Semisimple Hopf Algebras Research Articles

On Semisimple Hopf Algebras of Dimension pqn

AbstractLet p, q be prime numbers with p2 < q, n ∊ ℕ, and H a semisimple Hopf algebra of dimension pqn over an algebraically closed field of characteristic 0. This paper proves that H must possess one of the following two structures: (1) H is semisolvable; (2) H is a Radford biproduct R#kG, where kG is the group algebra of group G of order p and R is a semisimple Yetter–Drinfeld Hopf algebra in of dimension qn.

Character tables and normal left coideal subalgebras

We continue studying properties of semisimple Hopf algebras H over algebraically closed fields of characteristic 0 resulting from their generalized character tables. We show that the generalized character table of H reflects normal left coideal subalgebras of H. These are the Hopf analogues of normal subgroups in the sense that they arise from Hopf quotients. We apply these ideas to prove Hopf analogues of known results in group theory. Among the rest we prove that columns of the character table are orthogonal and that all entries are algebraic integers. We analyze ‘semi-kernels’ and their relations to the character table. We prove a full analogue of the Burnside–Brauer theorem for almost cocommutative H. We also prove the Hopf algebras analogue of the following (Burnside) theorem: If G is a non-abelian simple group then {1} is the only conjugacy class of G which has prime power order.

Rationality of the Hilbert series of Hopf-invariants of free algebras

It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function whenever the ground field has zero characteristic.

THE CALABI–YAU PROPERTY OF TWISTED SMASH PRODUCTS

Let H be a finite-dimensional cocommutative semisimple Hopf algebra and A * H a twisted smash product. The Calabi–Yau (CY) property of twisted smash product is discussed. It is shown that if A is a CY algebra of dimension dA, a necessary and sufficient condition for A * H to be a CY Hopf algebra is given.

Semisimple Hopf actions on commutative domains

Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen.The main results of this article extend to working over k of positive characteristic. On the other hand, we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.

Are we counting or measuring something?

Let H be a semisimple Hopf algebras over an algebraically closed field k of characteristic 0. We define Hopf algebraic analogues of commutators and their generalizations and show how they are related to H′, the Hopf algebraic analogue of the commutator subgroup. We introduce a family of central elements of H′, which on one hand generate H′ and on the other hand give rise to a family of functionals on H. When H=kG, G a finite group, these functionals are counting functions on G. It is not clear yet to what extent they measure any specific invariant of the Hopf algebra. However, when H is quasitriangular they are at least characters on H.

EXAMPLES OF POINTED COLOR HOPF ALGEBRAS

We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras.

New examples of the Green functors arising from representation theory of semisimple Hopf algebras

A general Mackey-type decomposition for representations of semisimple Hopf algebras is investigated. We show that such a decomposition occurs in the case that the module is induced from an arbitrary Hopf subalgebra and it is restricted back to a group subalgebra. Some other examples when such a decomposition occurs are also constructed. They arise from gradings on the category of corepresentations of a semisimple Hopf algebra and provide new examples of the Green functors in the literature.

Kernels of representations of Drinfeld doubles of finite groups

Abstract A description of the commutator of a normal subcategory of the fusion category of representation Rep A of a semisimple Hopf algebra A is given. Formulae for the kernels of representations of Drinfeld doubles D(G) of finite groups G are presented. It is shown that all these kernels are normal Hopf subalgebras.

On the central charge of a factorizable Hopf algebra

For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel’d element differs from the value that it takes on the Drinfel’d element itself by at most a fourth root of unity. This can be reformulated by saying that the central charge of the Hopf algebra is an integer. If the dimension of the Hopf algebra is odd, we show that these two values differ by at most a sign, which can be reformulated by saying that the central charge is even. We give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs. To formulate our results, we use the language of modular data.

Amitsur’s conjecture for associative algebras with a generalized Hopf action

We prove the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of several generalizations of polynomial identities for finite dimensional associative algebras over a field of characteristic 0, including G-identities for any finite (not necessarily Abelian) group G and H-identities for a finite dimensional semisimple Hopf algebra H. In addition, we prove that the Hopf PI-exponent of Sweedler’s 4-dimensional algebra with the action of its dual equals 4.

Hopf subalgebras and Hopf ideals of certain semisimple Hopf algebras

In the paper, for semisimple Hopf algebras that have only one non-one-dimensional irreducible representation, all Hopf ideals are described and, under some restriction concerning the number of group elements in the dual Hopf algebra, some series of Hopf subalgebras are found. Moreover, the quotient Hopf algebras of these semisimpleHopf algebras are described.

On semisimple Hopf algebras of dimension [formula omitted

Let q be a prime number, k an algebraically closed field of characteristic 0, and H a semisimple Hopf algebra of dimension 2q3. This paper proves that H is always semisolvable. That is, such Hopf algebras can be obtained by (a number of) extensions from group algebras or duals of group algebras.

Structure Theorems for Semisimple Hopf Algebras of Dimension pq 3

Let p, q be prime numbers with p > q 3, and k an algebraically closed field of characteristic 0. In this article, we obtain the structure theorems for semisimple Hopf algebras of dimension pq 3.

Depth One Extensions of Semisimple Algebras and Hopf Subalgebras

Depth one extensions of finite dimensional semisimple algebras are completely characterized in terms of their algebra centers. For extensions of semisimple Hopf algebras this characterization translates into a trivial monoidal action of the dual fusion category Rep(A*) on Rep(B).

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1–13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.

QUANTUM ISOMETRY GROUPS OF SYMMETRIC GROUPS

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of Snon the choice of generators in the case n = 3. We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

Maschke-Type Theorem, Duality Theorem, and the Global Dimension for Weak Crossed Products

The notion of a crossed product with a Hopf algebroid was introduced by Böhm and Brzeziński [12]. It is well-known that weak Hopf algebras (see Böhm et al. [10]) is an example of Hopf algebroids. Then we get the definition and some property of a weak crossed product A#σ H over an algebra A and a weak Hopf algebra H. Next we give a Maschke-type theorem for the weak crossed product over a semisimple weak Hopf algebra H. Furthermore, we obtain an analogue of the Nikshych's duality theorem for weak crossed products. Finally, using this duality theorem, we prove that the global dimension of A equals to the global dimension of A#σ H if H and H* are both semisimple.

The Calabi–Yau property of smash products

Let H be a semisimple Hopf algebra and R a braided Hopf algebra in the category of Yetter–Drinfeld modules over H. When R is a Calabi–Yau algebra, a necessary and sufficient condition for R#H to be a Calabi–Yau Hopf algebra is given. Conversely, when H is the group algebra of a finite group and the smash product R#H is a Calabi–Yau algebra, we give a necessary and sufficient condition for the algebra R to be a Calabi–Yau algebra.

Structure of Semisimple Hopf Algebras of Dimension p 2 q 2

Let p, q be prime numbers with p 4 < q, and k an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension p 2 q 2 can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct R#kG, where kG is the group algebra of group G of order p 2, R is a semisimple Yetter–Drinfeld Hopf algebra in of dimension q 2. As an application, the special case that the structure of semisimple Hopf algebras of dimension 4q 2 is given.