On the combinatorial rank of a graded braided bialgebra.
The author studies \(\mathbb N\)-graded braided bialgebras \(B\), \(\mathbb N\) the non-negative integers. \(B\) is called strongly \(\mathbb N\)-graded if (as coalgebra) the \((i,j)\)-homogeneous component of \(B^{i+j}\to B^i\otimes B^j\) is a monomorphism for all \(i,j\in\mathbb N\). The author shows that \(B\) is strongly \(\mathbb N\)-graded if and only if \(B\) has no homogeneous primitive elements of degree at least two. This is rephrased in terms of the algebra \(S(B)\) obtained by dividing \(B\) by the ideal generated by the space \(E(B\)) spanned by the homogeneous primitive elements of degree at least two. Then \(S(B)\) is strongly \(\mathbb N\)-graded if and only if the projection of \(B\) onto \(S(B)\) is an isomorphism. It is shown that \(S(B)\) is a graded braided bialgebra. Then the author studies the sequence \(B,S(B),S^{[2]}(B)=S(S(B))\), etc., whose direct limit is a graded braided bialgebra which is strongly \(\mathbb N\)-graded, and which coincides with the image of the canonical homomorphism of \(B\) into the cotensor coalgebra of \(B^1\) over \(B^0\). It also coincides with \(B^0[B^1]\), a certain braided bialgebra, when \(B\) is strongly \(\mathbb N\)-graded as an algebra. \textit{V. K. Kharchenko} defined the combinatorial rank \(\kappa(B)\) of \(B\) to be \(n\) if the sequence \(B,S(B),S^{[2]}(B),\dots\) stabilizes after exactly \(n\) steps [J. Algebra 238, No. 2, 534-559 (2001; Zbl 0987.16031)]. The author studies possible upper bounds for \(\kappa(B)\). He shows that if the set of non-zero homogeneous components of the limit of the \(S^{[n]}(B)\) is finite, then \(\kappa(B)\) is finite. Another result is that if \(B\) is strongly graded as an algebra, and \(B^0[B^1]\) divides out the tensor algebra of \(B^1\) over \(B^0\) by relations of degree not greater than \(N\), then \(\kappa(B)\) is at most \(N-1\). He then specializes to \(B=T(V,c)\), the braided tensor algebra on a braided vector space \((V,c)\), giving various classes of examples when \(\kappa(V,c)=\kappa(T(V,c))\) is at most one. He also points out examples in the literature when \(\kappa(V,c)\) is greater than one.
scientific article; zbMATH DE number 5899113
scientific article