Essential Semigroups and Branching Rules

TL;DR

This paper introduces a new algebraic approach using essential semigroups to systematically determine branching rules for representations of semisimple Lie algebras when restricted to subalgebras, with practical algorithms and examples.

Contribution

It develops a method based on essential semigroups and branching algebras to compute branching rules, including an algorithm for finitely generated cases and explicit examples.

Findings

Derived branching rules for classical Lie algebra pairs.

Established a connection between semigroup algebras and toric degenerations.

Provided an algorithm for describing the essential semigroup in specific cases.

Abstract

Let $\mathfrak{g}$ be a semisimple complex Lie algebra of finite dimension and $\mathfrak{h}$ be a semisimple subalgebra. We present an approach to find the branching rules for the pair $\mathfrak{g}\supset\mathfrak{h}$. According to an idea of Zhelobenko the information on restriction to $\mathfrak{h}$ of all irreducible representations of $\mathfrak{g}$ is contained in one associative algebra, which we call the \emph{branching algebra}. We use an \emph{essential semigroup} $\Sigma$, which parametrizes some bases in every finite-dimensional irreducible representations of $\mathfrak{g}$, and describe the branching rules for $\mathfrak{g}\supset\mathfrak{h}$ in terms of a certain subsemigroup $\Sigma'$ of $\Sigma$. If $\Sigma'$ is finitely generated, then the semigroup algebra corresponding to $\Sigma'$ is a toric degeneration of the branching algebra. We propose the algorithm to find a description of $\Sigma'$ in this case. We give examples by deriving the branching rules for $A_n\supset A_{n-1}$, $B_n\supset D_n$, $G_2\supset A_2$, $B_3\supset G_2$, and $F_4\supset B_4$.