Invariant polynomials on truncated multicurrent algebras

TL;DR

This paper constructs invariant polynomials for truncated multicurrent Lie algebras, providing generators and a transversal slice, and explores the center of their universal enveloping algebras acting on finite-dimensional representations.

Contribution

It introduces a method to construct invariant polynomials and a transversal slice for truncated multicurrent algebras, extending classical invariant theory to these structures.

Findings

Constructed algebraically independent generators for invariant polynomials.

Described a transversal slice to regular orbits in the algebra.

Showed the center acts trivially on finite-dimensional irreducible representations.

Abstract

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form $\mathfrak{g} \otimes_\mathbb{F} \mathbb{F}[t_1,\dotsc,t_\ell]/I$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra over a field $\mathbb{F}$ of characteristic zero, and $I$ is a finite-codimensional ideal of $\mathbb{F}[t_1,\dotsc,t_\ell]$ generated by monomials. In particular, when $\mathfrak{g}$ is semisimple and $\mathbb{F}$ is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in $\mathfrak{g} \otimes_\mathbb{F} \mathbb{F}[t_1,\dotsc,t_\ell]/I$. As an application of our main result, we show that the center of the universal enveloping algebra of $\mathfrak{g} \otimes_\mathbb{F} \mathbb{F}[t_1,\dotsc,t_\ell]/I$ acts trivially on all irreducible finite-dimensional representations provided $I$ has codimension at least two.