On the n-symplectic structure of faithful irreducible representations

Each faithful irreducible representation of an N-dimensional vector space V-1 on an n-dimensional vector space V-2 is shown to define a unique irreducible n-symplectic structure on the product manifold V-1 x V-2. The basic details of the associated Poisson algebra are developed for the special case N = n(2), and 2n-dimensional symplectic submanifolds are shownto exist. Published by AIP Publishing.