Left-symmetric algebras for 𝔤𝔩(𝔫)

Abstract

Abstract. We study the classification problem for left-symmetric algebras with commutation Lie algebra gl(n) in characteristic 0. The problem is equivalent to the classification ofétale affine representations of gl(n). Algebraic invariant theory is used to characterize those modules for the algebraic group SL(n) which belong to affineétale representations of gl(n). From the classification of these modules we obtain the solution of the classification problem for gl(n). As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.

Crystallographic actions on Lie groups and post-Lie algebra structures

“…For more results and details concerning the reductive case and étale affine representations of reductive groups see [6], [15], [28], [29]. On the other hand, if g is abelian, then we ask which Lie algebras exactly admit an LR-structure.…”

ÉTALE REPRESENTATIONS FOR REDUCTIVE ALGEBRAIC GROUPS WITH FACTORS Spn OR SOn

“…The existence of an affine étale representation for a given group G implies the existence of a left-invariant flat affine connection on G, and these structures appear in many different contexts in mathematics. For the specifics of this relationship and a survey of applications, see Burde [2,3], Baues [1] and the references therein. The primary motivation for the present work is Popov's study of linearizable subgroups of the Cremona group on affine n-space (those that are conjugate to linear group within the Cremona group).…”

“…1 , implying that the open G-orbit is also an open R-orbit. Suppose W 0 is not a direct summand in W 1 .…”

Left-symmetric Structures on Complex Simple Lie Superalgebras

“…Actually, if G is a connected and simply connected Lie group over the field of real numbers whose Lie algebra is g, then there is a left-invariant flat and torsion free connection, that is, an affine structure on G if and only if g has a compatible left-symmetric algebra ( [13,14]). There are a lot of papers addressing the compatible left-symmetric algebras on a given Lie algebra (see [8,12,3,6,2]). In particular, an important result given by Chu [5] asserts that there do not exist any compatible left-symmetric algebra on a complex finite-dimensional semisimple Lie algebra.…”