Random matrices beyond the Cartan classification

It is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of Lie groups, or else with tangent spaces of the same. This results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables. In this paper we show that a large number of non–hermitean random matrix ensembles defined by physically motivated symmetries – chiral symmetry, time reversal invariance, space rotation invariance, particle–hole symmetry, or different reality conditions – can likewise be identified with symmetric spaces. We give explicit representations of the random matrix ensembles identified with lateral algebra subspaces, and of the corresponding symmetric subalgebras spanning the group of invariance. Among the ensembles listed we identify as special cases all the Cartan classes of symmetric spaces Ensembles of random matrices appear with increasing frequency in various branches of physics. In a typical application, an appropriately chosen ensemble of random matrices (hermitean or not) replaces an ensemble of physical quantum operators with known global