Let H be a finite-dimensional semisimple Hopf algebra. Recently it was shown in [LM] that a version of the Frobenius-Schur theorem holds for Hopf algebras, and thus that the Schur indicator ν(χ) of the character χ of a simple H-module is well-defined; this fact for the special case of Kac algebras was shown in [FGSV]. In this paper we show that for an important class of non-trivial Hopf algebra...

In the present paper for two $mathfrak{A}$-module Banach algebras $A$ and $B$, we investigate relations between $varphi$-$mathfrak{A}$-module approximate amenability of $A$, $psi$-$mathfrak{A}$-module approximate amenability of $B$, and $varphioplus psi$-$mathfrak{A}$-module approximate amenability of $Aoplus B$ ($l^1$-direct sum of $A$ and $B$), where $varphiin$ Hom$_{mathfrak{A}}(A)$ and $psi...

in this paper we introduce a generalization of m-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . we will use the structure of the radical of a module in  and get some suitable results about this class of modules. also the relation between injective hull in  and this kind of modules will be investigated in this article.   for a module  we show...

Crane and Frenkel proposed a notion of a Hopf category [2]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations Uqn+ of universal enveloping algebras Un+ for some nilpotent Lie algebras n+ together with canonical bases of these braided Hopf algebras [4, 5, 6]. The elements of the canonical basis are identi...

Crane and Frenkel proposed a notion of a Hopf category [2]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations Uqn+ of universal enveloping algebras Un+ for some nilpotent Lie algebras n+ together with canonical bases of these braided Hopf algebras [4, 5, 6]. The elements of the canonical basis are identi...

This paper describes the system SYNHESYS, which is a shell for developing expert systems composed of both a connectionist module and a symbolic module. The shell allows for several types of interactions between the two modules, and for several degrees of coupling. SYNHESYS aims at facilitating the process of knowledge acquisition and at improving the performance of the symbolic module. We prese...

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule with the left and right module actions $pi_ell: Atimes Xrightarrow X$ and $pi_r: Xtimes Arightarrow X$, respectively. In this paper, we  study  the topological centers of the left module action $pi_{ell_n}: Atimes X^{(n)}rightarrow X^{(n)}$ and the right module action $pi_{r_n}:X^{(n)}times Arightarrow X^{(n)}$,  which inherit from th...

let $s$ be an inverse semigroup and let $e$ be its subsemigroup of idempotents. in this paper we define the $n$-th module cohomology group of banach algebras and show that the first module cohomology group $hh^1_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is zero, for every odd $ninmathbb{n}$. next, for a clifford semigroup $s$ we show that $hh^2_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is a banach space,...