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Casimir elements associated with Levi subalgebras of simple Lie algebras and their applications | Dmitri I. Panyushev
; | Date: |
1 Dec 2019 | Abstract: | Let $mathfrak g$ 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 | Services: | Forum | Review | PDF | Favorites |
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