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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)

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