NONASSOCIATIVE ALGEBRAS: A FRAMEWORK FOR DIFFERENTIAL GEOMETRY

• Published in: International Journal of Mathematics and Mathematical Sciences Vol. 2003; no. 60; pp. 3777 - 3795-282
• Main Author: Ionescu, Lucian M.
• Format: Journal Article
• Language: English
• Published: Hindawi Limiteds 2003; Hindawi Limited
• ISSN: 0161-1712; 1687-0425
• DOI: 10.1155/S0161171203303023

A nonassociative algebra endowed with a Lie bracket, called a torsion algebra , is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre‐Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature‐associator‐monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre‐Lie algebra of Hochschild cochains of a K ‐module, with Lie bracket induced by Gerstenhaber composition.