A curve of nilpotent Lie algebras which are not Einstein nilradicals

The only known examples of non-compact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, i.e. nilpotent Lie algebras which are nilradicals of the Lie algebras of Einstein solvmanifolds. Up to now, very few examples of \mathbb N{\mathbb N}-graded nilpotent Lie algebras that cannot be Einstein nilradicals have been found. In particular, in each dimension, there are only finitely many known. We exhibit in the present paper two curves of pairwise non-isomorphic nine-dimensional two-step nilpotent Lie algebras which are not Einstein nilradicals.     

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.     

Curvature properties of metric nilpotent Lie algebras which are independent of metric

This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra \(\mathfrak {g}\) (respectively, of the Grassmannian of two-planes of \(\mathfrak {g}\)) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on \(\mathfrak {g}\). In the second part we study the subsets of \(\mathfrak {g}\) which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closure of these subsets is the whole algebra \(\mathfrak {g}\), apart from two exceptional cases: when \(\mathfrak {g}\) is two-step nilpotent and when \(\mathfrak {g}\) contains a codimension one abelian ideal.     

Einstein metrics on compact Lie groups which are not naturally reductive

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D??Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ??? 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ??? 11), Sp(n) (n ??? 3), E ₆, E ₇, and E ₈.     

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 ).     

Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicals

Abstract

In this article, solvable Leibniz algebras, whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to the nilradical is proved.

Communicated by K. C. Misra     

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.     

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting \mathfrakg^* = V₁ ?V₂ {\mathfrak{g}^*} = {V_1} \oplus {V_2} , and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.     

Endotrivial modules for nilpotent restricted Lie algebras

Let $$mathfrak {g}$$ be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For $$pgeqslant 5$$, we show that every endotrivial $$mathfrak {g}$$-module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for $$p geqslant 5$$ has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.