Left-symmetric algebras, or pre-Lie algebras in geometry and physics

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quit...

Full description

Saved in:
Bibliographic Details
Published in:Open mathematics (Warsaw, Poland) Poland), 2006-09, Vol.4 (3), p.323-357
Main Author: Burde, Dietrich
Format: Article
Language:English
Subjects:
17-02
22-02 53-02
affine manifold
Algebra
cohomology of pre-Lie algebras
convex homogeneous cone
deformation complex
faithful representation
Homology
Lie groups
operad
Physics
Pre-Lie algebra
radical
rooted tree
vertex algebra
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.
ISSN:2391-5455
DOI:10.2478/s11533-006-0014-9