On filtered multiplicative bases of some associative algebras

TL;DR

This paper investigates conditions under which finite-dimensional associative algebras possess filtered multiplicative bases, focusing on inheritance properties through homomorphic images and associated graded algebras, with applications to group and enveloping algebras.

Contribution

It provides new criteria for the inheritance of filtered multiplicative bases in associative algebras and applies these results to specific algebra classes.

Findings

Filtered multiplicative bases are inherited by certain homomorphic images.

The associated graded algebra may or may not preserve the basis property.

Applications to group algebras and restricted enveloping algebras demonstrate practical relevance.

Abstract

We deal with the existing problem of filtered multiplicative bases of finite-dimensional associative algebras. For an associative algebra A over a field, we investigate when the property of having a filtered multiplicative basis is hereditated by homomorphic images or by the associated graded algebra of $A$. These results are then applied to some classes of group algebras and restricted enveloping algebras.