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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

On the orbit space of a compact linear Lie group with commutative connected component


Author: O. G. Styrt
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 70 (2009).
Journal: Trans. Moscow Math. Soc. 2009, 171-206
MSC (2000): Primary 22E45; Secondary 20C15, 22C05, 22E15
DOI: https://doi.org/10.1090/S0077-1554-09-00178-2
Published electronically: December 3, 2009
MathSciNet review: 2573640
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Abstract:

This paper is devoted to the study of topological quotients of compact linear Lie groups. More precisely, it investigates the question of when such a quotient is a topological or a smooth manifold.

The topological quotient of a finite linear group was studied by Mikhaĭlova in 1984. Here the connected component $G^0$ of the original Lie group $G$ is assumed to be a torus of positive dimension.

The main method used here is to consider the weight system corresponding to the decomposition of the representation of the commutative group $G^0$ into irreducible representations. In §8 an arbitrary linear group is reduced to a linear group of special type, namely, one with a $2$-stable weight system (for the definition and properties of $q$-stable sets of vectors, where $q\in \mathbb {N}$, see §§1, 4). The main results for a group with a $2$-stable weight system are stated in the Introduction (Theorems 1.3–1.8) and proved in §§6 and 7.


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Additional Information

O. G. Styrt
Affiliation: Moscow State University, Moscow, Russia

Keywords: Compact, linear group, Lie group, orbit space, smooth manifold
Published electronically: December 3, 2009
Article copyright: © Copyright 2009 American Mathematical Society