Associative $H$-pseudoalgebras with a semigroup

TL;DR

This paper introduces $ abla$-associative $H$-pseudoalgebras indexed by a semigroup, constructs them from various algebraic structures, and explores their cohomology and deformations, linking to quantum field theory renormalizations.

Contribution

It defines $ abla$-associative $H$-pseudoalgebras, constructs them from existing structures, and studies their cohomology and deformations, connecting algebraic theory with quantum physics applications.

Findings

Constructed $ abla$-associative $H$-pseudoalgebras from multiple algebraic frameworks.

Established cohomology theory for $ abla$-associative $H$-pseudoalgebras.

Linked first-order deformations to $ abla$-Poisson $H$-pseudoalgebras.

Abstract

Family algebraic structures indexed by a semigroup arise naturally in renormalizations of quantum field theory. In this paper, we first define the notion of $\Omega$-associative $H$-pseudoalgebra, where the operations are indexed by pairs of elements from a semigroup $\Omega$. Then we construct $\Omega$-associative $H$-pseudoalgebras from associative $H$-pseudoalgebras, $\Omega$-associative algebras, Rota-Baxter family algebras, $\Omega$-type $H$-pseudoalgebras and family-type $H$-pseudoalgebras. Moreover, we investigate the cohomology of $\Omega$-associative $H$-pseudoalgebras and establish that it both induces the cohomology of pseudo-$\mathcal{O}$-operator families and governs the associated formal deformations. As an application, we show that the first-order deformation of a commutative $\Omega$-associative $H$-pseudoalgebra yields an $\Omega$-Poisson $H$-pseudoalgebra.