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

Authors : Rıdvan GÜNER

Pages : 1146-1155

View : 11 | Download : 7

Publication Date : 2019-08-08

Article Type : Research Paper

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