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.     

On the index of certain nilpotent Lie algebras

We introduce a method of calculation of the index of Lie algebras that are factors of the unitriangular Lie algebra with respect to ideals spanned by subsets of root vectors. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Characteristically nilpotent Lie algebras

Translated from Algebra i Logika, Vol. 28, No. 6, pp. 722–737, November–December, 1989.     

On Einstein extensions of nilpotent metric Lie algebras

The main result of the article is as follows: If a nilpotent noncommutative metric Lie algebra (n, Q) is such that the operator Id ? trace(Ric) / trace(Ric²) Ric is positive definite then every Einstein solvable extension of (n, Q) is standard. We deduce several consequences of this assertion. In particular, we prove that all Einstein solvmanifolds of dimension at most 7 are standard.     

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.     

Representations of nilpotent Lie algebras

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L^*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A_￥ ]\Bbb Z [A_\infty ] or \Bbb Z [A_n]/(t)\Bbb Z [A_n]/(\tau ).     

Serial subalgebras of finitary Lie algebras

A Lie subalgebra of is said to be finitary if it consists of elements of finite rank. We show that, if acts irreducibly on , and if is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of acts irreducibly on too. When , it follows that the locally solvable radical of such is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic are hyperabelian.

     

Decompositions of nilpotent Lie algebras

It is proved that decompositions of nilpotent Lie algebras are global. In the complex case, nilpotency is also a necessary condition for every decomposition to be global. The results obtained are applied to the classification of complex homogeneous spaces of simply connected nilpotent Lie groups.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 27–30, January, 1978.In conclusion, the author would like to thank A. L. Onishchik for his interest in this research.     

Hyperplane subalgebras of real Lie algebras

The subalgebras of codimension one in a real Lie algebra are fully classified. Some applications are described.Helmut R. Salzmann zu seinem 60. Geburtstag     

Two generator subalgebras of Lie algebras

In [Thompson, J., 1968, Non-solvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74, 383–437.], Thompson showed that a finite group G is solvable if and only if every two-generated subgroup is solvable (Corollary 2, p. 388). Recently, Grunevald et al. [Grunewald et al., 2000, Two-variable identities in groups and Lie algebras. Rossiiskaya Akademiya Nauk POMI, 272, 161–176; 2003. Journal of Mathematical Sciences (New York), 116, 2972–2981.] have shown that the analogue holds for finite-dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two-generated subalgebras determine the structure of the algebra. It is to this question that this article is addressed. Here, we consider the classes of strongly-solvable and of supersolvable Lie algebras, and the property of triangulability.     

On the tensor square of non-abelian nilpotent finite-dimensional Lie algebras

For every finite p-group G of order p ⁿ with derived subgroup of order p ^m , Rocco [N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. 1 (1991), pp. 63–79] proved that the order of tensor square of G is at most p ^n(n?m). This upper bound has been improved recently by the author [P. Niroomand, On the order of tensor square of non abelian prime power groups (submitted)]. The aim of this article is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given n-dimensional non-abelian nilpotent Lie algebra L with derived subalgebra of dimension m we have dim(L???L)?≤?(n???m)(n???1)?+?2. Furthermore for m?=?1, the explicit structure of L is given when the equality holds.