On Hopf algebra structures over free operads
Ralf Holtkamp
TL;DR
This paper explores Hopf algebra structures over free operads, extending classical theorems like Poincare'-Birkhoff-Witt to new operadic contexts including non-associative and polytope-related operads.
Contribution
It generalizes Hopf algebra structures and PBW theorems to a broad class of free operads, including non-associative and polytope operads.
Findings
Established PBW-type theorems for free operads
Characterized operads of primitives Prim P
Analyzed characteristic functions of these operads
Abstract
The operad Lie can be constructed as the operad of primitives Prim As from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincare'-Birkhoff-Witt and Cartier-Milnor-Moore. We replace As by families of free operads P, which include the operad Mag freely generated by a noncommutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincare'-Birkhoff-Witt type theorems and collect information about the operads Prim P, e.g. in terms of characteristic functions.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology