Generalized tangent representations for groups and symmetric spaces of matrices

Abstract

Any two representations of dimensions n resp. r of a given group G allow the construction of a third representation φ in the space of rectangular n × r matrices Kn,r over the same ground field K. The φ-semidirect product of Kn,r and G then has (n + r) dimensional representation. The inhomogenizations of G and in case of matrix Lie groups G the tangent groups are special cases of this construction. The contragredient as well as the Lie algebraical versions of these results are included. In the final section the construction is generalized to symmetric spaces and their local algebraical structures, the Lie triples, by defining semidirect products resp. semidirect sums with respect to a representation