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[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) |
