Invariant theory and coefficient algebras of Lie algebras

TL;DR

This paper investigates the relationship between classical invariant theory and coefficient algebras of finite-dimensional complex Lie algebras, revealing isomorphisms with well-known algebraic structures and applications to characteristic polynomials.

Contribution

It establishes new isomorphisms between coefficient algebras of Lie algebras and classical invariant rings, providing explicit descriptions for standard, general linear, and special linear cases.

Findings

Coefficient algebra of upper triangular Lie algebra is isomorphic to symmetric polynomials.

Coefficient algebra of general linear Lie algebra matches the invariant ring under conjugation.

The coefficient algebra of special linear Lie algebra is generated by trace functions.

Abstract

The coefficient algebra of a finite-dimensional Lie algebra on a finite-dimensional representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We explore connections between classical invariant theory and the coefficient algebras of finite-dimensional complex Lie algebras on some representations. Specifically, we prove that with respect to any symmetric power of the standard representation: (a) the coefficient algebra of the upper triangular solvable complex Lie algebra is isomorphic to the ring of symmetric polynomials; (b) the coefficient algebra of the general linear complex Lie algebra is isomorphic to the invariant ring of the general linear group with the conjugacy action on the full space of matrices; and (c) the coefficient algebra of the special linear complex Lie algebra can be generated by classical trace functions. As an application, we determine the characteristic polynomial of the special linear complex Lie algebra on its standard representation.