On the role of semisimple subalgebras in the deformation theory of lie algebras and their homomorphisms general theory and applications

Abstract

Formal deformations of Lie algebras are determined by sequences of bilinear alternating maps, and those of their homomorphisms by sequences of linear maps. The question of the existence, in any equivalence class of formal deformations of Lie algebras and of their homomorphisms, of elements determined by well-behaved sequences is investigated in this paper. A satisfactory affirmative answer is given provided the Lie algebra to be deformed has a semisimple subalgebra different from {0}. The meaning of this result in the geometric approach to deformation theory is pointed out. Applications to the problem of coupling the Poincaré group and an internal symmetry group in a nontrivial way and to the study of deformations of irreducible finite-dimensional representations of E (3) are given.