Arbitrary Hopf Algebras Research Articles

Actions of Ore extensions and growth of polynomial H-identities

ABSTRACTWe show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexpH(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.

A Braided Version of Some Results of Skryabin

We extend the main result of Skryabin in [9] to Yetter-Drinfeld Hopf algebras over an arbitrary Hopf algebra H with bijective antipode.

Jordan-Hölder theorem for finite dimensional Hopf algebras

We show that a Jordan-Hölder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus’ butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Hölder theorem holds as well for lower and upper composition series, even though the factors of such series may not be simple as Hopf algebras.

Root systems and Weyl groupoids for Nichols algebras

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The ob- tained combinatorial structure fits perfectly into an existing frame- work of generalized root systems associated to a family of Cartan matrices, and provides novel insight into Nichols algebras. We demonstrate the power of our construction with new results on Nichols algebras over finite non-abelian simple groups and sym- metric groups.

Cyclic homology of Hopf crossed products

We obtain a mixed complex, simpler than the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E = A # f H , where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homology of E relative to K. The first one works in the general setting and the second one (which generalizes those previously found by several authors) works when f takes its values in K.

Lazy cohomology: An analogue of the Schur multiplier for arbitrary Hopf algebras

We propose a detailed systematic study of a group H L 2 ( A ) associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac–Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally, the explicit computation of H L 2 ( A ) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.

WIGNER–ECKART THEOREM FOR TENSOR OPERATORS OF HOPF ALGEBRAS

We prove Wigner–Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representations is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra Ut(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of Ut(su(2)) is given.

Dynamical Yang-Baxter Equation and Quantum Vector Bundles

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, etc. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and its Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups.

Morita theory for corings and cleft entwining structures

Using the theory of corings, we generalize and unify Morita contexts introduced by Chase and Sweedler [Hopf Algebras and Galois Theory, in: Lecture Notes in Math., vol. 97, Springer-Verlag, Berlin, 1969], Doi [Generalized smash products and Morita contexts for arbitrary Hopf algebras, in: J. Bergen, S. Montgomery (Eds.), Advances in Hopf algebras, in: Lecture Notes in Pure and Appl. Math., vol. 158, Dekker, New York, 1994], and Cohen, Fischman, and Montgomery [J. Algebra 133 (1990) 351]. We discuss when the contexts are strict. We apply our theory to corings arising from entwining structures, and this leads us to the notion of cleft entwining structure.

Integrals for (dual) quasi-Hopf algebras. Applications

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension ⩽1, and for a finite-dimensional Hopf algebra, this dimension is exactly one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode follows from the other axioms of a finite-dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.

ON THE GENERALIZED H-LIE STRUCTURE OF ASSOCIATIVE ALGEBRAS IN YETTER-DRINFELD CATEGORIES

ABSTRACT We study the structure of the generalized H -Lie algebras (i.e., the Lie algebras in the Yetter-Drinfeld category ) for any Hopf algebras and the H -Lie structure of an algebra A in . Let H be arbitrary Hopf algebra. Firstly, We show that if A is a sum of two H -commutative subalgebras, then the H -commutator ideal of A is nilpotent, generalizing the results from [1] for a cotriangular Hopf algebra to the case of any Hopf algebra. Secondly, We investigate the H -Lie ideal structure of A by showing that if A is H -simple, then any non-commutative H -Lie ideal I of A must contain , giving a positive answer to the question given in [[1], p. 42]. Finally, a partial analog of [7] is shown in a more general Hopf algebra setting.

Principal homogeneous spaces for arbitrary Hopf algebras

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.

Crossed Products and Inner Actions of Hopf Algebras

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if $\pi :H \to \overline H$ is a Hopf algebra epimorphism which is split as a coalgebra map, then $H$ is algebra isomorphic to $A{\# _\sigma }H$, a crossed product of $H$ with the left Hopf kernel $A$ of $\pi$. Given any crossed product $A{\# _\sigma }H$ with $H$ (weakly) inner on $A$, then $A{\# _\sigma }H$ is isomorphic to a twisted product ${A_\tau }[H]$ with trivial action. Finally, if $H$ is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of $A$ implies that of $A{\# _\sigma }H$; in particular this is true if the (weak) action of $H$ is inner.

Crossed products and inner actions of Hopf algebras

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if π : H → H ¯ \pi :H \to \overline H is a Hopf algebra epimorphism which is split as a coalgebra map, then H H is algebra isomorphic to A # σ H A{\# _\sigma }H , a crossed product of H H with the left Hopf kernel A A of π \pi . Given any crossed product A # σ H A{\# _\sigma }H with H H (weakly) inner on A A , then A # σ H A{\# _\sigma }H is isomorphic to a twisted product A τ [ H ] {A_\tau }[H] with trivial action. Finally, if H H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A A implies that of A # σ H A{\# _\sigma }H ; in particular this is true if the (weak) action of H H is inner.

Operators on Hopf Algebras

Let A be a dimensional Hopf algebra with antipode s over a field k. In (6) the structure of s and s2 is seen to be closely related to the important algebraic features of A and A*. This paper furthers the investigation. More generally finite bialgebra endomorphisms t:A ->A of an arbitrary Hopf algebra A are considered (this means A is the sum of the dimensional t-invariant subspaces). Applications are given to dimensional Hopf algebras. We show that the antipode s of an arbitrary Hopf algebra over a field is injective (bijective) if the same is true when it is restricted to the coradical. We prove that s is bijective if it is a locally operator, or if the coradical is dimensional. Examples of dimensional Hopf algebras over Q (which can be defined over Z) are constructed with antipode of order 6 and 8, respectively. These examples show that the relationship between the order of the antipode and the primitive roots of unity in the ground field is not as intricate as indicated by some of the results of (6).