ar X iv : q - a lg / 9 70 40 07 v 1 8 A pr 1 99 7 QUANTUM GROUPS AND REPRESENTATIONS WITH HIGHEST WEIGHT

We consider a special category of Hopf algebras, depending on parameters Σ which possess properties similar to the category of representations of simple Lie group with highest weight λ. We connect quantum groups to minimal objects in this categories—they correspond to irreducible representations in the category of representations with highest weight λ. Moreover, we want to correspond quantum groups only to finite dimensional irreducible representations. This gives us a condition for λ: λ— is dominant means the minimal object in the category of representations with highest weight λ is finite dimensional. We put similar condition for Σ. We call Σ dominant if the minimal object in corresponding category has polynomial growth. Now we propose to define quantum groups starting from dominant parameters Σ. 1. Definitions and examples 1.1 Torus. Let us fix an n-dimensional torus H (i.e. an algebraic group isomorphic to C * n). We denote by S the Hopf algebra of regular functions on H. Let Λ be the lattice of characters of H. Then Λ ⊂ S is a basis in S. The dual algebra S * can be realized as the algebra of functions on the lattice Λ. We denote byˆH the group algebra of H: for any element h ∈ H we denote the corresponding generator inˆH byˆh (ˆ h 1 ˆ h 2 = h 1 h 2). The Hopf algebrâ H is a subalgebra in S * (ˆ h(λ) = λ(h)) on which the comultiplication is well defined: ∆ ˆ h = ˆ h ⊗ ˆ h. The Hopf algebrâ H is too small for some of our future purposes. In order to be able to define the comultiplication on S * we have to complete S * ⊗ S * to (S ⊗ S) *. We need the comultiplication to define an action of S * on the tensor product of two S *-modules. We are interested only in those S *-modules W for which the S *-module structure is inherited from some S-comodule structure. That means, we should consider only S *-modules W which are algebraic representations of H. Under this condition the completed comultiplication on S * would allow us to define the action of S * on the tensor product of two such S *-modules (see [B-Kh]). We set S * ˆ ⊗S * := (S ⊗ S) *. The algebra S * ˆ ⊗S * could be realized …