On Gelfand-Zetlin modules

Drozd, Yu. A. ; Ovsienko, S. A. ; Futorny, V. M.

Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1991), p. [143]-147 / Harvested from

Résumé

[For the entire collection see Zbl 0742.00067.]Let ${\mathfrak {g}}_k$ be the Lie algebra ${\mathfrak {g}l}(k,\mathcal {C})$, and let $U_k$ be the universal enveloping algebra for ${\mathfrak {g}}_k$. Let $Z_k$ be the center of $U_k$. The authors consider the chain of Lie algebras ${\mathfrak {g}}_n\supset {\mathfrak {g}}_{n-1}\supset \dots \supset {\mathfrak {g}}_1$. Then $Z=\langle Z_k\mid k=1,2,\dots n\rangle $ is an associative algebra which is called the Gel’fand-Zetlin subalgebra of $U_n$. A ${\mathfrak {g}}_n$ module $V$ is called a $GZ$-module if $V=\sum _x\oplus V(x)$, where the summation is over the space of characters of $Z$ and $V(x)=\lbrace v\in V\mid (a-x(a))^mv=0$, $m\in \mathcal {Z}_+$, $a\in \mathcal {Z}\rbrace $. The authors describe several properties of $GZ$- modules. For example, they prove that if $V(x)=0$ for some $x$ and the module $V$ is simple, then $V$ is a $GZ$-module. Indecomposable $GZ$- modules are also described. The authors give three conjectures on $GZ$- modules and!

Détail

MR MR1151899 | Zbl 0754.17005

EUDML-ID : urn:eudml:doc:220503
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