Representations of Groups as Automorphisms on Orthomodular Lattices and Posets

Abstract

In this paper we study the problem of representing groups as groups of automorphisms on an orthomodular lattice or poset. This problem not only has intrinsic mathematical interest but, as we shall see, also has applications to other fields of mathematics and also physics. For example, in the “quantum logic” approach to an axiomatic quantum mechanics, important parts of the theory can not be developed any further until a fairly complete study of the representations of physical symmetry groups on orthomodular lattices is accomplished [1].We will consider two main topics in this paper. The first is the analogue of Schur's lemma and its corollaries in this general setting and the second is a study of induced representations and systems of imprimitivity.