Polynomials associated to Lie algebras

TL;DR

This paper introduces a new sequence of polynomials linked to semisimple Lie algebras, derived from special values of a modified Witten's zeta function, providing a novel algebraic invariant.

Contribution

It defines a new family of polynomials associated to Lie algebras based on Witten's zeta function, expanding the algebraic tools for studying Lie algebra isomorphism classes.

Findings

Polynomials uniquely associated to Lie algebra isomorphism classes.

Connection to special values of Witten's zeta function.

Comparison with previously defined polynomials by Komori, Matsumoto, and Tsumura.

Abstract

We associate to a semisimple complex Lie algebra $\mathfrak{g}$ a sequence of polynomials $P_{\ell,\mathfrak{g}}(x)\in\mathbb{Q}[x]$ in $r$ variables, where $r$ is the rank of $\mathfrak{g}$ and $\ell=0,1,2,\ldots $. The polynomials $P_{\ell,\mathfrak{g}}(x)$ are uniquely associated to the isomorphism class of $\mathfrak{g}$, up to re-numbering the variables, and are defined as special values of a variant of Witten's zeta function. Another set of polynomials associated to $\mathfrak{g}$ were defined in 2008 by Komori, Matsumoto and Tsumura using different special values of another variant of Witten's zeta function.