Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-23T19:29:56.416Z Has data issue: false hasContentIssue false
  • English
  • Français

On involutive Lie algebras having a Cartan decomposition

Published online by Cambridge University Press:  17 April 2009

A. J. Calderón Martín
A. J. Calderón Martín
Affiliation:
Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Cádiz e-mail: ajesus.calderon@uca.es
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce the concept of Cartan decomposition relative to a Cartan subalgebra H in the sense of Y. Billig and A. Pianzola for involutive complex Lie algebras L of arbitrary dimension. If L has such a decomposition and is infinite dimensional and simple, we show it is *-isomorphic to a direct limit of classical finite dimensional simple involutive Lie algebras of the same type A, B, C, or D.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 2 , April 2004 , pp. 191 - 202
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Balachandran, V.K., ‘Simple Systems of Roots in L*-algebras’, Trans. Amer. Math. Soc. 130 (1968), 513–524.Google Scholar
[2]Billig, Y. and Pianzola, A., ‘On Cartan subalgebras’, J. Algebra 171 (1995), 397–412.CrossRefGoogle Scholar
[3]Calderón, A.J. and Martín, C., ‘Direct limits of L*-triples’, Algebras Groups Geom. 18 (2001), 223–232.Google Scholar
[4]Calderón, A.J. and Martín, C., ‘Hilbert space methods in the theory of Lie triple systems’, in Recent Progress in Functional Analysis, (Bierstedt, K.D., Bonet, J., Maestre, M. and Schmets, J., Editors), North-Holland Math. Studies 189 (North-Holland Publishing Co., Amsterdam, 2001), pp. 309–319.Google Scholar
[5]Calderón, A.J. and Forero, M., ‘Roots and roots spaces of compact Lie algebras’, Irish Math. Soc. Bull. 49 (2002), 15–22.CrossRefGoogle Scholar
[6]Calderón, A.J. and Forero, M., ‘On infinite dimensional Lie algebras having a Cartan decomposition’, (preprint, Universidad de Cádiz).Google Scholar
[7]Cuenca, J.A., García, A. and Martín, C., ‘Structure theory for L*-algebras’, Math. Proc. Cambridge Philos. Soc. 107 (1990), 361–365.CrossRefGoogle Scholar
[8]Jacobson, N., Lie algebras (Dover Publications, Inc., New York, N.Y., 1979).Google Scholar
[9]Moody, R. and Pianzola, A., Lie algebras with triangular decomposition, Canad. Math. Soc. Series of Monographs and Advanced Texts (J. Wiley & Sons, New York, 1995).Google Scholar
[10]Neeb, K.-H., ‘Locally finite Lie algebras with unitary highest weight representations’, Manuscripta Math. 104 (2001), 359–381.CrossRefGoogle Scholar
[11]Neeb, K.-H. and Stumme, N., ‘The classification of locally finite split simple Lie algebras’, J. Reine Angew Math. 533 (2001), 25–53.Google Scholar
[12]Neher, E., ‘Cartan-involutionen von halbeinfachen rellen Jordan tripelsystemen’, Math. Z. 169 (1979), 271–292.CrossRefGoogle Scholar
[13]Schue, J.R., ‘Hilbert Space methods in the theory of Lie algebras’, Trans. Amer. Math. Soc. 95 (1960), 69–80.CrossRefGoogle Scholar
[14]Schue, J.R., ‘Cartan decompositions for L*-algebras’, Trans. Amer. Math. Soc. 98 (1961), 334–349.Google Scholar
[15]Stumme, N., ‘The structure of locallyfinite split Lie algebras’, J. Algebra 220 (1999), 664–693.CrossRefGoogle Scholar
[16]Stumme, N., ‘Automorphisms and conjugancy of compact real forms of the classical infinite-dimensional matrix Lie algebras’, Forum Math. 13 (2001), 817–851.CrossRefGoogle Scholar