
TL;DR
This paper introduces a new operad framework for Poisson pseudoalgebras associated with cocommutative Hopf algebras, extending classical structures and developing related cohomology theories.
Contribution
It constructs a novel operad $\\mathcal{P}^{cl}_H(V)$ linking Hopf algebra modules to Poisson pseudoalgebras and introduces two cohomology theories for these structures.
Findings
Operad $\\mathcal{P}^{cl}_H(V)$ generalizes classical operads for Poisson structures.
Morphisms from Lie operad characterize Poisson vertex algebra structures.
Two new cohomology theories extend variational and classical cohomology.
Abstract
For any cocommutative Hopf algebra and a left -module , we construct an operad , which in the special case when is the algebra of polynomials in one variable reduces to the classical operad . Morphisms from the Lie operad to correspond to Poisson vertex algebra structures on . Likewise, our operad gives rise to the notion of a Poisson pseudoalgebra; thus extending the notion of a Lie pseudoalgebra. As a byproduct of our construction, we introduce two cohomology theories for Poisson pseudoalgebras, generalizing the variational and classical cohomology of Poisson vertex algebras.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons
