On the meromorphic potential for a harmonic surface in a \(k\)-symmetric space
type: Item
instance of: scholarly article
title: On the meromorphic potential for a harmonic surface in a \(k\)-symmetric space
zbMATH Open document ID: 0903.58005
DOI: 10.1007/BF02678186
author: Ian McIntosh
author: Hongyou Wu
author: Josef F. Dorfmeister
author: Franz Pedit
published in: Manuscripta Mathematica
publication date: 1997-07-16T00:00:00Z
full work available at URL: https://eudml.org/doc/156257
review text: The paper deals with harmonic maps \(\psi: \Omega\to G/K\), where \(\Omega\) is the complex plane or the Poincaré disc and \(G/K\) denotes the homogeneous space of a compact semisimple Lie group \(G\) generated by a periodic automorphism \(\sigma\in \Aut (G)\) of order \(k\geq 2\) (thus, \(G/K\) is a \(k\)-symmetric space in the sense of \textit{O. Kowalski} [`Generalized symmetric spaces' (Springer, Berlin) (1980; Zbl 0431.53042)]). In the paper under review, there are given canonical ways of constructing primitive harmonic maps from holomorphic and meromorphic potentials. For example, the holomorphic potential is defined by means of some loop group \(L_iG^\mathbb{C}_\sigma\) and its Lie algebra \(L_i{\mathfrak g}^\mathbb{C}_\sigma\) (defined in the article) as follows. Let \[ \Lambda_{-1,\infty} =\{X\in L_1 {\mathfrak g}^\mathbb{C}_\sigma \mid \lambda X\quad \text{extends holomorphically into } I_0\} \] \((I_0\) is a disc) and define the holomorphic potential as a holomorphic section \(\xi\) of \(T^*_{1,0} \otimes \Lambda_{-1, \infty}\). In the same fashion, the meromorphic potentials are defined using meromorphic sections of \(T^*_{1,0} \otimes {\mathfrak g}_{-1}\) \(({\mathfrak g}_i\) denote subspaces of the grading determined by \(\sigma)\). Some formulae for meromorphic and holomorphic potentials are given.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1019705
zbMATH Keywords: \(k\)-symmetric space
zbMATH Keywords: harmonic maps
zbMATH Keywords: meromorphic potentials
zbMATH Keywords: holomorphic potentials
MaRDI profile type: Publication
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OpenAlex ID: W2042830085
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description: scientific article; zbMATH DE number 1019705