Essential Semigroups and Branching Rules

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} $\mathrm{\Sigma }$, 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 ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ of $\mathrm{\Sigma }$. If ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is finitely generated, then the semigroup algebra corresponding to ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is a toric degeneration of the branching algebra. We propose the algorithm to find a description of ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ 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 $F_4\supset B_4$.