Functorial, operadic and modular operadic combinatorics of circuit algebras

2024/12/28 by Sophie Raynor, Raynor, Sophie · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2412.20260

openalex publication_date 2024/12/28 · openalex created_date 2025/01/01 · openalex updated_date 2026/07/28

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.

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• Operads, modules over walled Brauer categories, and Koszul complexes

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