Left-symmetric Algebras Research Articles

On bi-symmetric algebras

LEFT-symmetric algebra is a new kind of algebra system obtained from the studying of Lie al-gebra, Lie group and differential geometry. It is very useful for many topics in geometry andalgebra. In this note, we discuss a special kind of left-symmetric algebra which is verymeaningful--bi-symmetric algebra.

The five-dimensional complete left-symmetric algebra structures compatible with an abelian Lie algebra structure

The classification of five-dimensional complete left-symmetric algebras is known to be a hard problem. In this paper we establish an explicit classification of the five-dimensional complete left-symmetric algebras over R , having an abelian associated Lie algebra. A detailed list of all possible isomorphism types is presented.

Uniqueness theorems for left-symmetric enveloping algebras

AbstractLet L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

Left-Invariant Affine Structures on Reductive Lie Groups

We describe left-invariant affine structures (that is, left-invariant flat torsion-free affine connections ∇) on reductive linear Lie groupsG. They correspond bijectively to LSA-structures on the Lie algebra g ofG. Here LSA stands for left-symmetric algebra. If g has trivial or one-dimensional center z then the affine representation α=λ⊕1 of g, induced by any LSA-structure gλon g isradiant, i.e., the radiance obstructioncα∈H1(g,gλ) vanishes. If dimz=1 we prove that g=s⊕z, where s is split simple, admits LSA-structures if and only if s is of typeAl, that is, g=gln.Here we have the associative LSA-structure given by ordinary matrix multiplication corresponding to the bi-invariant affine structure on GL(n), which was believed to be essentially the only possible LSA-structure on gln. We exhibit interesting LSA-structures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSA-structures on glnusing a result of Baues. Forn=2 we compute all structures explicitly over the complex numbers.

Sharply transitive linear algebraic groups

We study Zariski-closed linear groupsG ≤ GL n (k) over fieldsk of characteristic 0 which act sharply transitively on the non-zero vectors ofk n . For square-freen, orn⩽15, or ifk has cohomological dimension ⩽1 we obtain a complete classification (i.e. a reduction to questions about associative division algebras). The main tools are representation theory of Lie algebras over algebraically closed and non-closed fields, and results about simple associative algebras in order to control the interplay between linear Lie algebras and the associative algebras generated by them. The relation to nearfields and left-symmetric division algebras is also discussed.

Inductive Construction of Homogeneous Cones

A method is explained how to construct all homogeneous cones in a unique way out of lower dimensional ones. The infinitesimal automorphisms of such a cone and its associated left-symmetric algebras are described in terms of the lower dimensional constituents of the cone. It is characterized when a homogeneous cone is self-dual or a sum of homogeneous cones.

Inductive construction of homogeneous cones

A method is explained how to construct all homogeneous cones in a unique way out of lower dimensional ones. The infinitesimal automorphisms of such a cone and its associated left-symmetric algebras are described in terms of the lower dimensional constituents of the cone. It is characterized when a homogeneous cone is self-dual or a sum of homogeneous cones.