A note on linearization of actions of finitely semisimple Hopf algebras on local algebras

Let H be a Hopf algebra over a field k and let H A → A, h a → h.a, be an action of H on a commutative local Noetherian kalgebra (A, m). We say that this action is linearizable if there exists a minimal system x₁, …, x_n of generators of the maximal ideal m such that h.x_i ε kx₁ + …+ kx_n for all h ε H and i = 1, …, n. In the paper we prove that the actions from a certain class are linearizable (see Theorem 4), and we indicate some consequences of this fact.     

The theorem for semisimple Hopf algebras

We give an algebraic version of a result of G. I. Kac, showing that a semisimple Hopf algebra of dimension , where is a prime and , over an algebraically closed field of characteristic 0 contains a non-trivial central group-like. As an application we prove that, if , is isomorphic to a group algebra.

     

Properties of semisimple Hopf algebras

Semisimple finite-dimensional Hopf algebras all of whose non-one-dimensional irreducible modules of the same dimension are isomorphic are considered. The one-dimensional submodules of the tensor squares of irreduciblemodules are indicated and the almost cocommutative Hopf algebras and the left ideal coideals are described.     

On semisimple finite-dimensional Hopf algebras

This paper considers conditions under which the construction of a certain form is a Hopf algebra.     

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.     

A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

Schlichting’s theorem for approximate subgroups

We prove Schlichting’s theorem for approximate subgroups: if $${mathcal {X}}$$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with $${mathcal {X}}$$.     

Group elements of some semisimple finite-dimensional Hopf algebras

V. A. Artamonov and I. A. Chubarov proved a criterion under which an element of some semisimple finite-dimensional Hopf algebra is group-like. The studied Hopf algebra has only one nonone- dimensional irreducible representation. Let n be a dimension of this representation. It is shown in this paper that for odd prime n the set of group-like elements of these algebras is a cyclic group of order 2n.     

On the simplicity of some semisimple Hopf algebras

Two new series of Hopf algebras were discovered by V. A. Artamonov. The following paper gives a negative answer to N. Andruskiewitsch’s question about simplicity of these series.     

Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories

Given an action of a finite group G on a fusion category we give a criterion for the category of G-equivariant objects in to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non-group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in [7]. Namely, we show that certain /2-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami [26] are equivalent to representation categories of non-group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.      

The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras

With an aim of exploring homological algebra for weak Hopf modules, this paper investigates the HOM-functor and presents the structure theorem for endomorphism algebras of weak two-sided (A,H)-Hopf modules, and gives the duality theorem for weak “big” smash products.     

A note on Fung-Fu's theorem

The axiomatic approach to rational decision malting in a fuzzy environment proposed by Fung and Fu (1975) lays down a too restrictive family of aggregation rules. Because of such axioms, the independence of the size of the groups under aggregation is assumed; moveover, in a general problem, these groups are fuzzy sets.     

A note on Potter’s theorem for quasi-commutative matrices

We discuss the converse of a theorem of Potter stating that if the matrix equation AB=ωBA is satisfied with ω a primitive qth root of unity, then A^q+B^q=(A+B)^q. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and q=3 and a conjecture for the general case. We also present some further partial results and conjectures.     

A variant of Yano’s extrapolation theorem on Hardy spaces

In this note, a variant of Yano’s classical extrapolation theorem for sublinear operators acting on analytic Hardy spaces over the torus is obtained.     

Hua’s theorem on five squares of primes

We give an alternative proof of Hua’s theorem that each large N≡5 (mod 24) can be written as a sum of five squares of primes. The proof depends on an estimate of exponential sums involving the Möbius function.     

A note on a theorem of minc on irreducible non-negative matrices

A short proof is given for Mine's partial converse to a fundamental theorem of Frobenius on irreducible nonnegative matrices.     

A note on a theorem of minc on irreducible non-negative matrices

A short proof is given for Mine's partial converse to a fundamental theorem of Frobenius on irreducible nonnegative matrices.