On the combinatorial rank of a graded braided bialgebra.
type: Item
instance of: scholarly article
title: On the combinatorial rank of a graded braided bialgebra.
zbMATH Open document ID: 1231.16025
author: Alessandro Ardizzoni
published in: Journal of Pure and Applied Algebra
publication date: 2011-05-23T00:00:00Z
review text: 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…
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 16T10
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 5899113
zbMATH Keywords: graded braided bialgebras
zbMATH Keywords: combinatorial ranks
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.11.014
OpenAlex ID: W2009873941
cites work: Q4944223
cites work: Q3530254
cites work: Q4944224
cites work: From racks to pointed Hopf algebras
cites work: Lifting of quantum linear spaces and pointed Hopf algebras of order \(p^3\)
cites work: Finite quantum groups and Cartan matrices
cites work: Q3153375
cites work: Braided Bialgebras of Type One
cites work: Braided bialgebras of Hecke-type.
cites work: Quadratic Lie Algebras
cites work: Hochschild Homology in a Braided Tensor Category
cites work: On Nichols algebras of low dimension
cites work: Q4356241
cites work: Q4242303
cites work: Skew primitive elements in Hopf algebras and related identities
cites work: Q5702515
cites work: Q3998175
cites work: Construction of quantized enveloping algebras by cocycle deformation
cites work: Pointed indecomposable Hopf algebras over Coxeter groups.
cites work: Q4274976
cites work: Bialgebras of type one*
cites work: A characterization of the borel-like subalgebras of quantum enveloping algebras
cites work: Q5590089
cites work: Survey of braided Hopf algebras
DOI: 10.1016/J.JPAA.2010.11.014
description: scientific article; zbMATH DE number 5899113