期刊名称：International Journal of Mathematics and Mathematical Sciences
印刷版ISSN：0161-1712
电子版ISSN：1687-0425
出版年度：2003
卷号：2003
DOI：10.1155/S0161171203303023
出版社：Hindawi Publishing Corporation
摘要：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.
