Classification and construction of finite dimensional irreducible representations of the graded algebras; application to the (Sp(2<i>n</i>); 2<i>n</i>) algebra

Abstract

A method, which enables us to construct the finite-dimensional representations of the graded Lie algebras [explicitly the GLA (Sp(2N); 2N)] on the irreducible tensors is suggested. Those tensors are constructed with the aid of the specifically symmetrized products of vectors from the fundamental [i.e., (2N+1) -dimensional] representation space of the graded Lie algebra. The tensors, on which it is possible to represent the graded algebra (Sp(2N); 2N) irreducibly, represent a generalization of tensors which are known from the general representation theory of the symplectic Lie algebra Sp(2N). The knowledge of the irreducible tensors of the algebra Sp(2N) gives us then the possibility of solving the problems of classification as well as construction of the irreducible tensors of the graded algebra (Sp(2N); 2N). For illustration, by using the suggested method of tensors the irreducible representations of the simplest graded algebra, i.e., of the algebra (Sp(2); 2) are constructed.