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Semi-simple splittings for solvable Lie groups and polynomial structures

Semi-simple splittings for solvable Lie groups and polynomial structures Abstract. In this paper we construct a semi-simple splitting for all connected and simply connected solvable Lie groups G. Such a semi-simple splitting G is itself a connected and simply connected solvable Lie group, containing G, and moreover, G splits over its nilradical. The construction we present is the continuous analogue of a similar construction for polycyclic groups, due to D. Segal. Finally we use this semi-simple splitting to show that any G admits a polynomial structure of degree dim(G ). 1991 Mathematics Subject Classi®cation: 22E25. 1 Introduction The purpose of this paper is to continue our study of polynomial structures on solvable groups (see below). In order to be able to do so, we need much knowledge concerning the structure of solvable Lie groups. Any simply connected, connected solvable Lie group G ®ts in a short exact sequence 1 3 N 3 G 3 Rs 3 1 where N, the nilradical of G, is a connected and simply connected nilpotent subgroup of G. This is very comfortable because the simple connected nilpotent Lie groups are pretty well understood and behave in many ways quite nicely (e.g. the exponential map is an analytic di¨eomorphisms in this case). Unfortunately, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Semi-simple splittings for solvable Lie groups and polynomial structures

Dekimpe, Karel
Forum Mathematicum , Volume 12 (1) – Dec 8, 1999
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References (12)

Publisher
de Gruyter
Copyright
Copyright © 1999 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1999.030
Publisher site
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Abstract

Abstract. In this paper we construct a semi-simple splitting for all connected and simply connected solvable Lie groups G. Such a semi-simple splitting G is itself a connected and simply connected solvable Lie group, containing G, and moreover, G splits over its nilradical. The construction we present is the continuous analogue of a similar construction for polycyclic groups, due to D. Segal. Finally we use this semi-simple splitting to show that any G admits a polynomial structure of degree dim(G ). 1991 Mathematics Subject Classi®cation: 22E25. 1 Introduction The purpose of this paper is to continue our study of polynomial structures on solvable groups (see below). In order to be able to do so, we need much knowledge concerning the structure of solvable Lie groups. Any simply connected, connected solvable Lie group G ®ts in a short exact sequence 1 3 N 3 G 3 Rs 3 1 where N, the nilradical of G, is a connected and simply connected nilpotent subgroup of G. This is very comfortable because the simple connected nilpotent Lie groups are pretty well understood and behave in many ways quite nicely (e.g. the exponential map is an analytic di¨eomorphisms in this case). Unfortunately,

Journal

Forum Mathematicum – de Gruyter

Published: Dec 8, 1999

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