Reducible good representations of semisimple Lie algebras $A_r$ and $B_r$ Part I

Year 2019, Volume: 48 Issue: 4, 1146 - 1155, 08.08.2019

Rıdvan Güner

Abstract

Given a semisimple (preferably simple) complex Lie algebra $L$, we consider the monoid $\Gamma=\Gamma(L)$ of equivalence classes of the finite dimensional reducible complex representations of $L$. Here $\Gamma$ is identified with the lattice of the corresponding highest weights. (This equips $\Gamma$ with the monoid structure.) For $\pi\in\Gamma$ one considers the symmetric algebra $\displaystyle S(\pi)=\bigoplus_{n=0}^{\infty}S^n(\pi)$ (here regarded as a representation). The elements of $\Gamma$ ``occurring'' in $S(\pi)$ -- i.e., which are the highest weights of some irreducible component of the representation $S(\pi)$ -- form a subsemigroup $M(\pi)$ of $\Gamma$. Such a $M(\pi)$ has a naturally defined rank $r(\pi)$ with $1\leq r(\pi)\leq r = \text{rank of }L$. In this paper we give a classification, for all the simple $L=A_r$ and $L=B_r$ of all the $\pi$ with $r(\pi)< r$.

Keywords

Lie algebra, Lie group, reducible representation, classification

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