General algebraic properties of the algebras of vector fields over quantum linear groups GLq(N) and SLq(N) are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The qanalogues of the Newton relations connecting two different generating s...

The Cayley-Hamilton-Newton theorem which underlies the Newton identities and the Cayley-Hamilton identity is reviewed, first, for the classical matrices with commuting entries, second, for two q-matrix algebras, the RTTalgebra and the RLRL-algebra. The Cayley-Hamilton-Newton identities for these q-algebras are related by the factorization map. A class of algebras M(R̂, F̂ ) is presented. The alge...

The aim of this paper is to introduce the notions of (∈,∈ ∨ q)fuzzy p-ideals, (∈,∈ ∨ q)-fuzzy q-ideals and (∈,∈ ∨ q)-fuzzy a-ideals in BCIalgebras and to investigate some of their properties. Several characterization theorems for these generalized fuzzy ideals are proved and the relationship among these generalized fuzzy ideals of BCI-algebras is discussed. It is shown that a fuzzy set of a BCI...

Part I of this paper develops various general concepts in generic representation and cohomology theories. Roughly speaking, we provide a general theory of orders in non-semisimple algebras applicable to problems in the representation theory of nite and algebraic groups, and we formalize the notion of a \generic" property in representation theory. Part II makes new contributions to the non-descr...

We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A case. The presentation is compatible with Lusztig’s modified form : U of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from : U, that this gives them a cellular structure, and thus the...

Let H be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If H is not semisimple and dim(H) = 2n for some odd integers n, then H or H * is not unimodular. Using this result, we prove that if dim(H) = 2p for some odd primes p, then H is semisimple. This completes the classification of Hopf algebras of dimension 2p. In recent years, there has been some pro...

We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gln. Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto. Mathem...

Q is any quasivariety. A congruence relation 0 on a member A of Q is a Q-congruence if A/0 G Q. The set CouqA. of all Qcongruences is closed under arbitrary intersection and hence forms a complete lattice Cong A. Q. is relatively congruence-distributive if ConjA is distributive for every A e Q.. Relatively congruence-distributive quasivarieties occur naturally in the theory of abstract data typ...

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K n quadratic number fields.

We continue the study of family algebras introduced by the author. In this paper we describe completely the structure of quantum family algebras for two cases of representations with a simple spectrum. 1. Generalities about family algebras 1.1. Basic definitions. A new class of associative algebras related to simple complex Lie algebras (or root systems) was introduced and studied in [K]. They ...