Essential semigroups and branching rules

Abstract

Let $g$ be a semisimple complex Lie algebra of finite dimension and $h$ be a semisimple subalgebra. We present an approach to find the branching rules for the pair $g\supset h$. According to an idea of Zhelobenko the information on restriction to $h$ of all irreducible representations of $g$ is contained in one associative algebra, which we call the branching algebra. We use an essential semigroup Σ, which parametrizes certain bases in every finite-dimensional irreducible representations of $g$, and describe the branching rules for $g\supset h$ in terms of a certain subsemigroup ${\Sigma }^{\prime }$ of Σ. If ${\Sigma }^{\prime }$ is finitely generated, then the semigroup algebra corresponding to ${\Sigma }^{\prime }$ is a toric degeneration of the branching algebra. We propose an algorithm to find a description of ${\Sigma }^{\prime }$ 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 $F_4\supset B_4$.