Functorial, operadic and modular operadic combinatorics of circuit algebras

TL;DR

This paper develops a comprehensive framework for understanding circuit algebras through operadic and modular operadic perspectives, providing formal definitions, equivalences, and applications to classical group algebras.

Contribution

It introduces a formal definition of a broad class of circuit algebras and establishes three equivalent descriptions using operads, modular operads, and Brauer diagram categories.

Findings

Provided a dictionary translating various incarnations of circuit algebras.

Established formal equivalences between different descriptions of circuit algebras.

Applied the framework to characterize algebras over orthogonal and symplectic groups.

Abstract

Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.