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December 2022 Framed Matrices and A-Bialgebras *
Samson Saneblidze, Ronald Umble
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Adv. Studies: Euro-Tbilisi Math. J. 15(4): 41-140 (December 2022). DOI: 10.32513/asetmj/19322008230


We complete the construction of the biassociahedra KK, construct the free matrad H, realize H as the cellular chains of KK, and define an A-bialgebra as an algebra over H. We construct the bimultiplihedra JJ, construct the relative free matrad rH as a H-bimodule, realize rH as the cellular chains of JJ, and define a morphism of A-bialgebras as a bimodule over H. We prove that the homology of every A-bialgebra over a commutative ring with unity admits an induced A-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of A-bialgebras and determine the A-bialgebra structure of H*ΩΣX;. For each n2, we construct a space Xn and identify an induced nontrivial A-bialgebra operation ω2n:H*ΩXn;22H*ΩXn;2n.

Funding Statement

*This research was funded in part by grant SRNSF/217614 and multiple Millersville University Research Grants.

Version Information

The current pdf replaces the original pdf file, first available on 22 December 2022. The new version corrects the DOI prefix to read 10.32513.


The second author wishes to thank the Institute of Mathematics of the Czech Academy of Sciences and the Institute of Mathematics of University of Seville (IMUS) for providing financial support and residential accommodations during various stages of this project.


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Samson Saneblidze. Ronald Umble. "Framed Matrices and A-Bialgebras *." Adv. Studies: Euro-Tbilisi Math. J. 15 (4) 41 - 140, December 2022.


Received: 22 June 2022; Accepted: 20 September 2022; Published: December 2022
First available in Project Euclid: 22 December 2022

Digital Object Identifier: 10.32513/asetmj/19322008230

Primary: 55P35 , 55P48 , 55P99
Secondary: 52B05 , 52B11

Keywords: A∞-bialgebra , biassociahedron , bimultiplihedron , bipartition , framed matrix , matrad , relative matrad

Rights: Copyright © 2022 Tbilisi Centre for Mathematical Sciences


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Vol.15 • No. 4 • December 2022
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