The q-deformed boson realization of representations of quantum universal enveloping algebras for q a root of unity. I The case of UqSL(L)

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Abstract
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 UqSL(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 UqSL(2) and UqSL(3). In particular, the Lusztig operator extension of UqSL(2) is discussed in an explicit form.

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