Canonical Filtrations of Finite-Dimensional Algebras

TL;DR

This paper introduces a new invariant called canonical filtrations for finite-dimensional algebras, derived via GIT, and explores their properties, computation methods, and implications for algebra stability and structure.

Contribution

It defines canonical filtrations using GIT destabilizing subgroups and establishes their fundamental properties, linking algebra stability to semisimplicity.

Findings

Algebra is semisimple iff GIT semistable.

Provides a method to compute canonical filtrations.

Analyzes the structure of associated graded algebras.

Abstract

We study canonical filtrations of finite-dimensional associative algebras and Lie algebras. These filtrations are defined via optimal destabilizing one-parameter subgroups in the sense of geometric invariant theory (GIT), and appear to be a new invariant of finite-dimensional algebras. We establish some fundamental properties of these filtrations, and show that an algebra is semisimple if and only if it is GIT semistable. We give a method to compute canonical filtrations of algebras whose automorphism group or module of derivations is sufficiently rich, and use this to compute these filtrations in examples when the automorphism group contains a sufficiently large torus. We also obtain some results on the structure of the associated graded algebra of the canonical filtration of an associative algebra.