Entrelacement d'algèbres de Lie [Wreath products for Lie algebras]
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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.
A moderately detailed english summary of the paper can be found on pages 9 to 15, in "Produit d'entrelacement et action triangulaire d'algèbres de Lie", arXiv:math.RT/0704.3840v1
A moderately detailed english summary of the paper can be found on pages 9 to 15, in "Produit d'entrelacement et action triangulaire d'algèbres de Lie", arXiv:math.RT/0704.3840v1