{"title":"Noncommutative Networks on a Cylinder","authors":"S. Arthamonov, N. Ovenhouse, M. Shapiro","doi":"10.1007/s00220-023-04873-9","DOIUrl":null,"url":null,"abstract":"<p>In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder <span>\\(\\Sigma \\)</span>, which gives rise to the quasi Poisson bracket of G. Massuyeau and V. Turaev on the group algebra <span>\\(\\textbf{k}\\pi _1(\\Sigma ,p)\\)</span> of the fundamental group of a surface based at <span>\\(p\\in \\partial \\Sigma \\)</span>. This bracket also induces a noncommutative Goldman Poisson bracket on the <i>cyclic space</i> <span>\\(\\mathcal C_\\natural \\)</span>, which is a <span>\\({\\textbf{k}}\\)</span>-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative <i>r</i>-matrix formalism. This gives a more conceptual proof of the result of Ovenhouse (Adv Math 373:107309, 2020) that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on <span>\\(\\mathcal C_\\natural \\)</span>.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-023-04873-9","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder \(\Sigma \), which gives rise to the quasi Poisson bracket of G. Massuyeau and V. Turaev on the group algebra \(\textbf{k}\pi _1(\Sigma ,p)\) of the fundamental group of a surface based at \(p\in \partial \Sigma \). This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space\(\mathcal C_\natural \), which is a \({\textbf{k}}\)-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of Ovenhouse (Adv Math 373:107309, 2020) that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on \(\mathcal C_\natural \).
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.