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Let $p$ and $q$ be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $pq^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac-Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix., Remark 6.9 added; notation change in equation (5.7); some typos corrected and references updated. Accepted in J. Pure Appl. Algebra

We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnar's smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A. We study the category of modules of the smash coproduct., 12 pages

We give a direct combinatorial proof that a certain n-dimensional algebra A is a comodule algebra over the Taft Hopf algebra of dimension n 2.

We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dim H. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator., additional references, fixed some typos, minor additions including a questions and some remarks

In this paper we describe all gradings by a finite abelian group G G on the following Lie algebras over an algebraically closed field F F of characteristic p ≠ 2 p\neq 2 : s l n ( F ) \mathfrak {sl}_n(F) ( n n not divisible by p p ), s o n ( F ) \mathfrak {so}_n(F) ( n ≥ 5 n\geq 5 , n ≠ 8 n\neq 8 ) and s p n ( F ) \mathfrak {sp}_n(F) ( n ≥ 6 n\geq 6 , n n even).

Assume that H is a finite-dimensional Hopf algebra over a field k and that A is an H-module algebra satisfying a polynomial identity (PI). We prove that if H is semisimple and A is H-semiprime, then A#H is semiprime. If H is cosemisimple, we show that the prime radical of A is H-stable.

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂ A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t -Lying Over (t -LO), and Going Up (GU), analogous to the classical Krull relations for prime ideals; we also defined three new “dual” Krull relations. We say that H itself is said to have one of the Krull relations if the relation holds for all faithfully flat H -Galois extensions. We showed in [MS] that H has one of the three “dual” Krull relations if and only if the dual Hopf algebra H ∗ of H has the original relation (hence the name). An important example of Hopf Galois extensions is given by Hopf crossed products A = Rσ # H . Moreover, Galois extensions can be useful in studying crossed products, since they satisfy a “transitivity” property which crossed products lack. That is, if K is a normal Hopf subalgebra of H with Hopf quotient H , then, in general, one cannot write A = Rσ # H = (Rσ K)#τ H , by an example of [S2]. Another basic example of a Hopf Galois extension is given by a Hopf algebra A with a normal Hopf subalgebra R of finite index such that A is faithfully flat over R: for then R ⊂ A is faithfully flat H -Galois.

In this paper we show that for an important class of non-trivial Hopf algebras, the Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself.

In this paper we extend a well-known theorem of M. Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.

We prove a version of the Frobenius–Schur theorem for a finite-dimensional semisimple Hopf algebra H over an algebraically closed field; if the field has characteristic p not 0, H is also assumed to be cosemisimple. Then for each irreducible representation V of H, we define a Schur indicator for V, which reduces to the classical Schur indicator when H is the group algebra of a finite group. We prove that this indicator is 0 if and only if V is not self-dual. If V is self dual, then the indicator is positive (respectively, negative) if and only if V admits a nondegenerate bilinear symmetric (resp., skew-symmetric) H-invariant form. A more general result is proved for algebras with involution.

For a finite-dimensional Hopf algebraH, we study the prime ideals in a faithfully flatH-Hopf-Galois extensionR ⊂A. One application is to quotients of Hopf algebras which arise in the theory of quantum groups at a root of 1. For the Krull relations betweenR andA, we obtain our best results whenH is semisolvable; these results generalize earlier known results for crossed products for a group action and for algebras graded by a finite group. We also show that ifH is semisimple and semisolvable, thenA is semiprime providedR isH-semiprime.

In this paper we find necessary and sufficient conditions on a finite-dimensional Lie superalgebra under which any associative PI-envelope of L L is finite-dimensional. We also extend M. Scheunert’s result which enables one to pass from color Lie superalgebras to the ordinary ones, to the case of gradings by an arbitrary abelian group.

(1999). On yetter-drinfeld categories and H-commutativity. Communications in Algebra: Vol. 27, No. 3, pp. 1321-1345.

We prove that if A A is a Noetherian Z \mathbf Z -graded algebra, then the Jacobson radical of A A is nilpotent under mild hypotheses on A 0 A_0 . We also consider affine PI-algebras graded by torsion groups. Finally we prove a Nullstellensatz-type theorem for enveloping algebras of Lie color algebras.

We consider when extensions S ⊂ R S\subset R of subalgebras of a Hopf algebra are β \beta -Frobenius, that is Frobenius of the second kind. Given a Hopf algebra H H , we show that when S ⊂ R S\subset R are Hopf algebras in the Yetter-Drinfeld category for H H , the extension is β \beta -Frobenius provided R R is finite over S S and the extension of biproducts S ⋆ H ⊂ R ⋆ H S\star H\subset R\star H is cleft. More generally we give conditions for an extension to be β \beta -Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants. We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.

We study the structure of Lie algebras in the category H MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA in H MA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain.

We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C. This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element.

We study the higher Frobenius-Schur indicators of the representations of the Drinfel'd double of a finite group G, in particular the question as to when all the indicators are integers. This turns out to be an interesting group-theoretic question. We show that many groups have this property, such as alternating and symmetric groups, PSL_2(q), M_{11}, M_{12} and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5^6 with non-integer indicators., Comment: 17 pages

In this article, we attempt to generalize the result that for a commutative ring R R the outer automorphism group of R R -automorphisms of M n ( R ) {M_n}(R) is abelian of exponent n n . It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive dimension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring.

We study actions of a Hopf algebraH on an algebraR such that the action is twisted by an invertible mapσ:H⊗H →R; the biinvertible condition means that these actions also have both an inverse and an antiinverse in Hom(H, EndR). WhenR is an ordinaryH-module algebra, the action is biinvertible if the antipose is bijective. As a new example we show that if theH-action is twisted and the coradical ofH is cocommutative, then the action is biinvertible. After studying the continuity of these actions with respect to the filter of ideals ofR with zero annihilator, we consider when the actions may be extended to the symmetric Martindale quotient ring ofR and itsH-analog. Our results can be applied to crossed productsR# σ .

We consider a Hopf Galois extension B A, for A a comodule algebra over the Hopf algebra H with coinvariant algebra B . After giving a number of examples, we discuss Galois extensions with additional properties, such as having a normal basis. We then consider when there is a category equivalence between the category of modules over B and the category of “relative Hopf modules” for A and H . Finally we discuss more recent work of van Oystaeyen and Zhang and of Schauenburg on obtaining correspondence theorems between suitable subalgebras of A and Hopf ideals of H . 16W30; 16S40, 16S34

This note first describes the basic properties of the skew derivations on the polynomial ringk[X]. As a consequence of these properties it is proved that theq-analogue of the enveloping algebra of sl(2),U q(sl(2)), has a unique action on C[X], where “action” means that C[X] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra ofU q(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it.

We study the representations and their Frobenius-Schur indicators of two semisimple Hopf algebras related to the symmetric group $S_n$, namely the bismash products $H_n = k^{C_n}# kS_{n-1}$ and its dual $J_n = k^{S_{n-1}}# kC_n = (H_n)^*,$ where $k$ is an algebraically closed field of characteristic 0. Both algebras are constructed using the standard representation of $S_n$ as a factorizable group, that is $S_n = S_{n-1}C_n = C_nS_{n-1}$. We prove that for $H_n$, the indicators of all simple modules are +1. For the dual Hopf algebra $J_n = k^{S_{n_1}}# kC_n$, the indicator can have values either 0 or 1. When $n = p$, a prime, we obtain a precise result as to which representations have indicator +1 and which ones have 0; in fact as $p \to \infty$, the proportion of simple modules with indicator 1 becomes arbitrarily small. We also prove a result about Frobenius-Schur indicators for more general bismash products $H =k^G# kF$, coming from any factorizable group of the form $L = FG$ such that $F\cong C_p.$ We use the definition of Frobenius-Schur indicators for Hopf algebras, as described in work of Linchenko and the second author, which extends the classical theorem of Frobenius and Schur, for a finite group $G.$

The Hopf order of an element $h$ of a Hopf algebra $H$ is the least $n$ such that the $n$-th Hopf power of $h$ is trivial. For some bismash product Hopf algebras obtained from factorizable groups (including Drinfeld doubles of some groups) we compute the dimensions of the spaces of elements with trivial $n$-th Hopf powers, and determine which Hopf orders of elements are possible. We discuss the behavior under twists and duality., 28 pages

In this paper we consider generalized Lie algebra structures on graded associative algebras. We are interested in the situation where such structures are solvable and even commutative. We first prove some theorems in the case of solvable structures on semiprime, prime or simple associative algebras. Then we describe finite-dimensional associative algebras over an algebraically closed field of characteristic zero graded by a finite elementary abelian group which are generalized commutative under a skew-symmetric bicharacter on the grading group.

If G is any finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group, with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgroup of the Weyl group W, then all representations of Cw(E) are defined over Q.

(1981). Prime ideals in fixed rings. Communications in Algebra: Vol. 9, No. 4, pp. 423-449.

LetL be a Lie algebra over a fieldK which acts asK-derivations on aK-algebraR. Then this action determines a crossed productR *U(L) whereU(L) is the enveloping algebra ofL. The goal of this paper is to describe the Jacobson radical ofR * U(L) forL≠0. We are most successful whenR is a p.i. algebra or Noetherian. In more general situations we at least obtain upper and lower bounds forJ(R * U(L)) which are ideals extended fromR. Furthermore, we offer an interesting example in all characteristics of a commutativeK-algebraC which admits a derivationδ such thatC isδ-prime but not semiprime.

Let R be a ring with involution*. In this paper, we study additive subgroups A of R which are invariant under all mappings of the form ϕ x : a → xax*. That is, xAx* ⊆ A for all x ∈ R. Obvious examples of such subgroups A are ideals of R, the set of symmetric elements, and the set of skew-symmetric elements. We will prove that when R is *-prime, these examples are essentially the only ones.

Let A A be a k k -algebra graded by a finite group G G , with A 1 {A_1} the component for the identity element of G G . We consider such a grading as a “coaction” by G G , in that A A is a k [ G ] ∗ k{[G]^ \ast } -module algebra. We then study the smash product A # k [ G ] ∗ A\# k{[G]^ \ast } ; it plays a role similar to that played by the skew group ring R ∗ G R\, \ast \,G in the case of group actions, and enables us to obtain results relating the modules over A , A 1 A,\,{A_1} , and A # k [ G ] ∗ A\# k{[G]^ \ast } . After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of A A and A 1 {A_1} . In particular we generalize Lorenz and Passman’s theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.

(1987). Prime ideals in fixed rings of free algebras. Communications in Algebra: Vol. 15, No. 11, pp. 2209-2234.

A classical theorem of E. Noether asserts that if R is a commutative ring, finitely generated over a field k, and G is any finite group of k- automorphisms of R, then the fixed ring (or ring of invariants) R” is also finitely generated. The question naturally arises as to what extent Noether’s theorem can be generalized to the noncommutative case. If R is also Noetherian and IG/ ~ ’ E k, all is well: RG is finitely generated, a result of Montgomery and Smail [6]. However, it is false in general, even for Pf rings [6]. Moreover, a recent result in Dicks and Formanek [l] (and, somewhat later, Kharchenko [4]), shows that almost the opposite of Noether’s theorem holds in the free aIgebra. That is, they prove that if G acts linearly on the free algebra F= k(xj,..., x,>, then P is finitely generated if and only if G acts by scalar matrices. In the present paper we consider the analogous problem for an algebra of generic matrices. That is, let U = k(X i,..., X,f be the generic matrix algebra generated over a field k by the m x m (m 2 2) generic matrices X1,..., X, (da 2). Let G act linearly on U; that is, for each gE G, Xp = c, cliiXi, for uij E k. Thus g corresponds to the dxd matrix A = (a,). If G consists of scalar matrices and \G/ -’ E k, then UC is always finitely generated. For, consider the free algebra F= k(x, ,..., xd> with the same action; since I/= ir’, a homomorphic image of F, it follows that v =

LetR be a prime ring andL a Lie algebra acting onR as “Q-outer” derivations (if charR=p≠0, assume thatL is restricted). We study ideals and the center of the smash productR #U(L) (respectivelyR #u(L) ifL is restricted) and use these results to study the relationship betweenR and the ring of constantsR L . More generally, for any finite-dimensional Hopf algebraH acting onR such thatR #H satisfies the “ideal intersection property”, we useR #H to study the relationship betweenR and the invariant ringR H .

Recently Britten has proven an analog of Goldie’s theorem for the Jordan ring S S of symmetric elements in a ring with involution of characteristic not 2. In this paper we first extend Britten’s theorem to the situation where R R is an arbitrary ring and the Jordan ring is only an ample subring of the symmetric elements. We apply this result to show that if S S has ACC on quadratic ideals, then the (Jordan) nil radical of S S is nilpotent.