  We consider a special category of Hopf algebras, depending on parameters
$\Sigma$ which possess properties similar to the category of representations of
simple Lie group with highest weight $\lambda$. We connect quantum groups to
minimal objects in this categories---they correspond to irreducible
representations in the category of representations with highest weight
$\lambda$. Moreover, we want to correspond quantum groups only to finite
dimensional irreducible representations. This gives us a condition for
$\lambda$: $\lambda$--- is dominant means the minimal object in the category of
representations with highest weight $\lambda$ is finite dimensional. We put
similar condition for $\Sigma$. We call $\Sigma$ dominant if the minimal object
in corresponding category has polynomial growth. Now we propose to define
quantum groups starting from dominant parameters $\Sigma$.
