Pseudokähler forms on complex Lie groups
type: Item
instance of: scholarly article
title: Pseudokähler forms on complex Lie groups
zbMATH Open document ID: 0982.32021
author: Ralph J. Bremigan
published in: Documenta Mathematica
publication date: 2000-11-30T00:00:00Z
full work available at URL: https://eudml.org/doc/121709
full work available at URL: 17.html
review text: Let \(G\) be a semisimple complex Lie group and \(G_{\mathbf R}\) its real form. Define \(f: i\mathfrak g_{\mathbf R}\times G_{\mathbf R}\to G\) by \(f(iX,r) = \exp(iX)r\) and denote by \(M\subset i\mathfrak g_{\mathbf R}\times G_{\mathbf R}\) the subset of nonsingular points of \(df\). Then \(M\) is a complex manifold, with complex structure \(J\) induced by the complex structure on \(G\). The form \(\omega = -i\partial\bar\partial\phi\) on \(M\), where \(\phi(iX,r) = \kappa(X,X)\), \(\kappa\) being the Killing form on \(\mathfrak g\), is considered. It is proved that \((A,B)\mapsto\omega(JA,B) + i\omega(A,B), A,B\in T_mM\), is a pseudo-Kähler form on \(M\) that is invariant under the \(G_{\mathbf R}\times G_{\mathbf R}\)-action on \(M\) given by \((r_1,r_2)(iX,r) = (i\text{Ad}r_1(X),r_1rr_2^{-1})\). Using \(f\), one can regard it as a \(G_{\mathbf R}\times G_{\mathbf R}\)-invariant pseudo-Kähler form on a neighborhood of the identity in \(G\). The signature of this form is equal to that of the Hermitian form \((Z,W)\mapsto\kappa(Z,\overline W)\). The moment map of the symplectic form \(\omega\) is also calculated.
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 1537285
zbMATH Keywords: pseudo-Kähler form
zbMATH Keywords: symplectic form
zbMATH Keywords: moment map
zbMATH Keywords: Lie group
zbMATH Keywords: cotangent bundle
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1537285