On the Combinatorial Rank of a Graded Braided Bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V. K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write k(B) = n. We investigate conditions guaranteeing that k(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V; c) associated to a braided vector space (V; c), providing meaningful examples such that k(T(V; c))<2.