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

Dietrich Burde (2006)

Open Mathematics

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...

Leibniz A -algebras

David A. Towers (2020)

Communications in Mathematics

A finite-dimensional Lie algebra is called an A -algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras.

Lie solvable groups algebras of derived length three.

Meena Sahai (1995)

Publicacions Matemàtiques

Let K be a field of characteristic p > 2 and let G be a group. Necessary and sufficient conditions are obtained so that the group algebra KG is strongly Lie solvable of derived length at most 3. It is also shown that these conditions are equivalent to KG Lie solvable of derived length 3 in characteristic p ≥ 7.

Maximal solvable extensions of filiform algebras

Libor Šnobl (2011)

Archivum Mathematicum

It is already known that any filiform Lie algebra which possesses a codimension 2 solvable extension is naturally graded. Here we present an alternative derivation of this result.

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