Left symmetric algebras over a real reductive Lie algebra
The author studies left-symmetric algebra structures on real reductive Lie algebras \({\mathfrak g}\). A bilinear product \((x,y)\mapsto x.y\) is called a left-symmetric algebra structure on \({\mathfrak g}\) if \([x,y]=x.y-y.x\) and \(x.(y.z)-y.(x.z)-[x,y].z=0\) for all \(x,y,z \in {\mathfrak g}\). The name pre-Lie algebra structure is also used. These structures appear in the context of affine manifolds, convex homogeneous cones, Hochschild cohomology, rooted tree algebras and other topics. If the left-symmetric algebra has a right identity and if the canonical trace form \(h(x,y)=\text{tr} R(xy)\) of the right multiplication \(R\) is nondegenerate, then the algebra is called regular. The paper under review studies the structure of regular left-symmetric algebras \(A\) over a reductive Lie algebra. Under certain conditions, \(A=B\oplus \overline{B}\) with a minimal ideal \(B\) in \(A\) and a suitable subalgebra \(\overline{B}\) with \(B \overline{B}=0\). Some examples of left-symmetric algebra structures over \({\mathfrak gl}_2(\mathbb R)\) are discussed. They coincide with the examples given in [\textit{D. Burde}, J. Algebra 181, No. 3, 884-902 (1996; Zbl 0860.53020)].
scientific article; zbMATH DE number 1679376