Let L be a Lie superalgebra over a field of characteristic p≠2 with enveloping algebra U(L) or let L be a restricted Lie superalgebra over a field of characteristic p>2 with restricted enveloping algebra u(L). In this note, we establish when u(L) or U(L) is bounded Lie Engel.

Let L be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of L is a principal ideal ring if and only if L is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p.

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie solvable, Lie nilpotent, or bounded Lie Engel.

Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the symmetric and skew-symmetric elements of u(L) are also discussed. Moreover, a new theorem about an upper bound for the Lie nilpotency class of u(L) is proved.

We investigate the conditions under which the smash product of an (ordinary or restricted) enveloping algebra and a group algebra is Lie solvable or Lie nilpotent.

Lie solvable restricted enveloping algebras were characterized by Riley and Shalev except when the ground field is of characteristic 2. We resolve the characteristic 2 case here which completes the classification. As an application of our result, we obtain a characterization of ordinary Lie algebras over any field whose enveloping algebra is Lie solvable.

A full characterization is given of ordinary and restricted enveloping algebras which are normal with respect to the principal involution.

We deal with the existing problem of filtered multiplicative bases of finitedimensional associative algebras. For an associative algebra A over a field, we investigate when the property of having a filtered multiplicative basis is hereditated by homomorphic images or by the associated graded algebra of A. These results are then applied to some classes of group algebras and restricted enveloping algebras.

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satises an Engel condition.

In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of Togo on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gaschutz on the existence of $p$-power automorphisms of $p$-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.

For a restricted Lie algebra L, the conditions under which its restricted enveloping algebra u(L) is semiperfect are investigated. Moreover, it is proved that u(L) is left (or right) perfect if and only if L is finite-dimensional.

Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restrictedenveloping algebra. We establish when the symmetric or skew elements of u(L) under the principal involution are Lie metabelian.

Let L be a non-abelian restricted Lie algebra over a field of characteristic p>0 and let u(L) denote its restricted enveloping algebra. In 2006 it was proved that if u(L) is Lie solvable then the Lie derived length of u(L) is at least $lceillog_2(p+1)rceil$. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound.

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the onedimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by $p^{MT(L)}$, where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.

In this paper we investigate the relation between the multiplicities of split strongly abelian p-chief factors of nite-dimensional restricted Lie algebras and first degree restricted cohomology. As an application we obtain a characterization of solvable restricted Lie algebras in terms of the multiplicities of split strongly abelian p-chief factors. Moreover, we derive some results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of finite-dimensional solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module. The analogues of these results are well known in the modular representation theory of finite groups.

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As an application we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module.

We consider some questions about subnormal subgroups of a group in the setting of Lie superalgebras. In particular, the analogues of Nilpotence Join Theorem and Roseblade's Theorem for Lie superalgebras are proved.