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

By Dietrich Burde. 2005

366 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/243a846dd6df044c83ac1d6274a145af1dc727fb

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In Algebra and Number Theory

1. Li Guo
2. Chengming Bai
3. Wolfgang Alexander Moens
4. Alice Fialowski
5. Xianglong Ni
6. Dietrich Burde
7. Kurusch Ebrahimi-Fard
8. William F. Keigher

In Geometry and Topology

1. Chengming Bai
2. Wolfgang Alexander Moens
3. Alice Fialowski
4. Li Guo
5. Xianglong Ni
6. Dietrich Burde
7. Kurusch Ebrahimi-Fard
8. William F. Keigher

In Mathematical Physics

1. Wolfgang Alexander Moens
2. Alice Fialowski
3. Chengming Bai
4. Xianglong Ni
5. Li Guo
6. Dietrich Burde
7. Loïc Foissy
8. Manuel Ceballos

Citations in the corpus

• Maximally homogeneous para-CR manifolds of semisimple type (2006, neutral): “rs play an important role in the study of left invariant flat affine structures on Lie groups, see section 5. Left symmetric algebras also appear in several other mathematical and physical contexts, see [25] for a recent survey. 3.2 Development representation of affinities Let(M,˜ ∇) be theuniversalcover”
• Jacobi–Jordan algebras (2014, methodology): “We mention also pre-Lie algebras, Novikov algebras and LR-algebras, which we studied in a geometric context, see [3], [2], [4].”
• Affine actions on Lie groups and post-Lie algebra structures (2011, background): “A complex semisimple Lie algebra does not admit a pre-Lie algebra structure, see [5], [8].”
• On the Lie Enveloping Algebra of a Post-Lie Algebra (2014, neutral): “We begin by defining pre-Lie algebras, see [1, 5, 17] for further details.”
• Commutative post-Lie algebra structures on Lie algebras (2015, methodology): “We rediscovered post-Lie algebras as a natural common generalization of pre-Lie algebras [11, 12, 17, 2, 3, 4] and LR-algebras [6, 7] in the geometric context of nil-affine actions of Lie groups.”
• Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras (2009, background): “, is the Koszul dual of ComTrias, the operad of commutative trialgebras. (ii) If the bracket [,] in the definition of PostLie algebra happens to be trivial, t hen a PostLie algebra is a pre-Lie algebra[11]. Lemma 5.3. Let (L,[,],◦) be a PostLie algebra. Define ρ : L → gl(L) by ρ(x)y = x ◦ y for any x,”