Journal article
Extensions and Index of Hermitian Representations
Publications of the Research Institute for Mathematical Sciences, Vol.25(6), pp.923-945
1989
DOI: 10.2977/prims/1195172511
Abstract
We study Hermitian representations of algebras A with involution, a→a*. A representation π of A on a vector space D is said to be Hermitian (or to be a positive-energy-representation) if D carries a positive definite inner product (·, ·) such that (π(a)u, v)=(u, π(a*)v), a∈A, u, v∈D. Such representations arise in quantum field theory and in the study of unitary representations of Lie groups. They were introduced (in this general context) by Powers. We show that main features of von Neumann's index theory (for single Hermitian operators) carry over to representations. Moreover, we get explicit index-information directly from the representation theory, and this is applied to the study of representations of semisimple Lie algebras \mathfrak{g} (with Cartan decomposition \mathfrak{g}=\mathfrak{k}+\mathfrak{p}). For a certain class (to be specified below) of positive energy representations π, we show that the index may be computed from the restriction of π to the compact subalgebra \mathfrak{k}. Our results are then applied to the integrability problem for representations of semi-simple Lie algebras. Several classes of examples are included.
Details
- Title: Subtitle
- Extensions and Index of Hermitian Representations
- Creators
- Palle E.T Jorgensen
- Resource Type
- Journal article
- Publication Details
- Publications of the Research Institute for Mathematical Sciences, Vol.25(6), pp.923-945
- DOI
- 10.2977/prims/1195172511
- ISSN
- 0034-5318
- eISSN
- 1663-4926
- Language
- English
- Date published
- 1989
- Academic Unit
- Mathematics
- Record Identifier
- 9983985913902771
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