Integral Structures in Real Semisimple Lie Algebras and Iwasawa N-Groups

Details Integral Structures in Real Semisimple Lie Algebras and Iwasawa N-Groups Integral Structures in Real Semisimple Lie Algebras and Iwasawa N-Groups

2012-02-02

arxiv.org/abs/1202.0451v1

Mathematics

Representation Theory

24 pages, 10 figures

Scientific paper

We construct a convenient basis for all real semisimple Lie algebras by means of an adapted Chevalley basis of the complexification. It determines (half-)integer structure constants which we express only in terms of the root system and its automorphism defining the real structure. Provided the real algebra admits one, the basis exhibits an explicit complex structure. Part of the basis spans the nilpotent algebra of an Iwasawa decomposition. This shows that Iwasawa N-groups have lattices. We give explicit realizations of all Iwasawa N-groups in the real rank one case and we construct coordinate charts for all symmetric spaces of noncompact type in a uniform way.

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