On a locally nilpotent radical Jacobson for special Lie algebras

O. A. Pikhtilkova ; E. V. Mescherina ; A. N. Blagovisnaya ; E. V. Pronina ; O. A. Evseeva

Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 234-272

Résumé

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.

Détail

Keywords: Lie algebra, primitive Lie algebra, special Lie algebra, irreducible $PI$-representation, Jacobson radical, locally nilpotent radical, reductive Lie algebra, almost locally solvable Lie algebra.