THE Q-DEFORMED BOSON REALIZATION OF REPRESENTATIONS OF QUANTUM UNIVERSAL ENVELOPING-ALGEBRAS FOR Q A ROOT OF UNITY - (I) THE CASE OF UQSL(L)

The properties of q-deformed boson operators with non-generic q (q is a root of unity) are analysed by using the representation theory method and their finite-dimensional representations are thereby obtained. Based on this discussion, reducibilities and decompositions of q-deformed boson-realized representations of quantum universal enveloping algebra U(q)SL(l) are studied for non-generic cases. The explicit matrix elements of some indecomposable representations are obtained on the q-deformed Fock spaces. Necessary details are provided for U(q)SL(2) and U(q)SL(3). In particular, the Lusztig operator extension of U(q)SL(2) is discussed in an explicit form.