Functional Analysis in Lie Theory
by
Antonio-Jesús Calderón-Martín
Universidad de Cádiz
Coauthors: Cándido Martín González (Universidad de Málaga)

In [1], Lister introduces the concept of Lie triple system. He classified the finite-dimensional simple Lie triple systems over an algebraically closed field of characteristic zero. Neher also studies Lie triple systems and their relations with Jordan triple systems in [2]. In order to study infinite-dimensional Lie triple systems, we introduce the notion of L^*-triple, as a mixture between a Lie triple system and a Hilbert space, and obtain a classification of L^*-triples admitting a two-graded L^*-algebra envelope. However, the problem on the existence of L^*-algebra envelopes is still open. We prove, using Jordan H^*-pairs techniques, that every infinite-dimensional topologically simple L^*-triple, verifying an additional property, has a two-graded L^*-algebra envelope and then, we classify them.

References:

1. W.G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217-242.

2. E. Neher, On the classification of Lie and Jordan triple systems, Comm. in Algebra. Vol. no. 13 (12), 2615-2667.

MSC: 46K70, 46L70

(T)

Date received: April 13, 2000