n , y , . . . , y n ) where q = β ( u, t ) , satisfies f ( x , . . . , x n , y , . . . , y n ) = X σ ∈ Sym( n ) λ σ [ x , y σ (1) ] q · · · [ x n , y σ ( n ) ] q = 0 , λ σ ∈ F , for any x , . . . , x n ∈ L u , y , . . . , y n ∈ L t . Then any n elements in L t are linearlydependent modulo ∆ mt,u ( L ) . Repeated application of this theorem together with Theorem 5.2 above on bi-linear maps allows one to obtain an L + -submodule M ⊂ L − , with the desired ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 9 properties in the case where U ( L ) satisfies a polynomial identity and char F = 0(see Theorem 4.4 as well as its proof in the last section). In this case also U ( L + )is a prime algebra and so Posner’s Theorem 2.8 applies, which easily implies that L + must be abelian.Also, using Theorem 2.5 in the case of U ( L ) F G , with x , . . . , x n ∈ G togetherwith additional Delta sets allows one to prove in Theorem 4.4 the condition on theexistence of the abelian subgroup A . Delta sets in Hopf algebras.
The definition of Delta sets in general Hopf alge-bras was given in [8], as follows.Let H be a Hopf algebra over a field F of arbitrary characteristic. Let us definethe “Delta set” H fin by setting(5.5) H fin = { x ∈ H | dim F (ad H )( x ) < ∞} . Since the adjoint action is a measuring, H fin is an F -subalgebra of H , but notnecessarily a Hopf subalgebra.If T is a subset of H which generates H as an F -algebra then x ∈ H fin if andonly if x is contained in an ad T -stable finite-dimensional subspace of H .The Delta sets H fin are fairly closely related to the Delta sets defined earlier, forgroups and Lie algebras. Proposition 5.6 (J. Bergen - D.S. Passman [8]) . The following are true: (1) If H = F G is a group algebra of a group G then H fin = F ∆( G ) ; (2) If L is a restricted Lie algebra over a field F with char F = p > and H = u ( L ) its restricted enveloping algebra then H fin = u (∆( L )) ; (3) Let L be a Lie algebra over a field F with char F = 0 , H = U ( L ) itsenveloping algebra and ∆ L the join of all finite-dimensional ideals of L .Then H fin = U (∆ L ) . The authors of [8] mention that U ( L ) fin can be appreciably smaller than U (∆( L )).An explicit example is given in the case char F = p > A H ,where A is an H -module algebra, H a Hopf algebra, under the condition that A is generated by an H -submodule V , which generates A as a unital algebra. Thisrestriction is natural in all cases of pointed Hopf algebras considered here. He sets δ mH ( V ) = { v ∈ V | dim( H · v ) ≤ m } , δ H ( V ) = [ m δ mH ( V ) ,δ mV ( H ) = { h ∈ H | dim( h · V ) ≤ m } , δ V ( H ) = [ m δ mV ( H ) . Some properties of these Delta sets are similar to those of Delta sets in the caseof group algebras, enveloping algebras and their smash products. For instance,(1) For any α, β ∈ F , x ∈ δ iH ( V ), y ∈ δ j ( V ), one has αx + βy ∈ δ i + jH ( V );(2) All δ mH ( V ) are H -invariant sets;(3) If ϕ : V → V is a homomorphism of H -modules then ϕ ( δ mH ( V )) ⊂ δ mH ( V );(4) If α, β ∈ F , h ∈ δ iV ( H ), k ∈ δ jV ( H ) then αh + βk ∈ δ i + jV ( H );(5) All δ mV ( H ) are invariant under the left and the right multiplication by theelements of H .As before, δ V ( H ) is a two-sided ideal in H but not necessarily a Hopf ideal; δ H ( V )is an H -submodule in V . Since Delta sets do not need to be subspaces, we give the following definition.Suppose that W is a subset in a vector space V . We say that W has finite codimen-sion in V if there exist v , . . . , v m ∈ V such that V = W + Span { v , . . . , v m } . If m is the minimum possible integer with such property then we set dim V /W = m .We also introduce the notation m · W = { w + · · · + w m | w i ∈ W } , m ∈ N . Thefollowing is [7, Lemma 6.3]. Lemma 5.7.
Let L be a vector space. Suppose that a subset W is stable undermultiplication by scalars and such that dim F L/W ≤ m . Then Span F W = 4 m · W . Using the generating set V for A one can consider the associated graded algebragr A defined by the degree filtration { A m } ∞ m =0 of A with respect to V . Here A = F m >
0, we have A m = A m − + V m . Thengr A = M gr m A, where gr m A = A m /A m − . All information about the action of H on A is contained in the action of H on gr A .Let us denote by S ( V ) the symmetric algebra of V , Λ( V ) the Grassman algebra of V and, if char F = p > V [ p ] will stand for the subspace in S ( V ) spanned by all v p ,where v ∈ V . We quote the following result of M. Kochetov [13, Proposition 2.4]. Theorem 5.8.
Let A be an H -module algebra generated by an H -submodule V asa unital algebra. Assume that the associated graded algebra gr A is isomorphic toone of S ( V ) , Λ( V ) or S ( V ) / ideal( V [ p ] ) . If the polynomial identity (2.6) holds forany h , . . . , h n ∈ H and y , . . . , y n ∈ V then (1) dim V /δ n H ( V ) < n ; (2) dim H/δ n V ( H ) < n . If H = F G is a group algebra and A = U ( L ) where G is a group and L is a colorLie superalgebra then this theorem can be used to obtain Theorem 4.4. Details arein the last section, making the observation that the theorem generalizes to colorLie superalgebras A .6. Identities and Delta sets in Radford’s algebra F ( q ) A very particular case of algebras described in the previous section are the Hopfalgebras F ( q ) introduced by D. Radford [25]. The algebra H = F ( q ) , q ∈ F , isgenerated by one group-like element a and one skew-primitive element x such that(6.1) ax = qxa. The algebras F ( q ) are standard 2-generator subalgebras in the pointed Hopf alge-bra U ( D , λ ) introduced in the next section. In this section we study polynomialidentities and Delta sets in F ( q ) . Polynomial identities in F ( q ) .Theorem 6.2. The algebra H = F ( q ) satisfies a nontrivial polynomial identity ifand only if q is a root of 1. If q is a primitive n th root of unity, then H = F ( q ) satisfies the standard identity S n ( x , . . . , x n ) = X σ ∈ Sym(2 n ) sgn( σ ) x σ (1) · · · x σ (2 n ) = 0 , and n is the minimum degree of identities in F ( q ) . ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 11
Proof.
In [6, Section 4.1.6], the authors construct an irreducible module for a colorpolynomial algebra in two variables. Similarly, one should consider here a vectorspace V with basis { v i | i ∈ Z } . We will make V an F ( q ) -module by setting(6.3) a ◦ v i = v i − , x ◦ v i = q i v i +1 for any i ∈ Z . Proposition 6.4. If q is not a root of 1, then V is an irreducible F ( q ) − modulewhose centralizer is F id V .Proof. Let W be a nonzero submodule in V . If some v i is in W then clearly W = V .Otherwise, let us take a nonzero element w in W , which has the form w = m X i =0 µ i v t i , where 0 = µ i ∈ F , m ≥ , t i = t j , if i = j. We apply induction on m starting with m = 0 to show that W = V . Indeed, let usconsider ( ax ) ◦ w = m X i =0 q t i µ i v t i for all 0 ≤ s ≤ m. Subtracting this element from q t m w , we obtain q t m w − ( ax ) ◦ w = m − X i =0 µ i ( q t m − q t i ) v t i ∈ W. Since all coefficients in this new element are nonzero, induction applies and hence V is irreducible. If ϕ is in the centralizer of this module then ϕ ( a ◦ v i ) = a ◦ ( ϕ ( v i ) and ϕ ( x ◦ v i ) = x ◦ ( ϕ ( v i ) for all i ∈ Z . If ϕ ( v i ) = P j ∈ Z t ji v j then we have, for all i, j , that t ji = t j − i − and q j t ji = q i t j +1 i +1 .Since q j − i = 1 if and only if j = i we have ϕ ( v i ) = t ii v i . But also all t ii are equal,and so ϕ = λ id V , for some λ ∈ F . Thus the centralizer of F ( q ) -module V is F id V ,as claimed. (cid:3) Note that, in a quite similar way to the proof of Proposition 6.4, if q is a primitive n th root of unity, one can construct an irreducible module V n for F ( q ) , whosedimension is n . One simply has to take V n = h v i | i ∈ Z n i and use the same formulas(6.3), but view addition modulo n . The same proof shows that the following is true. Proposition 6.5. If q is a primitive n th root of 1, then V n is an irreducible F ( q ) -module of dimension n . The centralizer of V n is F id V n . ✷ Continuing with the proof of the theorem, let us assume that q is not root of1. Since H = F ( q ) has an irreducible module of infinite dimension, by the DensityTheorem, for any n , a homomorphic image of F ( q ) contains a subalgebra isomorphicto the matrix algebra M n ( F ) of order n . By Amitsur-Levitzky’s Theorem, theminimum degree of identities satisfied by M n ( F ) equals 2 n . As a result, F ( q ) is nota PI-algebra.If q is a primitive n th root of 1, then it follows from (6.1) that both a n and x n arecentral elements in H = F ( q ) . Moreover, it follows from [25] (PBW-basis for F ( q ) )that H is a free left module of rank n over the commutative subalgebra K = F [ a, x n ].(One could also take K = F [ a n , x ], etc.) Let r u denote right multiplication in H by u ∈ F ( q ) , an endomorphism of the free left K -module H . Since 1 · r u = u and r u u = r u r u , the mapping u r u is an injective homomorphism of H toEnd K ( H ). Thus H embeds in the matrix algebra M n ( K ) and so satisfies a standardpolynomial identity of degree 2 n .On the other hand, since H has an irreducible module V n , by the Density Theo-rem its homomorphic image is isomorphic to M n ( F ), hence by Amitsur-Levitzky’sTheorem F ( q ) cannot satisfy an identity of degree less than 2 n . (cid:3) Corollary 6.6. If q is a primitive n th root of 1, then F ( q ) satisfies an identity ofdegree n but no identity of degree less than n . Delta sets in F ( q ) . Let F be an arbitrary field, q ∈ F × . Let H = F ( q ) as above(see [25, § x ) = x ⊗ a + 1 ⊗ x and S ( x ) = − xa − . Recall the definition of Delta sets from Section 5, under the adjoint action of H on itself: δ iH ( H ) = { h ∈ H | dim F (ad H )( h ) ≤ i } and H fin = ∞ [ i =0 δ iH ( H ) = { h ∈ H | dim F (ad H )( h ) < ∞} . We will prove that H fin = H if, and only if, q is a root of unity, and that if q is aprimitive n th root of unity, then δ nH ( H ) = H . First we need a calculational lemma. Lemma 6.7.
For all integers i ≥ and all integers j , (i) (ad x i )( a j ) = i − Y l =0 (1 − q j − l ) x i a j − i and (ad a i )( a j ) = a j , (ii) (ad x i )( x j ) = 0 and (ad a i )( x j ) = q ij x j .Proof. We will prove the first formula in part (i) by induction on i . First note that(ad x )( a j ) = xa j a − + a j S ( x ) = xa j − − a j xa − = (1 − q j ) xa j − . Next assume that (ad x i − )( a j ) = i − Y l =0 (1 − q j − l ) x i a j − i for some i ≥
2. Then(ad x i )( a j ) = (ad x )(ad x i − )( a j )= i − Y l =0 (1 − q j − l ) xx i − a j − i +1 a − − x i − a j − i +1 xa − )= i − Y l =0 (1 − q j − l )( x i a j − i ) . Therefore the first formula in part (i) holds. Clearly the second formula also holds.Note that (ad x )( x j ) = xx j a − − x j xa − = 0. Thus the first formula in part (ii)holds. The second is straightforward. (cid:3) Theorem 6.8.
Let H = F ( q ) . Then H fin = H if, and only if, q is a root of unity.Moreover, if q is a primitive n th root of unity, then δ nH ( H ) = H . ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 13
Proof.
Suppose q is a primitive n th root of unity. Then by Lemma 6.7, for each i, j , (ad x n + i )( a j ) = 0 . Therefore (ad x n + i a i ′ )( a j ) = 0 for all i, i ′ , and similarly for any linear combi-nation of elements of the form x n + i a i ′ . It now follows from Lemma 6.7 thatdim k ((ad H )( a j )) = n , and so a j ∈ δ n ( H ). Further, dim k ((ad H )( x j )) = 1, so x j ∈ δ ( H ). Consequently, dim k ((ad H )( x i a j )) = n for all i ≥ j ≥
1, sincethe adjoint action is multiplicative, so x i a j ∈ δ n ( H ). Since the adjoint action is k -linear, it now follows that δ nH ( H ) = H fin = H .Now suppose that q is not a root of unity. By Lemma 6.7, for all i ,(ad x i )( a ) = i − Y l =0 (1 − q − l ) x i a − i . It follows that a H fin . Therefore H fin = H . (cid:3) The pointed Hopf algebras U ( D , λ ) of Andruskiewitsch - Schneider In this section we consider some infinite dimensional pointed Hopf algebras in-troduced by Andruskiewitsch and Schneider in their work on classification of finitedimensional pointed Hopf algebras [2]. Our algebras are somewhat more general,with fewer restrictions on the groups of grouplike elements.Let F be a field of characteristic 0. Let Γ be a group and ˆΓ its group of characters,that is group homomorphisms to F × , with identity element denoted ε . Let θ bea positive integer and ( a ij ) ≤ i,j ≤ θ a Cartan matrix of finite type. For each i ,1 ≤ i ≤ θ , let g i be in the center of Γ and χ i ∈ ˆΓ such that χ i ( g i ) = 1 and χ j ( g i ) χ i ( g j ) = χ i ( g i ) a ij for all i, j . Let q ij = χ j ( g i ). In case Γ is finite abelian, we call(7.1) D = (Γ , ( g i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , ( a ij ) ≤ i,j ≤ θ )a datum of finite Cartan type .Let V be the vector space with basis x , . . . , x θ . Define an action of Γ on V by g ( x i ) = χ i ( g ) x i and a coaction by δ ( x i ) = g i ⊗ x i for all i . Then V is a Yetter-Drinfeld moduleover F Γ. The tensor algebra T ( V ) is a braided Hopf algebra in the Yetter-Drinfeldcategory F Γ F Γ YD, where the braided coproduct takes x i to x i ⊗ ⊗ x i for all i .Define braided commutators:ad c ( x i )( y ) = x i y − g i ( y ) x i for all y ∈ T ( V ). Choose scalars λ ij , 1 ≤ i, j ≤ θ , i j (i.e. i, j not in the sameconnected component of the Dynkin diagram) for which λ ij = 0 if g i g j = 1 or χ i χ j = ε . Let H = U ( D , λ ) be the Hopf algebra defined as in [2] to be the quotientof the smash product T ( V ) k Γ by relations (Serre) (ad c x i ) − a ij ( x j ) = 0 ( i = j, i ∼ j ) , (linking) (ad c x i )( x j ) = λ ij (1 − g i g j ) ( i < j, i j ) . The coalgebra structure on U ( D , λ ) is given by∆( g ) = g ⊗ g, ∆( x i ) = x i ⊗ g i ⊗ x i , ε ( g ) = 1, ε ( x i ) = 0, S ( g ) = g − , and S ( x i ) = − g − i x i for all g ∈ Γ, 1 ≤ i ≤ θ . Thuswe see that ad c as defined above agrees with the adjoint action of the Hopf algebra H = U ( D , λ ) on itself.Using the root system Φ associated with the Cartan matrix ( a ij ), one can buildfinitely many elements x β , β ∈ Φ + , as braided commutators in the algebra T ( V ). IfΓ is a finite abelian group, then by [2, Theorem 3.3], H has a PBW basis determinedby the root vectors corresponding to positive roots. We use this theorem to establishthe following. Theorem 7.2.
For arbitrary Γ , if all χ i ( g j ) are roots of unity, then (1) H has a PBW basis consisting of elements x a β x a β · · · x a p β p g, where a i ≥ , g ∈ Γ . (2) For any α, β ∈ Φ + , there is a positive integer N β and a scalar q αβ (givenby products of values of χ i ( g j ) ) such that the elements x N β β generate a Hopfideal and x α x N β β = q Nαβ x N β β x α . Moreover, a power (at least N β ) of x β commutes with any other element.Proof. Denote Γ χ = θ \ i =1 Ker( χ i ) . Let us consider the datum D given by(7.3) D = (Γ , ( g i ) ≤ i ≤ θ , ( χ i ) ≤ i ≤ θ , ( a ij ) ≤ i,j ≤ θ )where Γ is replaced by Γ = Γ / Γ χ , g = g Γ χ , χ i ( g ) = χ i ( g ). By the hypotheses andby its definition, Γ is necessarily finite abelian. One checks that this datum givesrise to an Andruskiewitsch-Schneider Hopf algebra H = U ( D , λ ), where the vectorspace is again V but with the natural Yetter-Drinfeld module structure over F Γ.Since Γ is finite abelian, we can invoke [2, Theorem 3.3], according to which H isof finite codimension over its braided center. More precisely, H has a PBW basis x a β x a β · · · x a p β p g, where a i ≥ , g ∈ Γ . Also [2, Theorem 3.3] ensures existence of the numbers N β such that x N β β is in thebraided center of H . In addition, since all χ i ( g j ) are necessarily of finite order,for each β there is some integer (at least N β ) such that x β raised to this powercommutes with any other element.Considering the defining relations for H and H , one easily checks that there isa “natural” homomorphism ϕ : H → H such that ϕ ( u g ) = u g . Using thishomomorphism, the PBW-basis of H and the numbers mentioned above for H , onefinds that both statements of our theorem are true. (cid:3) We give several small examples to illustrate the ubiquity of the Hopf algebras U ( D , λ ). Their polynomial identities and delta sets will be taken up in the nextsection. ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 15
Example 7.4. ( U q ( sl ) ≥ ) Let ℓ be an odd positive integer and Γ = Z /ℓ Z × Z /ℓ Z (respectively Γ = Z × Z ), with generators g , g . Let q be a primitive ℓ th root ofunity in C (respectively, let q be any element of C other than 0 , , − A Cartan matrix, (cid:18) − − (cid:19) Define characters χ , χ by χ ( g ) = q , χ ( g ) = q − ,χ ( g ) = q − , χ ( g ) = q . Then U ( D , ∼ = R C Γ where R is the algebra generated by x , x with relations x x = q − x x , x x = q − x x , where x := [ x , x ] c = x x − q − x x . The algebra R has PBW basis { x a x b x c | a, b, c ≥ } . In case Γ is infinite, this is the quantum group U q ( sl ) ≥ . In caseΓ is finite, a quotient by the ideal generated by powers of x , x , x is the smallquantum group u q ( sl ) ≥ . Example 7.5. ( U q ( sl )) Let ℓ be an odd positive integer and Γ = Z /ℓ Z (respec-tively Γ = Z ), with generator g . Let q be a primitive ℓ th root of unity in C (respectively, let q be any element of C other than 0 , , − A × A Cartan matrix, (cid:18) (cid:19)
Let g = g = g and define characters χ , χ by χ ( g ) = q − , χ ( g ) = q . Let λ = qq − q − . Then U ( D , λ ) is the quotient of the smash product T ( V ) C Γ, where V is a vector space with basis x , x , by the ideal generated by x x − qx x − λ (1 − g ) . In case Γ is infinite, this is isomorphic to U q ( sl ). (An isomorphism is given by x E , x K − F , g K − .) In case Γ is finite, a quotient by the idealgenerated by powers of x , x is the small quantum group u q ( sl ). Example 7.6. (Generalization of F ( q ) to more generators [1]) Let Γ = Z ×· · ·× Z ( n copies) with generators g , . . . , g n . Consider the type A × · · · × A Cartan matrix.Choose nonzero scalars q ij ( i ≤ j ). Define characters χ , . . . , χ n by χ i ( g j ) = q ji for i ≤ j and χ i ( g j ) = q − ij for i > j . Let λ ij = 0 for all i, j . Then U ( D , ∼ = R k Γwhere R is the algebra generated by x , . . . , x n with relations x i x j = q ij x j x i . The quotient by the ideal generated by all x N i i (where N i = o ( q ii )) is the quantumlinear space of [1].8. Identical relations and Delta sets in U ( D , λ )In this section, we determine necessary and sufficient conditions for U ( D , λ ),defined in Section 7, to satisfy a polynomial identity, or to be equal to its delta set. Identities in U ( D , λ ) .Theorem 8.1. An algebra H = U ( D , λ ) satisfies a nontrivial polynomial identityif and only if (1) Γ has an abelian normal subgroup e Γ of finite index, which contains thecenter of Γ , hence the elements g , . . . , g θ ; (2) the orders of all characters χ , . . . , χ θ are finite.Proof. Assume that H satisfies a nontrivial polynomial identity. Since C Γ is asubalgebra in H , one can use D. S. Passman’s Theorem [21] to conclude that such e Γ of finite index in Γ exists. So Condition (1) must hold. Now if the order of some χ i is infinite then for any m there is g ∈ Γ such that the order of q = χ i ( g ) isgreater than m . In this case, H contains a subalgebra isomorphic to F ( q ) such thatby Corollary 6.6 this subalgebra does not satisfy a polynomial identity of degreeless than 2 m . As a result, H does not satisfy a nontrivial polynomial identity. Thiscontradiction shows that Condition (2) must also hold.Now assume that both conditions are satisfied. By Theorem 7.2, H is a freeright module of finite type over a subalgebra K generated by { x N β β | β ∈ Φ + } , andΓ χ . At the same time C Γ χ is a free right module of finite type over a commutativesubalgebra C (Γ χ ∩ e Γ). If S is the subalgebra generated by { x N β β | β ∈ Φ + } andΓ χ ∩ e Γ, then S is commutative and H is a free right module of finite type over thecommutative subalgebra S . It follows that H is a PI-algebra. (cid:3) Delta sets in U ( D , λ ) . Recall from Section 5 that a group G is an F C -group ifall conjugacy classes in G are finite. In other words, ∆( G ) = G . Theorem 8.2.
Let H = U ( D , λ ) . Then H fin = H if and only (1) Γ is an FC-group; (2) the orders of all characters χ , . . . , χ θ are finite.Proof. Assume our Conditions (1) and (2) hold. Then all q ij are roots of unity. ByTheorem 7.2, there are positive integers M i such that the elements x M β , . . . , x M p β p are central in U ( D , λ ), and ad x M β , . . . ad x M p β p are zero operators. Since for each g ∈ G , ad g applied to any PBW basis element x a β · · · x a p β p is multiplication by aroot of unity, under conditions (1) and (2), { ad( x b β · · · x b p β p g )( h ) | b , . . . , b p ≥ g ∈ G } is a finite set for each h ∈ H . That is, H fin = H .Now assume H fin = H . Then also ( C G ) fin = C G . By Proposition 5.6, we havethat ( C G ) fin = C ∆( G ). So we must have C ∆( G ) = C G . It follows that G = ∆( G ),so Condition (1) is satisfied.If condition (2) does not hold, then as we saw in the proof of Theorem 8.1, forany m , H has a subalgebra isomorphic to F ( q ) where q has order greater than m .By Lemma 6.7(i), there exists an element of H that is not in δ mH ( H ). Therefore H fin = H . (cid:3) ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 17 Proof of Theorem 4.4
Sufficiency.
Let us first prove that the conditions of Theorem 4.4 guarantee thatan algebra R = U ( L ) F G is a PI-algebra. For this, we note that L + ⊕ M is a G -invariant ideal in L . Let us consider in R the subalgebra R = U ( L + ⊕ M ) F A. It follows from the PBW Theorem for color Lie superalgebras (see [6, Chapter 3,Theorem 2.2] that R is a left R -module generated by the elements of the form(9.1) x i · · · x i s g j , i < · · · < i s , s = 1 , , . . . , where { x i } is a basis of L − modulo M and { g j } is a right transversal for A in G .Since the number of elements of the form (9.1) is finite, it is sufficient to prove that R is a PI-algebra.Let us consider a two-sided ideal I of R generated by N = [ L + , M ]. Since N isan A -invariant ideal in the superalgebra L + ⊕ M , we can see that I = R N = N R . By the PBW Theorem, N dim N +1 = { } , so that I dim N +1 = { } , showing that I isa nilpotent ideal. Moreover, R /I ∼ = U ( L + ⊕ M/N ) F A ∼ = U ( L + ⊕ M/N ) ⊗ F A since the actions of A and of L + on M/N are trivial. Now L + ⊕ ( M/N ) satisfiesthe conditions of [6, Chapter 4, Theorem 1.2] and so U ( L + ⊕ M/N ) is a PI-algebra.Since F A is commutative, R is indeed a PI-algebra. Necessity. If U ( L ) F G is PI, then also U ( L ) and F G are PI and so we have anabelian subgroup A of finite index in G and an L + -submodule M in L − , satisfyingthe following conditions:(9.2) [ L + , L + ] = 0 , [ M, M ] = 0 , L − /M and [ L + , M ] are finite-dimensional.Now we need to consider the action of G on L . Without loss of generality, we mayassume that G is abelian and [ L − , L − ] = 0.First we look at U ( L + ) F G . Let us denote by p ∗ x the result of the action of p ∈ F G on x ∈ U ( L + ). The existence of a subgroup of finite index in G , whichtrivially acts on L + , can be recovered from the proof of [7, Theorem 2.3]. In thatproof just a couple of lines need changing. First of all, instead of a polynomialring U ( L ) we need to deal with a color polynomial ring U ( L + ), where L + is anabelian color Lie algebra. This is a T -graded vector space and xy = β ( t, u ) yx for homogeneous elements x of degree t and y of degree u . Now G acts by T -graded automorphisms and so for any p, q ∈ F G , p ∗ x is still homogeneous ofdegree t and q ∗ y is still homogeneous of degree u . Now U ( L + ) is still an integralring but β -commutative, not commutative. The only place in the the proof inquestion where the commutativity is used in that paper is on p. 374, after equation(4). In the lines of that paper that follow, one needs to show that if, for some p i , q i ∈ F G, x, y ∈ U ( L + ) one has( p ∗ x )( q ∗ y ) + · · · + ( p m ∗ x )( q m ∗ y ) = 0 , then one also has ( q ∗ x )( p ∗ y ) + · · · + ( q m ∗ x )( p m ∗ y ) = 0 . Choosing x, y homogeneous of degree t, u , respectively and using β -commutativityproves that this can be done in the case of U ( L + ). The triviality of the action ofa subgroup of finite index on all homogeneous elements of L + implies the trivialityof its action on the whole of L + .Thus we may assume that we deal with the smash products U ( L ) F G where G is abelian acting trivially on the whole of L + . Now we proceed to the studyof the action of G on L − . Since U ( L − ) F G ⊂ U ( L − ) F G , we may assume that U ( L − ) F G is a PI-algebra and so there is a nontrivial identical relation for theaction of G on U ( L − ) such that L − is a G -invariant subspace. Since [ L − , L − ] = 0,we know from the PBW Theorem for color Lie superalgebras (see [6, Chapter 3,Theorem 2.2] that U ( L ) is a color Grassmann algebra Λ β ( L − ), where β : T × T → F × is an alternating bicharacter such that β ( t, t ) = − t = 1 in T . We willuse the proof of [13, Theorem 2.14] (where L is an ordinary superalgebra, that is, L + = L and L − = L ) to prove the following more general result. Proposition 9.3.
Let L = L + ⊕ L − be a color Lie superalgebra over a field F ofcharacteristic zero such that R = U ( L − ) F G is a PI-algebra. If there is an identityholding in R of the form (9.4) X π ∈ Sym( n ) γ π ( h ∗ X π (1) ) · · · ( h n ∗ X π ( n ) ) = 0 where h , . . . , h n ∈ F G , X , . . . , X n ∈ L − , then there is a subgroup G of finiteindex in G , and a G -invariant subspace M of finite codimension in L − and N offinite dimension, such that the action of G on M/N is trivial.
In the proof of this result Kochetov uses Theorem 5.8. In this theorem onedeals with algebras A G generating a subspace V such that the associated gradedalgebra gr A is one of S ( V ), Λ( V ) or S ( V ) / ideal( V [ p ] ). However, the proof onlyuses the PBW-bases for A in question. Since the PBW Theorem holds for color Liesuperalgebras, without any changes the proof applies to A = U ( L ), in particular tocolor Grassmann algebras.So we have the following. Proposition 9.5.
If (9.4) holds in U ( L − ) for any h , . . . , h n ∈ F G and X , . . . , X n ∈ L − , then (1) dim L − /δ n H ( L − ) < n ; (2) dim H/δ n L − ( H ) < n . Note that in the case H = F G , the first inequality enables us to obtain a G -invariant subspace M of codimension at most n in L − , which is contained in δ n n F G ( L − ). Likewise, the second inequality enables us to obtain an ideal I of codi-mension at most n which is contained in δ n n L − ( F G ). If we apply P. M. Neumann’sTheorem 5.2 to the bilinear map I × M → I ∗ M , given by ( p, w ) p ∗ w , for p ∈ I , w ∈ M , we obtain(9.6) dim( I ∗ M ) ≤ n n . Some additional work is needed to obtain the subgroup G = δ L − ( G ) of finiteindex in G and a finite-dimensional G -invariant subspace N such that the actionof G on M/N is trivial. In the case of ordinary superalgebras, this is done in [13,Proposition 2.13]. The “color” version of this proposition, adjusted to our needs,is as follows.
ELTA SETS AND POLYNOMIAL IDENTITIES IN POINTED HOPF ALGEBRAS 19
Proposition 9.7.
Let G be an abelian group acting on A = Λ β ( V ) , where β is analternating bicharacter on a finite abelian group T , V a T -graded vector space. Ifthe identity of the action of degree n (9.4) holds for any h , . . . , h n ∈ F G and any X , . . . , X n ∈ A then [ G : δ V ( G )] < d where d depends only on n . The proof of Proposition 2.13 in [13] is based on Lemmas 2.9, 2.10, 2.11 andProposition 2.12, describing the restrictions on the T -graded action of a cyclicgroup ( g ) on A = Λ β ( V ) if the identity of the action is satisfied. The proofsof the lemmas remain unchanged if all the elements of V on which g acts areassumed T -homogeneous. If two elements x, y present in the calculations are ofthe same degree t ∈ T then yx = β ( t, t ) xy = − xy and so the calculations workin the same way as in the case of the ordinary Λ( V ). The final conclusion of apreliminary Proposition 2.12 in [13] word by word translates to the color situationand provides us with the number c , depending only on n , such that, for any g ∈ G ,dim(( g c − ∗ V ) < c .Finally, the derivation of Proposition 2.13, hence of our Proposition 9.7, trans-lates to the color case without problems. One only needs to keep in mind that thespaces, algebras and the actions appearing in the proofs in [13] are T -graded.Let us denote by G the subgroup δ L − ( G ), which is of finite index thanks toProposition 9.7. Remember the ideal I appearing in (9.6). If we denote by ( F G ) + the augmentation ideal of F G , thendim( F G I ∩ ( F G ) + ≤ dim( F G ) /I ≤ n. We can then choose g , . . . , g n ∈ G such that( F G ) + = Span( g − , . . . , g n −
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Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s, NL, A1C5S7, Canada
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