Naslov
A generalized global Cartan decomposition: a basic example

Autori
Širola, Boris

Izvornik
Communications in Algebra (0092-7872) 34 (2006), 9; 3267-3279

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
symmetric Lie algebra; symplectic group; symplectic Lie algebra; global Cartan decomposition; Pfaffian; J-twisted Pfaffian

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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