The adjacency algebra of an association scheme is defined over an arbitrary field. In general, it is always semisimple over a field of characteristic zero but not always semisimple over a field of positive characteristic. The structures of adjacency algebras over fields of positive characteristic have not been sufficiently studied. In this paper, we consider the structures of adjacency algebras...

We introduce the notion of a chopped and sliced cone in combinatorial geometry and prove two structure theorems for the number of integral points in the individual slices of such a cone. We observe that this notion applies to weight multiplicities of Kac-Moody algebras and Littlewood-Richardson coefficients of semisimple Lie algebras, where we obtain the corresponding results.

In this paper, we prove that the variety of modal FLew-algebras is generated by its finite simple members. The result is obtained by showing that every free modal FLew-algebra is semisimple and then showing that every variety generated by a simple modal FLew-algebra is generated by a set of finite simple modal FLew-algebras.

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of categories if, and only it non-degenerate. This includes case semisimple modular categories, but also non-semisimple examples such as representations small quantum group at good roots unity. Via cobordism hypothesis, we obtain new 4-dimensional framed topological field theories, which regard versio...

In this article, using an idea of the physics superselection principal, we study a modularity on vertex operator algebras arising from semisimple primary vectors. We generalizes the theta functions on vertex operator algebras and prove that the internal automorphisms do not change the genus one twisted conformal blocks.

Leibniz algebras are a non-anticommutative version of Lie algebras. They play an important role in different areas mathematics and physics have attracted much attention over the last thirty years. In this paper we investigate whether conditions such as being algebra, cyclic, simple, semisimple, solvable, supersolvable or nilpotent algebra preserved by lattice isomorphisms.

Permutation modules are fundamental in the representation theory of symmetric groups Sn and their corresponding Iwahori–Hecke algebras H = H (Sn). We find an explicit combinatorial basis for the annihilator of a permutation module in the “integral” case — showing that it is a cell ideal in G.E. Murphy’s cell structure of H . The same result holds whenever H is semisimple, but may fail in the no...