On the homological description of the Jacobson radical for Lie algebras and the locally nilpotent radical for special Lie algebras

S. A. Pikhtilkov ; O. A. Pikhtilkova ; A. A. Gorelik ; L. B. Usova

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 195-204.

Résumé

One way to study the properties of rings, algebras, Lie algebras and their ideals presupposes their description via the properties of modules over these rings, algebras, Lie algebras. This article deals with the study of radicals of Lie algebras. We discuss the possibility of homological descriptions of the Jacobson radical of Lie algebras and nilpotent radical of the special Lie algebra. The first section introduces the concepts of radicals of Lie algebras. The second section is devoted to the Jacobson radical of Lie algebras. It is proved that the intersection of all annihilators of irreducible modules over an arbitrary Lie algebra $L$ coincides with the intersection of the Lie algebras $L$ and the Jacobson radical of the universal enveloping algebra. This section contains examples that prove this fact. This examples allows to prove the equality of the nilpotent radical of $PI$-irreducible represented radical of finite-dimensional Lie algebra over a field of characteristic zero. We find the correlation between the locally nilpotent radical and others radicals of Lie algebras such that the irreducible represented radical, the $PI$-irreducible represented radical and the finitely irreducible represented radical. In the third section it is shown that the locally nilpotent radical is included in the $PI$-irreducible represented radical for an arbitrary special Lie algebra $L$ over a field $F$ of characteristics zero. We have proved that the prime radical is not included in the $PI$-irreducible represented radical. The reverse inclusion for these radicals does not hold. The $PI$-irreducible represented radical is not locally solvable in the general case. Shows an example of a special Lie algebra $L$ over a field $F$ with the locally nilpotent radical, which has is equal to zero. Bibliography: 15 titles.

Détail

Keywords: Lie algebra, special Lie algebra, the Jacobson radical of Lie algebras, the locally nilpotent radical of special Lie algebras.

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