Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · dg-ga/9703003

Twisted product of Lie groups

Michael A. Rudkovski
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself through the action of inner automorphisms is a group if and only if the initial group is a metabelian group. Further we will construct Lie algebra for Lie group of a twisted product of Lie groups. In the case of twisted product of Lie group by itself by means of the action of inner automorphisms we find the dependence of the scalar curvature for resulting Lie group on the scalar curvature for initial Lie group.

Explore related subjects

dg-gamath.DG

Keep this discovery

BibTeXRIS

Michael A. Rudkovski. 1997-03-03. Twisted product of Lie groups. https://arxiv.org/abs/dg-ga/9703003

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2013

Moduli spaces of PU(2)-monopoles

The goal of this article was the S^1-equivariant transversality-problem and the compactification-problem for the moduli spaces of (perturbed) PU(2)-monopoles. A substantially improved version entitled "Moduli spaces of PU(2)-monopoles (revised version)" which gives simpler, clearer proofs of the transversality results, has been published on arxiv in June 99 and appeared in Asian J. Math, see Moduli spaces of PU(2)-Monopoles, Asian J. Math. Vol. 4, No. 2 (2000), 391-436.

dg-ga↗
arXiv2004

Duality for Lie-Rinehart algebras and the modular class

We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincaré duality in Lie algebra cohomology and de Rham cohomology. We show that the duality isomorphisms can be given by a cap product with a suitable fundamental class and hence may be taken natural in any reasonable sense. Thereafter, for a Lie-Rinehart algebra satisfying duality, we introduce a certain intrinsic module which provides a crucial ingredient for the construction of the bilinear pairings. This module determines a certain class, called modular class of the Lie-Rinehart algebra, which lies in a certain Picard group generalizing the abelian group of flat line bundles on a smooth manifold. Finally, we show that a Poisson algebra having suitable properties determines a certain module for the corrresponding Lie-Rinehart algebra and hence modular class whose square yields the module and characteristic class for its Lie-Rinehart algebra mentioned before. In particular, this gives rise to certain bilinear pairings in Poisson homology.

dg-ga↗
arXiv2002

Seiberg-Witten Equations on Three-Manifolds with Euclidean Ends

We construct the Seiberg-Witten theory on 3-manifolds with Euclidean ends (connected sums of $\R^3$ and a compact manifold) with perturbations which approximate $*dx_3$ at infinity, and describe the structure of the moduli spaces. The setup is inspired by Taubes's program of relating the 4-dimensional Seiberg-Witten invariant with `singular Gromov invariants' and has related applications.

dg-ga↗