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.     

一类推广的Virasoro-like李代数

构造了一类无限维李代数,它是Virasoro-like李代数的推广.研究了这类李代数的两类自同构,这两类自同构均关于映射的合成构成自同构群,一类同构于对称群S3,另一类同构于Klein交换群.得到了这类李代数一些特殊的自同态、中心.证明了这类李代数不是半单李代数.     

On the homological finiteness properties of some modules over metabellian Lie algebras

We characterise the modulesB of homological typeFP _m over a finitely generated Lie algebraL such thatL is a split extension of an abelian idealA and an abelian subalgebraQ andA acts trivially onB. The characterisation is in terms of the invariant Δ introduced by R. Bryant and J. Groves and is a Lie algebra version of the generalisation [K 4, conjecture 1] of the still openFP _m -Conjecture for metabelian groups [Bi-G, Conjecture p. 367]. The casem=1 of our main result is treated separately, as there the characterisation is proved without restrictions on the type of the extension.     

C-Ideals of Lie Algebras

A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B ∩ C ≤ B _L , where B _L is the largest ideal of L contained in B. This is analogous to the concept of c-normal subgroup, which has been studied by a number of authors. We obtain some properties of c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also classify those Lie algebras in which every one-dimensional subalgebra is a c-ideal.     

Noetherianness of wreath products of Abelian Lie algebras with respect to equations of universal enveloping algebra

It is proved that a wreath product of two Abelian finite-dimensional Lie algebras over a field of characteristic zero is Noetherian w.r.t. equations of a universal enveloping algebra. This implies that an index 2 soluble free Lie algebra of finite rank, too, has this property.     

A solvability criterion for the Lie algebra of derivations of a fat point

We consider the Lie algebra of derivations of a zero-dimensional local complex algebra. We describe an inequality involving the embedding dimension, the order, and the first deviation that forces this Lie algebra to be solvable. Our result was motivated by and generalizes the solvability of the Yau algebra of an isolated hypersurface singularity.     

Simple left-symmetric algebras with solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie groupG correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we studysimple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied.     

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H ¹(W, W ? W) is trivial.     

Special polynomials in free framed Lie algebra

A framed Lie algebra is an algebra with two operations which is a Lie algebra with respect to one of these operations. A basic example is a Lie algebra of vector fields on a manifold with connection where the covariant derivative serves as an additional operation. In a free framed Lie algebra, we distinguish a set of special polynomials that geometrically correspond to invariantly defined tensors. A necessary condition of being special is derived, and we presume that this condition is also sufficient. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 571–583, September–October, 2008.     

Rigidity of group actions on solvable Lie groups

We establish analogs of the three Bieberbach theorems for a lattice in a semidirect product where is a connected, simply connected solvable Lie group and is a compact subgroup of its automorphism group. We first prove that the action of on is metrically equivalent to an action of on a supersolvable Lie group. The latter is shown to be determined by itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group as a lattice in a semidirect product with being a finite group whose order is bounded by a constant only depending on the dimension of . This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup that the quotient is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group. Received: 27 August 1999     

A Lie algebra attached to a projective variety

Abstract. Each choice of a K?hler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such K?hler classes is given, then we prove that the corresponding sl(2)-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperk?hler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra. Oblatum 21-V-1996 & 15-X-1996     

On the sum of two Lie algebras with finite-dimensional commutants

We prove that an infinite-dimensional Lie algebra over an arbitrary field which is decomposable into the sum of two of its subalgebras with finite-dimensional commutants is almost solvable.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1089–1096, August, 1995.     

Two explicit realizations of a Lie algebra

We introduce a Lie algebra whose some properties are discussed, including its proper ideals, derivations and so on. Then, we again give rise to its two explicit realizations by adopting subalgebra of the Lie algebra A₂ and a column-vector Lie algebra, respectively. Under the frame of zero curvature equations, we may use the realizations to generate the same Lax integrable hierarchies of evolution equations and their Hamiltonian structure.     

Simple left-symmetric algebras with¶solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group {G} correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied. Received: 10 June 1997 / Revised version: 29 September 1997     

Quasi-state rigidity for finite-dimensional Lie algebras

We say that a Lie algebra g is quasi-state rigid if every Ad-invariant continuous Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive Lie algebra, as well as the algebras C ⁿ ? u(n), n ≥ 1, are rigid. On the other hand, a Lie algebra which surjects onto the three-dimensional Heisenberg algebra is not rigid. For Lie algebras of dimension ≤ 3 and for solvable Lie algebras which split over a codimension one abelian ideal, we show that this is the only obstruction to rigidity.     

Computing the test rank of a free solvable Lie algebra

We computed the test rank of a free solvable Lie algebra of finite rank.     

Completeness of quasi-filiform Lie algebras

It is proved that for any quasi-filiform of non-zero rank the solvable Lie algebra obtained by adjoining a maximal torus of outer derivations is complete. Further, for any positive integer m, it is shown that there exist solvable complete Lie algebras with the second Chevalley–Eilenberg cohomology group of arbitrary dimension.     

The solution to the embedding problem of a (differential) Lie algebra into its Wronskian envelope

Any commutative algebra equipped with a derivation may be turned into a Lie algebra under the Wronskian bracket. This provides an entirely new sort of a universal envelope for a Lie algebra, the Wronskian envelope. The main result of this paper is the characterization of those Lie algebras which embed into their Wronskian envelope as Lie algebras of vector fields on a line. As a consequence we show that, in contrast to the classical situation, free Lie algebras almost never embed into their Wronskian envelope.