TL;DR
This paper introduces bigraft algebras, explores their combinatorial and algebraic structures, proves their Koszul property, and establishes a rigidity theorem for related bialgebras.
Contribution
It generalizes graft algebras, provides combinatorial descriptions, and proves Koszulness and rigidity results for bigraft algebra structures.
Findings
Defined the bigraft algebra and its free algebra structure.
Proved the bigraft operad is Koszul.
Established a rigidity theorem for infinitesimal bigraft bialgebras.
Abstract
In this paper, we introduce the notion of bigraft algebra, generalizing the notions of left and right graft algebras. We give a combinatorial description of the free bigraft algebra generated by one generator and we endow this algebra with a Hopf algebra structure, and a pairing. Next, we study the Koszul dual of the bigraft operad and we give a combinatorial description of the free dual bigraft algebra generated by one generator. With the help of a rewriting method, we prove that the bigraft operad is Koszul. Finally, we define the notion of infinitesimal bigraft bialgebra and we prove a rigidity theorem for connected infinitesimal bigraft bialgebras.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology