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 the corresponding category has polynomial growth. Now we propose to define quantum groups starting from dominant parameters Σ.
ArXiv: Quantum Algebra and Topology , q-alg/9704007
Comments: 6 pages
Revised December 1996