Lie identities on symmetric elements of 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 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.     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

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 satisfies an Engel condition.     

Engel conditions and symmetric tensors

In a recent study of Engel Lie rings, Serena Cicalò and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Our main result generalizes this in the language of tensor algebra, and describes a relatively small generating set for the module generated by all n-th tensor powers of elements of a finitely generated ?-module M, in terms of a generating set for M.     

Fixed points of automorphisms of Lie rings and locally finite groups

Let P be a locally finite group of prime exponent p. We prove that if P admits a finite soluble automorphism group G of order n coprime to p, such that the fixed point group C _P(G)is soluble of derived length d, then P is nilpotent of class bounded by a function of p, n, and d. A similar statement is shown to hold for Lie (p - 1)-Engel algebras; it is analogous to the Bergman-Isaacs theorem proved for associative rings, provided the condition of being soluble for an automorphism group is added. Our proof is based on a generalization of Kreknin's theorem concerning the solubility of Lie rings with a regular automorphism of finite order. This generalization, giving an affirmative answer to a question of Winter and extending one of his results to the case of infinitedimensional Lie algebras, is interesting in its own right. Moreover, we use a generalization of Higgins' theorem on the nilpotency of soluble Lie Engel algebras. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 706-723, November-December, 1995.Supported by RFFR grant No. 94-01-00048-a and by ISF grant NQ7000.     

The unitary closure property of the prime varieties of associative algebras

We prove that every prime variety of associative algebras over an infinite field of characteristic p>0 is generated by either a unital algebra or a nilalgebra of bounded index. We show that the Engel verbally prime T-ideals remain verbally prime as we impose the identity $ x^{p^N } = 0 $ x^{p^N } = 0 for sufficiently large N. We then describe all prime varieties in an interesting class of varieties of associative algebras.     

On the dualities between categories of k-completely regular modules

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S at least two distinct words of length n on these letters are equal in S. Let U(A) denote the group of units of an associative algebra A over an infinite field of characteristic p > 0. We show that if A is unitally generated by its nilpotent elements then the following conditions are equivalent: U(A) is collapsing; U(A) satisfies some semigroup identity; U(A) satisfies an Engel identity; A satisfies an Engel identity when viewed as a Lie algebra; and, A satisfies a Morse identity. The characteristic zero analogue of this result was proved by the author in a previous paper.     

On associative algebras satisfying the Engel condition

It is shown that every finitely generated associative algebra over a field of characteristicp>0 satisfying the Engel condition is Lie-nilpotent. It follows that the Engel condition is inherited from an algebraA to its group of units,U(A).     

Some Theorems on Leibniz Algebras

If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.     

On Schur’s theorem and its converses for n-Lie algebras

An n-Lie algebra analogue of Schur’s theorem and its converse as well as a Lie algebra analogue of Baer’s theorem and its converse are presented. Also, it is shown that, an n-Lie algebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to some finite dimensional n-Lie algebra.     

Restricted Enveloping Algebras with Minimal Lie Derived Length

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 Siciliano (Publ Math (Debr) 68:503–513, 2006) it was proved that if u(L) is Lie solvable then the Lie derived length of u(L) is at least ⌈log₂(p + 1)⌉. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound.     

ON LIE ALGEBRAS WHOSE NILRADICAL IS (n — p)-FILIFORM

We prove first that every (n — p)-filiform Lie algebra, p ≤ 3, is the nilradical of a solvable, nonnilpotent rigid Lie algebra. We also analize how this result extends to (n — 4)-filiform Lie algebras. For this purpose, we give a classificaction of these algebras and then determine which of the obtained classes appear as the nilradical of a rigid algebra.     

On Lie nilpotent modular group algebras

Let K be a field of characteristic p>0 and let KG be the group algebra of an arbitrary group G over K. It is known that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least p+1. The group algebras KG for which these indices are p+1 or 2p or 3p?1 or 4p?2 have already been determined. In this paper, we classify the group algebras KG for which the upper Lie nilpotency index is 5p?3, 6p?4 or 7p?5.     

Heisenberg Lie Color Algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(?) and the holomorph L of finite dimensional Heisenberg Lie color algebra ? graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(?) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.     

Lie algebras admitting a unique quadratic structure

We prove that any Lie algebra g over a field K of characteristic zero admitting a unique up to a constant quadratic structure is necessarily a simple Lie algebra. If the field K is algebraically closed, such condition is also sufficient.

Further, a real Lie algebra g admits a unique quadratic structure if and only if its complexification g^C is a simple Lie algebra over C     

On free p-superalgebras

In this article we consider several aspects of algebraic combinatorics and combinatorial algebra over fields of prime characteristics. P-super-Radford theorem gives the structure of the free associative algebra over a field of prime characteristic with the new multiplication given by the super shuffle product, we show that this algebra is isomorphic to the reduced free super commutative algebra on s-regular words. We prove the elimination theorem for free partially commutative color Lie p-superalgebras and obtain a Schreier type formula for free Lie p-superalgebras using formal power series techniques.     

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A{\mathcal {A}} . We determine explicitly all the A{\mathcal {A}} -module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A{\mathcal {A}} -module automorphisms of the (reduced) Dickson–Mùi algebras. The algebra of all A{\mathcal {A}} -module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A{\mathcal {A}} -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mùi algebra is an indecomposable A{\mathcal {A}} -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.     

Generators of Simple Lie Algebras and the Lower Rank of Some Pro-P Groups

Abstract

Let  be a simple classical Lie algebra over a field F of characteristic p > 7. We show that > d () = 2, where d() is the number of generators of . Let G be a profinite group. We say that G has lower rank ≤ l, if there are {G _α} open subgroups which from a base for the topology at the identity and each G _α is generated (topologically) by no more than l elements. There is a standard way to associate a Lie algebra L(G) to a finitely generated (filtered) pro-p group G. Suppose L(G) ?  ? tF _p [t], where  is a simple Lie algebra over F _p , the field of p elements. We show that the lower rank of G is ≤ d () + 1. We also show that if  is simple classical of rank r and p > 7 or p 2r ² ? r, then the lower rank is actually 2.     

Nondegenerate graded lie algebras with degenerate transitive subalgebras

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler is analyzed. Transitive irreducible graded Lie algebras over an algebraically closed field of characteristic p > 2 with classical reductive component L ₀ are considered. We show that if a nondegenerate Lie algebra L containes a transitive degenerate subalgebra L′such that dim L′₁ > 1, then L is an infinite-dimensional Lie algebra. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.