A rigidity theorem for Moor-bialgebras
Leroux Philippe
TL;DR
This paper introduces the Moor operad and Moor-bialgebras, establishing a rigidity theorem for connected structures and exploring their relation to permutative algebras, expanding the understanding of algebraic operad dualities.
Contribution
It defines Moor-bialgebras, proves a rigidity theorem for connected cases, and links free permutative algebras with Moor-cooperations, highlighting new algebraic structures.
Findings
Rigidity theorem for connected Moor-bialgebras.
Existence of Moor-cooperations on free permutative algebras.
Non-distributive compatibility relation in Moor-bialgebras.
Abstract
We introduce the operad Moor, dual of the operad NAP and the notion of Moor-bialgebras. We warn the reader that the compatibility relation linking the Moor-operation with the Moor-cooperation is not distributive in the sense of Loday. Nevertheless, a rigidity theorem (\`a la Hopf-Borel) for the category of connected Moor-bialgebras is given. We show also that free permutative algebras can be equipped with a Moor-cooperation whose compatibility with the permutative product looks like the infinitesimal relation.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models