Entrelacement d'alg\`ebres de Lie [Wreath products for Lie algebras]

TL;DR

This paper develops a comprehensive framework for the wreath product of Lie algebras, introducing new notions and formal actions, with the aim of eventually defining a corresponding Lie group structure.

Contribution

It defines the wreath product of Lie algebras using formal series and actions, and establishes foundational theorems analogous to classical extension results.

Findings

Definition of wreath product for Lie algebras

Description of triangular actions over product vector spaces

A Kaloujnine-Krasner type theorem for Lie algebra extensions

Abstract

Full details are given for the definition and construction of the wreath product of two arbitrary Lie algebras, in the hope that it can lead to the definition of a suitable Lie group to be the wreath product of two given Lie groups. In the process, quite a few new notions are needed, and introduced. Such are, for example : Formal series with variables in a vector space and coefficients in some other vector space. Derivation of a formal series relative to another formal series. The Lie algebra of a vector space. Formal actions of Lie algebras over vector spaces. The basic formal action of a Lie algebra over itself (as a formal version of the analytic aspect of the infinitesimal operation law of a Lie groupuscule). More generally, the wreath product of two Lie algebras is defined, relative to a formal action of the second onto an arbitrary vector space. Main features are : A description of the triangular actions of wreath products over product vector spaces, and a Kaloujnine-Krasner type theorem : In essence, it says that all Lie extensions of a given Lie algebra by another Lie algebra are, indeed, subalgebras of their wreath product.