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29 March 2024
 
  » arxiv » 1912.0341

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Casimir elements associated with Levi subalgebras of simple Lie algebras and their applications
Dmitri I. Panyushev ;
Date 1 Dec 2019
AbstractLet $mathfrak g$ be a simple Lie algebra, $mathfrak h$ a Levi subalgebra, and $C_{mathfrak h}in U(mathfrak h)$ the Casimir element defined via the restriction of the Killing form on $mathfrak g$ to $mathfrak h$. We study $C_{mathfrak h}$-eigenvalues in $mathfrak g/mathfrak h$ and related $mathfrak h$-modules. Without loss of generality, one may assume that $mathfrak h$ is a maximal Levi. Then $mathfrak g$ is equipped with the natural $mathbb Z$-grading $mathfrak g=igoplus_{iinmathbb Z}mathfrak g(i)$ such that $mathfrak g(0)=mathfrak h$ and $mathfrak g(i)$ is a simple $mathfrak h$-module for $i e 0$. We give explicit formulae for the $C_mathfrak h$-eigenvalues in each $mathfrak g(i)$, $i e 0$, and relate eigenvalues of $C_mathfrak h$ in $igwedge^ulletmathfrak g(1)$ to the dimensions of abelian subspaces of $mathfrak g(1)$. We also prove that if $mathfrak asubsetmathfrak g(1)$ is abelian, whereas $mathfrak g(1)$ is not, then $dimmathfrak ale dimmathfrak g(1)/2$. Moreover, if $dimmathfrak a=(dimmathfrak g(1))/2$, then $mathfrak a$ has an abelian complement. The $mathbb Z$-gradings of height $le 2$ are closely related to involutions of $mathfrak g$, and we provide a connection of our theory to (an extension of) the "strange formula" of Freudenthal-de Vries.
Source arXiv, 1912.0341
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