О локально нильпотентном радикале Джекобсона в специальных алгебрах Ли

Abstract

In the paper investigates the possibility of homological description of Jacobson radical and locally nilpotent radical for Lie algebras, and their relation with a $$PI$$ - irreducibly represented radical, and some properties of primitive Lie algebras are studied. We prove an analog of The F. Kubo \xa0theorem for almost locally solvable Lie algebras with a zero Jacobson radical. It is shown that the Jacobson radical of a special almost locally solvable Lie algebra $$L$$ over a field $$F$$ of characteristic zero is zero if and only if the Lie algebra $$L$$ has a Levi decomposition $$L=S\\oplus Z(L)$$, where $$Z(L)$$ is the center of the algebra $$L$$, $$S$$ is a finite-dimensional subalgebra $$L$$ such that $$J(L)=0$$. For an arbitrary special Lie algebra $$L$$, the inclusion of $$IrrPI(L)\\subset J(L)$$ is shown, which is generally strict. An example of a Lie algebra $$L$$ with strict inclusion \xa0$$J(L)\\subset IrrPI(L)$$ is given. It is shown that for an arbitrary special Lie algebra $$L$$ over the field $$F$$ of characteristic zero, the inclusion of $$N (L)\\subset IrrPI(L)$$, which is generally strict. It is shown that most Lie algebras over a field are primitive. An example of an Abelian Lie algebra over an algebraically closed field that is not primitive is given. Examples are given showing that infinite-dimensional commutative Lie algebras are primitive over any fields; a finite-dimensional Abelian algebra of dimension greater than 1 over an algebraically closed field is not primitive; an example of a non-Cartesian noncommutative Lie algebra is primitive. It is shown that for special Lie algebras over a field of characteristic zero $$PI$$-an irreducibly represented radical coincides with a locally nilpotent one. An example of a Lie algebra whose locally nilpotent radical is neither locally nilpotent nor locally solvable is given. Sufficient conditions for the primitiveness of a Lie algebra are given, and examples of primitive Lie algebras and non-primitive Lie algebras are given.