Essential Semigroups and Branching Rules
Andrei Gornitskii

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 be a semisimple complex Lie algebra of finite dimension and be a semisimple subalgebra. We present an approach to find the branching rules for the pair . According to an idea of Zhelobenko the information on restriction to of all irreducible representations of is contained in one associative algebra, which we call the \emph{branching algebra}. We use an \emph{essential semigroup} , which parametrizes some bases in every finite-dimensional irreducible representations of , and describe the branching rules for in terms of a certain subsemigroup of . If is finitely generated, then the semigroup algebra corresponding to is a toric degeneration of the branching algebra. We propose the algorithm to find a…
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Taxonomy
Topicssemigroups and automata theory
