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Left-symmetric algebras, or pre-Lie algebras in geometry and physics. (English) Zbl 1151.17301

A nonassociative algebra over a field is called left-symmetric if it satisfies the identity $\left(x,y,z\right)=\left(y,x,z\right)$, where $\left(x,y,z\right)=\left(xy\right)z-x\left(yz\right)$ is the associator. Left-symmetric algebras (LSAs in short) were introduced by Cayley in the context of rooted tree algebras. Forgotten for a long time, they appeared again in the work of E. B. Vinberg [see e.g. Tr. Mosk. Mat. O.-va. 12, 303–358 (1963; Zbl 0138.43301)] and J.-L. Koszul, Bull. Soc. Math. Fr. 89, 515–533 (1961; Zbl 0144.34002)] in the context of convex homogeneous cones and affinely flat manifolds.

From the 1960’s many articles on LSAs have been published, from quite different research areas, and under many different names, e.g. Vinberg algebras, Koszul algebras, or quasi-associative algebras. Right-symmetric algebras, satisfying $\left(x,y,z\right)=\left(x,z,y\right)$, are also called Gerstenhaber algebras or pre-Lie algebras. The article under review is a survey on the origin, theory and applications of left-symmetric algebras in geometry and in physics. The author considers relations with vector fields (relations with right Novikov algebras and Witt algebras), rooted tree algebras (relations with renormalizable quantum field theories and Feynman graphs), right-symmetric structure on algebras of words on an alphabet, vertex algebras, operads, deformation complexes of algebras, convex homogeneous cones, affine manifolds, left-invariant affine structures on Lie groups.

Furthermore the author studies the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. Finally, he discusses relations with the problem to find the minimal dimension of a faithful Lie algebra representation.

##### MSC:
 17A30 Nonassociative algebras satisfying other identities 17-02 Research monographs (nonassociative rings and algebras) 17B30 Solvable, nilpotent Lie (super)algebras 22E60 Lie algebras of Lie groups