Quantum groups and representations with highest weight

Abstract

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 $Σ$.