A new class of locally unital and finite dimensional algebras over an arbitrary algebraically closed field is discovered. Each them admits upper weakly triangular decomposition, a generalization split decomposition. It established that the category -lfdmod left -modules fully stratified in sense Brundan-Stroppel. Moreover, semisimple if only its centralizer subalgebras associated to certain ide...

We characterize rings R in which certain elements x have the property that CR(x) (resp. the set of zero divisors in CR(x)) is finite. We also explore the consequences of an assumption that certain x satisfy CR(x) = 〈x〉.

This paper discusses a computational problem arising in the study of the structure theory of Brauer's orthogonal and symplectic centralizer algebras. The problem is to compute the ranks of certain combinatorially defined matrices Zm k(x) (these matrices are presented in §2). This computation is difficult because the sizes of the matrices Zm k(x) are enormous even for small values of m and k . H...

An algebra extension A | B is right depth two in this paper if its tensor-square is A-B-isomorphic to a direct summand of any (not necessarily finite) direct sum of A with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obt...

In this paper, an algebra extension A |B is right depth two if its tensor-square is A-B-isomorphic to a direct summand of any (not necessarily finite) direct sum of A with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obt...

Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is very good. If X ∈ Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/Co both depend only on the Bala-Carter label of X; i.e. both are in...

Let G be a compact connected Lie group andH the centralizer of a oneparameter subgroup. We obtain a unified formula that expresses Steenrod operations on Schubert classes in the flag manifold G/H in term of Cartan