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.     

RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM

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 one dimensional 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.     

On the codimension growth of almost nilpotent Lie algebras

We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra L is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of L exists and is a positive integer.     

Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity

Let L be a Lie superalgebra with its enveloping algebra U(L) over a field F. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2×2 matrices over F. We characterize L when U(L) satisfies a non-matrix polynomial identity. We also characterize L when U(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.     

A Characterization of Stem Algebras in Terms of Central Derivations

Let L be a Lie algebra, and Der _z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der _z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der _z (L) is abelian if and only if L is a stem Lie algebra.     

Low dimensional Lie groups admitting left invariant flat projective or affine structures

We prove that any real Lie group of dimension ?5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension ?5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.     

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.     

Note on Co-split Lie Algebras

It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should be a weak co-split Lie algebra.A class of non-semi-simple co-split Lie algebras is given.     

Special Lie superalgebras with maximality condition for subalgebras

We consider finitely generated Lie superalgebras over a field of characteristic zero satisfying Capelli identities. We prove that any such an algebra with the maximality condition for abelian subalgebras is finite dimensional. In particular, any special Lie superalgebra with the maximality condition for its subalgebras has a finite dimension. We also prove that the universal enveloping algebra U(L) of special Lie superalgebra L is Noetherian if and only if $\dim L<\infty$ .     

Almost Primitive Elements of Free lie Algebras of Small Ranks

Let K be a field, X = {x1, . . . , xn}, and let L(X) be the free Lie algebra over K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free, A. I. Shirshov proved that subalgebras of free Lie algebras are free. A subset M of nonzero elements of the free Lie algebra L(X) is said to be primitive if there is a set Y of free generators of L(X), L(X) = L(Y ), such that M ? Y (in this case we have |Y | = |X| = n). Matrix criteria for a subset of elements of free Lie algebras to be primitive and algorithms to construct complements of primitive subsets of elements with respect to sets of free generators have been constructed. A nonzero element u of the free Lie algebra L(X) is said to be almost primitive if u is not a primitive element of the algebra L(X), but u is a primitive element of any proper subalgebra of L(X) that contains it. A series of almost primitive elements of free Lie algebras has been constructed. In this paper, for free Lie algebras of rank 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed.     

Centralizers in Lie Algebras

We determine the number of centralizers of different non-abelian finite dimensional Lie algebras over a specific field. Also, the concept of Lie algebras with abelian centralizers are studied and using a result of Bokut and Kukin [5], for a given residually free Lie algebra L, it is shown that L is fully residually free if and only if every centralizer of non-zero elements of L is abelian.     

Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center

A Lie algebra L is called 2-step nilpotent if L is not abelian and [L,L] lies in the center of L. 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.     

Lie metabelian restricted universal enveloping algebras

We characterize the restricted Lie algebras L whose restricted universal enveloping algebra u(L) is Lie metabelian. Moreover, we show that the last condition is equivalent to u(L) being strongly Lie metabelian.Received: 1 October 2004     

Poisson identities of enveloping algebras

Let L be a restricted Lie algebra. The symmetric algebra S_p(L) of the restricted enveloping algebra u(L) has the structure of a Poisson algebra. We give necessary and sufficient conditions on L in order for the symmetric algebra S_p(L) to satisfy a multilinear Poisson identity. We also settle the same problem for the symmetric algebra S(L) of a Lie algebra L over an arbitrary field. The first author was partially supported by MIUR of Italy. The second author was partially supported by Grant RFBR-04-01- 00739. Received: 31 October 2005     

Generalized Derivations of Lie Color Algebras

In this paper, we give some basic properties of the generalized derivation algebra GDer(L) of a Lie color algebra L. In particular, we prove that GDer(L) = QDer(L) + QC(L), the sum of the quasiderivation algebra and the quasicentroid. We also prove that QDer(L) can be embedded as derivations in a larger Lie color algebra.     

Filtered Multiplicative Bases of Restricted Enveloping Algebras

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.     

ON THE STRUCTURE OF COHOMOLOGY RINGS OF p-NILPOTENT LIE ALGEBRAS

In this paper the authors investigate the structure of the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose E ₂-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen–Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as a ring with the E ₂-term. We present criteria for the collapsing of this spectral sequence and provide some examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.     

Posets of infinite prinjective type and embeddings of Kronecker modules into the category of prinjective peak I-spaces

For a restricted Lie algebra L over a field of characteristic p > 0 we study the Lie nilpotency index t _L (u(L)) of its restricted universal enveloping algebra u(L). In particular, we determine an upper and a lower bound for t _L (u(L)). Finally, under the assumption that L is p-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of u(L) is maximal.

Communicated by I. Shestakov.