Essential Semigroups and Branching Rules

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. 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 some bases in every finite-dimensional irreducible representations of g, and describe the branching rules for g in terms of a certain subsemigroup Σ' of Σ. If Σ' is finitely generated, then the semigroup algebra corresponding to Σ' is a toric degeneration of the branching algebra. We propose the algorithm to find a description of Σ' in this case. We give examples by deriving the branching rules for Aₙ⊃ Aₙ₋₁, Bₙ⊃ Dₙ, G₂⊃ A₂, B₃⊃ G₂, and F₄⊃ B₄.