On a locally nilpotent radical Jacobson for special Lie algebras

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 theorem 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 $J(L)\subset IrrPI(L)$ 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.