A Generalized Global Cartan Decomposition: A Basic Example
| Autor: | Boris Širola |
|---|---|
| Rok vydání: | 2006 |
| Předmět: |
Discrete mathematics
Pure mathematics Algebra and Number Theory Simple Lie group Adjoint representation Cartan matrix Cartan decomposition Real form Cartan subalgebra symmetric Lie algebra symplectic group symplectic Lie algebra global Cartan decomposition Pfaffian J-twisted Pfaffian Killing form Kac–Moody algebra Mathematics |
| Zdroj: | Communications in Algebra. 34:3267-3279 |
| ISSN: | 1532-4125 0092-7872 |
| Popis: | Suppose that (G, G1) is a pair of complex linear simple Lie groups such that G contains G1. Let (g, g1), where g contains g1, be the corresponding pair of Lie algebras. For the orthogonal p of g1 in g with respect to the Killing form of g, we have a vector space direct sum g=g1+p which generalizes the classical Cartan decomposition on the Lie algebras level. In this paper we study the corresponding problem of a 'generalized global Cartan decomposition' on the Lie groups level for the concrete pair of groups (G, G1)=(SL(4, C), Sp(2, C)) ; here g=sl(4, C), G1=sp(2, C) and p={; ; ; ; ; X in g|X^#=X}; ; ; ; ; , where the map sending X to X^# is the symplectic involution. We prove that then G=G1exp(p)UiG1exp(p). The key point of the proof is to study in detail the set exp(p) ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n times 2n matrices X satisfying X^#=X, which is here a natural counterpart of the standard Pfaffian. |
| Databáze: | OpenAIRE |
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