Colloquium du 12-04-2023
Exposé de Alberto FacchiniLe 12-04-2023 à 15:00, en ligne.
We all have studied groups and rings, and have heard of other algebraic structures like semigroups, Lie algebras, Poisson algebras or Jordan algebras. But other very interesting non-associative or non-distributive algebraic structures have appeared on the scene, helping us to better understand the structure of operations: multiplicative lattices, skew braces, pre-Lie algebras, heaps, trusses,… In this Colloquium we will focus on pre-Lie algebras. Their very first appearance was in the sixties (Gerstenhaber and Vinberg, independently). We will present the first properties of pre-Lie algebras. More generally, we will present the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring k with identity. These notions will be applied to the study of pre-Lie k-algebras and Lie-admissible k-algebras. Associating with any algebra (A,.) its sub-adjacent anticommutative algebra (A,[-,-]) is a functor from the category of k-algebras with pre-morphisms to the category of anticommutative k-algebras. Pre-morphisms can be dualized in various ways to generalized morphisms (related to pre-Jordan algebras) and anti-pre-morphisms (related to anti-pre-Lie algebras).
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