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Title: On Gelfand-Zetlin modules (English)
Author: Drozd, Yu. A.
Author: Ovsienko, S. A.
Author: Futorny, V. M.
Language: English
Journal: Proceedings of the Winter School "Geometry and Physics"
Issue: 1990
Pages: [143]-147
Category: math
Summary: [For the entire collection see Zbl 0742.00067.]\par Let ${\germ g}\sb k$ be the Lie algebra ${\germ gl}(k,\mathcal{C})$, and let $U\sb k$ be the universal enveloping algebra for ${\germ g}\sb k$. Let $Z\sb k$ be the center of $U\sb k$. The authors consider the chain of Lie algebras ${\germ g}\sb n\supset {\germ g}\sb{n-1}\supset\dots\supset {\germ g}\sb 1$. Then $Z=\langle Z\sb k\mid k=1,2,\dots n\rangle$ is an associative algebra which is called the Gel'fand-Zetlin subalgebra of $U\sb n$. A ${\germ g}\sb n$ module $V$ is called a $GZ$-module if $V=\sum\sb x\oplus V(x)$, where the summation is over the space of characters of $Z$ and $V(x)=\{v\in V\mid(a-x(a))\sp mv=0$, $m\in\mathcal{Z}\sb +$, $a\in\mathcal{Z}\}$. 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! (English)
MSC: 17B10
MSC: 17B20
MSC: 17B35
idZBL: Zbl 0754.17005
idMR: MR1151899
Date available: 2009-07-13T21:26:58Z
Last updated: 2012-09-18
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