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Wreath products and Kaluzhnin-Krasner embedding for Lie algebras


Authors: V. M. Petrogradsky, Yu. P. Razmyslov and E. O. Shishkin
Journal: Proc. Amer. Math. Soc. 135 (2007), 625-636
MSC (2000): Primary 17B05, 17B35, 17B66, 11N45, 16W30
Published electronically: August 28, 2006
MathSciNet review: 2262857
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Abstract: The wreath product of groups $ A\wr B$ is one of basic constructions in group theory. We construct its analogue, a wreath product of Lie algebras.

Consider Lie algebras $ H$ and $ G$ over a field $ K$. Let $ U(G)$ be the universal enveloping algebra. Then $ \bar H=\operatorname{Hom}_K(U(G),H)$ has the natural structure of a Lie algebra, where the multiplication is defined via the comultiplication in $ U(G)$. Also, $ G$ acts by derivations on $ \bar H$ via the (left) coregular action. The semidirect sum $ \bar H \leftthreetimes G$ we call the wreath product and denote by $ H\wr G$. As a main result, we prove that an arbitrary extension of Lie algebras $ 0\to H\to L\to G\to 0$ can be embedded into the wreath product $ L\hookrightarrow H\wr G$.


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Additional Information

V. M. Petrogradsky
Affiliation: Faculty of Mathematics, Ulyanovsk State University, Lev Tolstoy 42, Ulyanovsk, 432970 Russia
Email: petrogradsky@rambler.ru, petrogradsky@hotbox.ru

Yu. P. Razmyslov
Affiliation: Department of Mechanics and Mathematics, Moscow State University, Moscow, 119992 Russia
Email: pankrat@shade.msu.ru

E. O. Shishkin
Affiliation: Department of Mechanics and Mathematics, Moscow State University, Moscow, 119992 Russia

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08502-9
Received by editor(s): June 14, 2005
Received by editor(s) in revised form: September 20, 2005
Published electronically: August 28, 2006
Additional Notes: This research was partially supported by Grant RFBR-04-01-00739
Communicated by: Martin Lorenz
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.