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On the index of horospherical subalgebras of semisimple Lie algebras. (Russian. English summary) Zbl 0626.17004

Let \(k\) be an algebraically closed field, \(\text{char}\;k=0\), and \(L\) a finite dimensional Lie \(k\)-algebra. Each \(f\in L^*\) defines a skew-symmetric bilinear form \(B_ f(x,y)=f([x,y])\) on \(L\). An index of \(L\) is \(\inf(\dim\;\text{Ker}\;B_ f\), \(f\in L^*)\). An horospherical subalgebra in a semisimple Lie algebra \(L\) is a nilpotent radical of a parabolic subalgebra in \(L\). The main result of the paper presents a calculation of indexes of horospherical subalgebras in simple Lie algebras. For algebras of types \(B_ n\), \(C_ n\), \(D_ n\) the result is given by formulas, for type \(A_ n^ -\) by an algorithm for calculation and for types \(E_ 6\), \(E_ 7\), \(E_ 8\), \(G_ 2\), \(F_ 4^ -\) by tables of results.
It is worth mentioning that for a nilpotent algebra the index equals the Gelfand-Kirillov dimension of the center in the universal enveloping algebra.
Reviewer: V.Artamonov

MSC:

17B20 Simple, semisimple, reductive (super)algebras
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