Quantum Algebra Uq(gl(3)) and Nonlinear Optics

Indecomposable representations are investigated for the Uq(gl(3)) quantum algebra. The matrix elements are explicitly determined for the elementary representations, and the extremal vectors which characterize invariant subspaces are given in explicit form. Quotient spaces are used to derive other representations from the elementary representations, including the finite-dimensional irreducible representations and infinite-dimensional representations which are bounded above. Applications to nonlinear-optical phenomena are discussed.