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We introduce Turaev bicategories and group braided Turaev bicategories. On one hand, the latter generalize the notions of braided Turaev categories, introduced at the turn of the millennium and originally called “braided crossed group categories” by Turaev himself, and on the other hand, the former generalize the notion of bicategories. For bimonads in 2-categories, which we defined in one of our previous papers, we introduce generalized Yetter-Drinfel’d modules in 2-categories. These generalize to the 2-categorical setting the generalized Yetter-Drinfel’d modules (over a field) of Panaite and Staic, and thus also in particular the anti Yetter-Drinfel’d modules, introduced by Hajac-Khalkhali-Rangipour-Sommerhauser as coefficients for the cyclic cohomology of Hopf algebras, defined by Connes and Moscovici. We construct a Turaev 2-category for bimonads in 2-categories as a Turaev extension of the 2-category of bimonads, generalizing the corresponding Turaev category of Panaite and Staic. We also prove in the 2-categorical setting their results on pairs in involution, which in turn go back to modular pairs in involution of Connes and Moscovici. Moreover, two examples of proper Turaev bicategories are constructed, one of which is group braided.

In this paper several classical facts known for group actions and group gradings on rings are extended to the case of a Noetherian H-module algebra A for a Hopf algebra H. When H is semisimple, a version of the Bergman-Isaacs result is proved, asserting the nilpotency of any onesided ideal of A whose intersection with the subalgebra of H-invariant elements A H is nilpotent. Under the additional assumption that A is H-semiprime, it is established that the classical quotient ring Q(A) of A is the Ore localization of A at the set of H-invariant regular elements. When H is finite-dimensional cosemisimple, the Jacobson radical of A is shown to be stable under the action of H. More generally, these results are obtained for algebras over an arbitrary commutative base ring under suitable restrictions on the Hopf algebra and its action.

Kashina, Montgomery and Ng introduced the $n$-th indicator $\nu_n(H)$ of a finite-dimensional Hopf algebra $H$ and showed that the indicators have some interesting properties such as the gauge invariance. The aim of this paper is to investigate the properties of $\nu_n$'s. In particular, we obtain the cyclotomic integrality of $\nu_n$ and a formula for $\nu_n$ of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra $u(\mathcal{D}, \lambda, \mu)$ introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of $u_q(\mathfrak{sl}_2)$ and show that if $p$ and $q$ are roots of unity of the same order, then $u_p(\mathfrak{sl}_2)$ and $u_q(\mathfrak{sl}_2)$ are gauge equivalent if and only if $q = p$, where $p$ and $q$ are roots of unity of the same odd order., Comment: The final version accepted for publication in Israel Journal of Mathematics. No essential changes from the version 2. The program I have used for computation is available as an ancillary file

The recent result of Brown and Zhang establishing Poincare duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podleś quantum 2-sphere.

Let K be an algebraically closed field of characteristic 0. We conclude the classification of finite-dimensional pointed Hopf algebras whose group of group-likes is $\mathbb{S}_4$ . We also describe all pointed Hopf algebras over $\mathbb{S}_5$ whose infinitesimal braiding is associated to the rack of transpositions.

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

The classical Clifford correspondence for normal subgroups is considered in the more general setting of semisimple Hopf algebras. We prove that this correspondence still holds if the extension determined by the normal Hopf subalgebra is cocentral., 12 pages

Let H be a Hopf algebra over a base field. If H has an ℕ-filtration such that the associated graded ring is connected graded noetherian and has enough normal elements, then H is Gorenstein. This gives a partial solution to a question of Brown and Brown-Goodearl. As a consequence, every quotient Hopf algebra of a generic quantized coordinate ring of a connected semisimple Lie group is Auslander-Gorenstein and Cohen-Macaulay. The last statement answers a question of Goodearl-Zhang.

This paper contributes to the classification problems of finite dimensional Hopf algebras H over an algebraically closed field k of characteristic zero. It is shown that for a non-semisimple Hopf algebra H of dimension 18 either H or H* is pointed.

We study the quasitriangular structures ofT-smash product Hopf algebrasB ⋈ H which are constructed by Caenepeel, Ion, Militaru and Zhu. The necessary and sufficient conditions for a class ofT-smash product Hopf algebras to be quasitriangular Hopf algebras are given. As applications of our results, some corollaries and examples are given as well.

We prove that the reflection equation (RE) algebra $\La_R$ associated with a finite dimensional representation of a quasitriangular Hopf algebra $\Ha$ is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that $\La_R$ is a module algebra over the twisted tensor square \twist{$\Ha$}{$\Ha$} and the double $\D(\Ha)$. We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces., 17 pages, AMS Latex

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.

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB.

A generalization of Hopf algebras (quantum groups), and braided-Hopf algebras (braided quantum groups) in which the multiplicativity axiom for the counit is dropped, is presented. The generalization overcomes an inherent geometrical inhomogeneity of standard quantum groups and braided quantum groups, in the sense of allowing completely ‘pointless’ objects. All braid-type equations appear as a consequence of deeper axioms. Braided counterparts of basic algebraic relations between fundamental entities of the standard theory are found.

In this paper we construct two families of non-trivial self-dual semisimple Hopf algebras of dimensionpq2 and investigate closely their (quasi) triangular structures. The paper contains also general results on finite-dimensional triangular Hopf algebras, unimodularity, semisimplicity and ribbon structures of finite-dimensional semisimple Hopf algebras.

The aim of this paper is to generalize Noether’s theorem for finite groups acting on commutative algebras, to finite-dimensional triangular Hopf algebras acting on quantum commutative algebras. In the process we construct a non-commutative determinant function which yields an analogue of the Cayley-Hamilton theorem for certain endomorphisms.

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 determine the isomorphic classes of 6 or 8 dimensional semisimple Hopf algebrasA over an algebraically closed field such that (dimA)1≠0.

LetH be a Hopf algebra andM a representation or a corepresentation ofH. In this paper we study semiinvariants ofM. This notion generalizes the known concept of weight spaces in the context of representations of Lie algebras. Our best results are attained for pointed Hopf algebras, and semiinvariants which are related to the coradical filtration ofH.

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 .

LetH be a finite-dimensional Hopf algebra over the fieldK and letA be anH-module algebra. In a previous paper, we defined the Connes spectrum CS(A, H) for the action ofH onA to be a certain subset of the set Irr(H) of irreducible representations ofH. In this paper, we compute a number of examples; specifically, we consider certain inner and outer actions and we take a closer look at the cocommutative situation. We discover that the information encoded in the Connes spectrum is rather subtle and elusive.

LetC be a neutral Tannakian category over a fieldk. By a theorem of Saavedra Rivano there exists a commutative Hopf algebraA overk such thatC is equivalent to the category of finite dimensional rightA-comodules. We review Saavedra Rivano’s construction of the bialgebraA and show thatA has still an antipode if the symmetry condition on the monoidal structure ofC is removed.

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described. Some applications of this structure to combinatorics and to differential operators are surveyed. Some possible future directions for this work are indicated.

LetH be a commutative, cocommutative and faithfully projective Hopf algebra over a commutative ringR. Using cohomological methods, we obtain a description of a subgroup of Long’s Brauer group ofH-dimodule algebras:Ψ where the mapψ is induced by the smash product, and where BDs (R, H) is the subgroup of the Brauer-Long group consisting of all elements which are split by a faithfully flat extension ofR. As an example, the Brauer-Long group of a free Hopf algebra of rank 2 is computed. The results are also applied to Orzech’s subgroup of the Brauer-Long group.

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.

LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.

We construct examples of bicrossproducts and double cross products of quantum groupsa(R) associated to general matrix solutionsR of the Quantum Yang-Baxter Equations. We also describe iterated double cross products of quantum groups. In the course of constructinga(R) we are led to introduce a suitable notion of mutually dual Hopf algebras and a dual quantum groupŬ(R).

LetT=R #H be a smash product whereH is a finite dimensional Hopf algebra. We show that ideals ofT invariant under the dualH* ofH are extended fromH-invariant ideals ofR. This allows us to transport the study of ideals inT to invariant ideals. When the Hopf algebra is pointed the relationship between an ideal and its invariant ideal is shown to be manageable. Restricting to prime ideals, this yields results on the prime spectra ofR andT. We obtain Krull relations forR ⊆T for someH, including Incomparability wheneverH is commutative (or more generally whenH* is pointed after base extension). The results generalize and unify a number of results known in the context of group and restricted Lie actions.

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field.

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗ B A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

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 .