MODULARITY-ASTERISK IN LIE-ALGEBRAS

A subalgebra Li of a Lie algebra L over a field F is called modular i n L if U satisfies the dual of the modular identities in the lattice o f subalgebras of L. Our aim is the study of the influence of the modul ar identities in the structure of the algebra. First we prove that if the modular conditions are imposed on an ideal of L then every eleme nt of L acts as an scalar on this ideal and if they are imposed on a n on-ideal subalgebra U of L then the largest ideal of L contained in Li also satisfies the modular identities. We determine Lie algebras hav ing a subalgebra which satisfies both the modular and modular identit ies, provided that either L is solvable or char(F) not equal 2, 3 As i mmediate consequences of this result we obtain that the existence of a co-atom satisfying the modular identities in the lattice L(L) forces that the lattice L(L) is modular and that the modular identities on any subalgebra U forces that U is quasi-abelian. In the case when L is supersolvable we obtain that the modular conditions on any non-ideal of L are enough to guarantee that L(L) is modular. For arbitrary fiel ds and any Lie algebra L, we prove that the modular conditions on eve ry co-atom of the lattice L(L) guarantee that L(L) is modular.