Matrix valued concomitants of $\mathrm{SL}_2(\mathbb{C})$

TL;DR

This paper studies the algebraic structure of matrix-valued polynomial maps associated with finite-dimensional representations of complex Lie groups, focusing on the case of $ ext{SL}_2( ext{C})$, and identifies minimal generators for these algebras.

Contribution

It determines minimal generating systems of the algebra of concomitants for irreducible finite-dimensional representations of $ ext{SL}_2( ext{C})$, both as an algebra and over its center.

Findings

Identified the center of the algebra of concomitants with invariant polynomial functions.

Determined minimal generating systems for the algebra of concomitants.

Analyzed the structure as an algebra and as a module over its center.

Abstract

To a finite dimensional representation of a complex Lie group $G$, an associative algebra of adjoint covariant polynomial maps from the direct sum of $m$ copies of the Lie algebra $\mathfrak{g}$ of $G$ into an algebra of complex matrices is associated. When the tangent representation of the given representation is irreducible, the center of this algebra of concomitants can be identified with the algebra of adjoint invariant polynomial functions on $m$-tuples of elements of $\mathfrak{g}$. For irreducible finite dimensional representations of $\mathrm{SL}_2(\mathbb{C})$ minimal generating systems of the corresponding algebras of concomitants are determined, both as an algebra and as a module over its center.