Some Quantum Analogues of Solvable Lie Groups

Introduction. In the papers [DK1-2],[DKP1-2] the quantized enveloping algebras introduced by Drinfeld and Jimbo have been studied in the case q = ε, a primitive l-th root of 1 with l odd (cf. §2 for basic definitions). Let us only recall for the moment that such algebras are canonically constructed starting from a Cartan matrix of finite type and in particular we can talk of the associated classical objects (the root system, the simply connected algebraic group G, etc.). For such an algebra the generic (resp. any) irreducible representation has dimension equal to (resp. bounded by) l where N is the number of positive roots and the set of irreducible representations has a canonical map to the big cell of the corresponding group G. In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of G. These algebras have the non-commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commutative rings (see e.g. [KP], [GL] and references there). In particular, we find maximal dimensions of their irreducible representations. Our results confirm the validity of the general philosophy that the representation theory is intimately connected to the Poisson geometry.