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Polynomials associated to Lie algebras

Matías Bruna, Alex Capuñay, Eduardo Friedman

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.

Polynomials associated to Lie algebras

Abstract

We associate to a semisimple complex Lie algebra a sequence of polynomials in variables, where is the rank of and . The polynomials are uniquely associated to the isomorphism class of , 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 were defined in 2008 by Komori, Matsumoto and Tsumura using different special values of another variant of Witten's zeta function.

Paper Structure

This paper contains 9 sections, 7 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The Shintani-Barnes zeta function $\zeta_{N,n}$
  3. Half-plane of convergence
  4. Analytic continuation of the zeta integral $\mathcal{Z}_{N,n}$
  5. Analytic continuation of the zeta series $\zeta_{N,n}$.
  6. Values of $\zeta_{N,n}$ and $\mathcal{Z}_{N,n}$ at $s=0,-1,-2,\ldots$
  7. Relations between zeta series and integrals
  8. Proof of claims concerning $P_{\ell,\mathfrak{g}}$ and $Q_{\ell,\mathfrak{g}}$
  9. Examples

Key Result

Theorem 1

Let $\mathfrak{g}$ be a semisimple complex Lie algebra of rank $r$, let $n$ be the number of positive roots in a root system for $\mathfrak{g}$, let $\ell=0,1,2,\ldots$, and let $\zeta_\mathfrak{g}(s,x)$ be as in GenHurwDef. Then $P_{\ell,\mathfrak{g}}(x):=\zeta_\mathfrak{g}(-\ell,x)$ is a polynomia $\mathrm{(iv)}$$P_{\ell,\mathfrak{g}}(\mathbf{1} - x)=(-1)^{n\ell+r}P_{\ell,\mathfrak{g}}(x),$ wher

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2
  • Proposition 3
  • proof
  • proof : Proof of Lemma $\mathrm{\ref{['MeromorphyZ']}}$.
  • Proposition 4
  • proof
  • Theorem 5
  • Corollary 6
  • proof
  • ...and 4 more