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 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.

[1]  Jean-Louis Koszul,et al.  Domaines bornés homogènes et orbites de groupes de transformations affines , 1961 .

[2]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1 , 1997, The Mathematical Gazette.

[3]  O. Baues.  Left-symmetric algebras for $\mathfrak{gl}(n)$ , 1999 .

[4]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[5]  J. Dixmier,et al.  Derivations of Nilpotent Lie Algebras , 1957 .

[6]  J. Smillie.  An obstruction to the existence of affine structures , 1981 .

[7]  O. Baues,et al.  LEFT-SYMMETRIC ALGEBRAS FOR GL(N) , 1999 .

[8]  V. Kac.  Vertex algebras for beginners , 1997 .

[9]  Hyuk Kim.  Complete left-invariant affine structures on nilpotent Lie groups , 1986 .

[10]  G. Meigniez,et al.  Inexistence de structures affines sur les fibrés de Seifert , 1993 .

[11]  L. Auslander.  SIMPLY TRANSITIVE GROUPS OF AFFINE MOTIONS. , 1977 .

[12]  W. Malfait,et al.  Affine structures on a class of virtually nilpotent groups , 1996 .

[13]  Murray Gerstenhaber,et al.  The Cohomology Structure of an Associative Ring , 1963 .

[14]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1 , 1997, The Mathematical Gazette.

[15]  N. Jacobson,et al.  Schur’s theorems on commutative matrices , 1944 .

[16]  V. Kac,et al.  Structure theory of finite conformal algebras , 1998 .

[17]  James Lepowsky,et al.  On Axiomatic Approaches to Vertex Operator Algebras and Modules , 1993 .

[18]  N. Jacobson.  A note on automorphisms and derivations of Lie algebras , 1955 .

[19]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[20]  J. Milnor.  On fundamental groups of complete affinely flat manifolds , 1977 .

[21]  A. Ogg,et al.  Review: Igor Frenkel, James Lepowsky and Arne Meurman, Vertex operator algebras and the Monster , 1991 .

[22]  Frederic Chapoton,et al.  Pre-Lie algebras and the rooted trees operad , 2000 .

[23]  J. Lepowsky,et al.  Introduction to Vertex Operator Algebras and Their Representations , 2003 .

[24]  D. Burde.  Simple left-symmetric algebras with¶solvable Lie algebra , 1998 .

[25]  J. P. Benzecri.  Variétés localement affines , 1960 .

[26]  D. Kreimer.  Structures in Feynman Graphs -Hopf Algebras and Symmetries , 2002, hep-th/0202110.

[27]  A. Dzhumadil'daev.  Conomologies and deformations of right-symmetric algebras , 1998, math/9807065.

[28]  Alain Connes,et al.  Hopf Algebras, Renormalization and Noncommutative Geometry , 1998 .

[29]  J. Helmstetter.  Radical d'une algèbre symétrique à gauche , 1979 .

[30]  D. Burde.  AFFINE STRUCTURES ON NILMANIFOLDS , 1996 .

[31]  R. Borcherds.  VERTEX ALGEBRAS FOR BEGINNERS (University Lecture Series 10) , 1998 .

[32]  Left-Invariant Affine Structures on Reductive Lie Groups , 1996 .

[33]  C. Löfwall,et al.  Trees, free right-symmetric algebras, free Novikov algebras and identities , 2002 .

[34]  F. Grunewald,et al.  Modules for certain Lie algebras of maximal class , 1995 .

[35]  T. Nagano,et al.  THE AFFINE STRUCTURES ON THE REAL TWO-TORUS (I) , 1974 .

[36]  Hyuk Kim,et al.  RADICALS OF A LEFT-SYMMETRIC ALGEBRA ON A NILPOTENT LIE GROUP , 2004 .

[37]  Nguiffo B. Boyom.  Sur les structures affines homotopes à zéro des groupes de Lie , 1990 .

[38]  C. Bai,et al.  A Lie algebraic approach to Novikov algebras , 2003 .

[39]  D. Segal.  The structure of complete left-symmetric algebras , 1992 .

[40]  Hyuk Kim,et al.  On radicals of a left-symmetric algebra , 1999 .

[41]  D. Burde.  Estimates on binomial sums of partition functions , 2000 .

[42]  Bruce Reed.  Representations of solvable Lie algebras. , 1969 .

[43]  J. Schur.  Zur Theorie der vertauschbaren Matrizen. , 1905 .

[44]  D. Burde.  Left-invariant affine structures on nilpotent Lie groups , 1999 .

[45]  Affine structures on 4-step nilpotent lie algebras , 1997 .

[46]  D. Burde.  AFFINE COHOMOLOGY CLASSES FOR FILIFORM LIE ALGEBRAS , 2006 .

[47]  R. Borcherds.  Vertex algebras, Kac-Moody algebras, and the Monster. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[48]  New mathematical structures in renormalizable quantum field theories , 2002, hep-th/0211136.

[49]  A. Nijenhuis.  The Graded Lie Algebras of an Algebra , 1967 .

[50]  Y. Benoist.  Une nilvariété non affine , 1992 .

[51]  J. Osborn.  Infinite Dimensional Novikov Algebras of Characteristic 0 , 1994 .