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On Gelfand-Zetlin modules. (English) Zbl 0754.17005
Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 143-147 (1991).
[For the entire collection see Zbl 0742.00067.]
Let $${\mathfrak g}_ k$$ be the Lie algebra $${\mathfrak gl}(k,\mathbb{C})$$, 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 $$V(x)=\{v\in V\mid(a-x(a))^ mv=0$$, $$m\in\mathbb{Z}_ +$$, $$a\in\mathbb{Z}\}$$. 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 show that for $$n\leq 3$$ they are true. There are many notions and notations in this short paper which makes it difficult to read.
Reviewer: A.Klimyk (Kiev)

##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras 17B35 Universal enveloping (super)algebras