Faithful representations of minimal dimension of current Heisenberg Lie algebras

TL;DR

This paper determines the minimal dimension of faithful representations for current Heisenberg Lie algebras over a field of characteristic zero, providing an explicit formula involving the degree of the polynomial defining the algebra.

Contribution

It introduces a precise formula for the minimal faithful representation dimension of current Heisenberg Lie algebras based on polynomial degree, extending understanding of their representation theory.

Findings

Derived an explicit formula for (\u211d_{m,p})

Connected representation dimension to polynomial degree

Enhanced understanding of current Heisenberg Lie algebra representations

Abstract

Given a Lie algebra $\mathfrak{g}$ over a field of characteristic zero $k$, let $\mu(\mathfrak{g})=\min\{\dim \pi: \pi\text{is a faithful representation of}\mathfrak{g}\}$. Let $\mathfrak{h}_{m}$ be the Heisenberg Lie algebra of dimension $2m+1$ over $k$ and let $k[t]$ be the polynomial algebra in one variable. Given $m\in\mathbb{N}$ and $p\in k[t]$, let $\mathfrak{h}_{m,p}=\mathfrak{h}_m\otimes k[t]/(p)$ be the current Lie algebra associated to $\mathfrak{h}_m$ and $k[t]/(p)$, where $(p)$ is the principal ideal in $k[t]$ generated by $p$. In this paper we prove that $ mu(\mathfrak{h}_{m,p}) = m \deg p + \left \lceil 2\sqrt{\deg p} \right\rceil$.