Minimal algebras and 2-step nilpotent Lie algebras in dimension 7

We use the methods of Bazzoni and Muñoz (Trans Am Math Soc 364:1007–1028, 2012) to give a classification of 7-dimensional minimal algebras, generated in degree 1, over any field ${\mathbf{k}}$ of characteristic ${{\rm char}(\mathbf{k})\neq 2}$ , whose characteristic filtration has length 2. Equivalently, we classify 2-step nilpotent Lie algebras in dimension 7. This classification also recovers the real homotopy type of 7-dimensional 2-step nilmanifolds.     

The classification of complete Lie algebras with commutative nilpotent radical

The work in this paper is a continuation of an earlier paper of the second author (Acta Math. 34 (1991), 191-202). We discuss the properties of finite-dimensional complete Lie algebras with abelian nilpotent radical over the complex field . We solve the problems of isomorphism, classification and realization of complete Lie algebras with commutative nilpotent radical.

     

Complete commutative sets of polynomials for solvable Lie algebras of dimension six and nilpotent Lie algebras of dimension seven

A complete commutative set of polynomials is constructed using Sadetov’s method on the coalgebra of each real 6-dimensional solvable non-nilpotent Lie algebra and of each real 7-dimensional nilpotent Lie algebra.     

On dimension of the Schur multiplier of nilpotent Lie algebras

Let L be an n-dimensional non-abelian nilpotent Lie algebra and $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) where M(L) is the Schur multiplier of L. In [Niroomand P., Russo F., A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra (in press)] it has been shown that s(L) ≥ 0 and the structure of all nilpotent Lie algebras has been determined when s(L) = 0. In the present paper, we will characterize all finite dimensional nilpotent Lie algebras with s(L) = 1; 2.     

Uncertainty principles on two step nilpotent Lie groups

We extend an uncertainty principle due to Cowling and Price to two step nilpotent Lie groups, which generalizes a classical theorem of Hardy. We also prove an analogue of Heisenberg inequality on two step nilpotent Lie groups.     

Characteristically nilpotent Lie algebras

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

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 ).     

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.     

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.     

A universal dimension formula for complex simple Lie algebras

We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as several other universal formulas. These formulas generalize formulas of Vogel and Deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients. We discuss consequences of the formulas including a relation with Scorza varieties.