Irreducible Modules of Finite Dimensional Quantum Algebras of type A at Roots of Unity

The quantum groupUq(g) associated with a simple Lie algebra g is an associative algebra over the rational function field C(q) (q is an indeterminate) and we can define its ”integral” form over the Laurant polynomial ring C[q, q], which enables us to specialize q to any non-zero complex number ε. We are going to see two types of such integral forms and accordingly, we obtain two types of specializations, one is called the ’restriced specialization’ denoted by U res ε , and the other is called the ’non-restricted specialization’ denoted by Uε. Both coincide if ε is trancendental. But we are interested in the case that ε is the l-th primitive root of unity, where l is an odd integer greater than 1. In the case, they do not so. The former is initiated by Lusztig [4],[5] and the latter is introduced in [3] by DeConcini and Kac. Their representation theories are quite different: Irreducible U res ε -modules are highest weight modules in some sense and the classification of the irreducible modules is same as the one for simple Lie algebras or ordinary quantum algebras (see Theorem 3.5 below). Furthermore, irreducble modules possess the remarkable property “tensor product theorem” (see Theorem 3.6 below), which claims that arbitray irreducible highest weight module V (λ) with the highest weight λ is devided into tensor product of two irreducible modules V (λ) and V (lλ) where λ and λ are as in Theorem 3.6. Here the module V (λ) is identified with the irreducible U ε -module, wehre U ε is some finite dimenstional subalgebra of U res ε (see 2.2) and the module V (lλ) can be identified with the irreducible highest weight U(g)module V (λ), whose structure is known very well. Thus, if the structure of V (λ) is clarified, we can analize the detailed feature of V (λ). Indeed, the character of V (λ) is given by the famous Kazhdan-Lusztig formula. But structures as a module, e.g., explicit descriptions of basis vectors or actions of the generators on them, are not still clear. On the other hand, irreducible Uε-modules are not necessarily highest or lowest weight modules. They are characterized by many continious parameters and if they are “generic”, their dimensions are all same (see [3],[1]). But if we specialize the parameters properly, the modules become reducible. In [2], Date, Jimbo, Miki and Miwa constrcuted such Uε-modules for An-type explicitly, which is called the ’maximal cyclic representations’ that is realized in the vector space V := (C) 1 2. They contains the continious parameters and it is shown that if those parameters are generic, they are irreducible. Here we