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.

     

Finite-dimensional solvable Lie algebras generated by normal operators are commutative

It is proved that any finite-dimensional solvable Lie algebra generated by normal operators is commutative, and some other related results are given.     

Cohomological properties of unimodular six dimensional solvable Lie algebras

In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture of Guan about step of nilpotency of a symplectic solvmanifold finding that it is true for all six dimensional unimodular solvable Lie algebras. Finally, we consider some cohomologies for symplectic manifolds introduced by Tseng and Yau in the context of symplectic Hogde theory and we use them to determine some six dimensional solvmanifolds for which the Hard Lefschetz property holds.     

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.     

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 ).     

The classification of two step nilpotent complex Lie algebras of dimension 8

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.     

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.     

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.     

On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension

The Ricci curvature of solvable metric Lie algebras is studied. In particular, we prove that the Ricci operator of any metric nonunimodular solvable Lie algebra of dimension not exceeding 6 has at least two negative eigenvalues, that generalizes the known results.     

Spectra for solvable Lie algebras of bundle endomorphisms

The aim of the paper is to investigate spectral properties of the Lie algebras corresponding to the symmetry groups of certain flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of -algebras. The flag manifolds in -algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles. Received: 8 October 2001 / Revised version: 4 February 2002 / Published online: 6 August 2002 Research supported from the contract ICA1–CT–2000–70022 with the European Commission.     

Lower bounds for algebraic algorithms for nilpotent and solvable lie algebras

We obtain the lower bounds for the tensor rank for the class of nilpotent and solvable Lie algebras (in terms of dimensions of certain quotient algebras). These estimates, in turn, give lower bounds for the complexity of algebraic algorithms for this class of algebras. We adduce examples of attainable estimates for nilpotent Lie algebras of various dimensions.