A lower bound for faithful representations of nilpotent Lie algebras

TL;DR

This paper establishes a new explicit lower bound for the minimal dimension of faithful representations of finite-dimensional nilpotent Lie algebras, using quadratic optimization and algebraic invariants.

Contribution

It introduces a quadratic optimization-based lower bound for faithful representation dimensions of p-step nilpotent Lie algebras, incorporating filtrations and algebraic invariants.

Findings

Provides explicit lower bounds involving algebraic dimensions and centers.

Applies bounds to specific families of nilpotent Lie algebras.

Offers estimates that improve understanding of representation minimality.

Abstract

In this paper we present a lower bound for the minimal dimension $\mu(\mathfrak{n})$ of a faithful representation of a finite dimensional $p$-step nilpotent Lie algebra $\mathfrak{n}$ over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary $p$ and takes into account a given filtration of $\mathfrak{n}$. We present some estimates of this minimum which leads to a very explicit lower bound for $\mu(\mathfrak{n})$ that involves the dimensions of $\mathfrak{n}$ and its center. This bound allows us to obtain $\mu(\mathfrak{n})$ for some families of nilpotent Lie algebras.