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

Abstract

Abstract. 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.

Rota–Baxter Operators on Pre-Lie Superalgebras

“…They play an important role in the study of symplectic and complex structures on Lie groups and Lie algebras [5,22,24,25,44], phases spaces of Lie algebras [8,42], certain integrable systems [16], classical and quantum Yang-Baxter equations [26], combinatorics [27], quantum field theory [23] and operads [19]. See [17] for a survey. Recently, pre-Lie superalgebras, the Z 2 -graded version of pre-Lie algebras also appeared in many others fields; see, for example, [19,34,52].…”

Realization of Homogeneous Cones Through Oriented Graphs

“…Another motivation is that we wanted to write up basics of homogeneous cones thoroughly in terms of clans, which, introduced also by Vinberg [23], are left-symmetric algebras (LSAs) with additional conditions (see Section 2.1 for the precise definition). LSAs are more widely known compared with the other two algebras, T -algebras and N-algebras, introduced by Vinberg [23] as is seen from the survey articles of Burde [5] and Manchon [19]. Moreover, an LSA A is Lie-admissible in the sense of Albert (cf.…”

“…Let be the homogeneous cone corresponding to V . The source homogeneous cones [4] and [5] are nine-dimensional and 10-dimensional, respectively, and their unique existence is guaranteed by [17]. According to (3.3), (3.9), (3.11) and (4.11), we have 0 12) where I 2 is the 2 × 2 identity matrix, 0 2 ∈ R 2 is the column zero vector, e 1 = 1 0 ∈ R 2 , and x 51 = ∈ R 2 .…”

“…The weighted oriented subgraphs [4] and [5] formed by N out [4] and N out [5], respectively, are both of the type S 5 4 in the notation used by Kaneyuki and Tsuji [17, p. 14]. We note here that to translate what they call skeletons to our weighted oriented graphs, we first replace each vertex name j with r + 1 − j (r is the order of the graph) in the skeleton, and we add arrows to the edges by obeying our ordering †.…”

Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory

“…A detailed survey of occurrences of pre-Lie algebras in geometry, physics, and the theory of formal languages can be found in [3].…”