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Noncommutative Networks on a Cylinder

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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 \(\mathcal C_\natural \).

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Notes

  1. Note that concatenation of paths is commonly written in the opposite order to the composition of morphisms in \(\mathcal C\), i.e. \(fg=g\circ f\).

  2. One can further assume that \({\overline{g}}=0\) for every \(g\in \mathcal C\) which is not a loop. We do not need this convention in our text, however, it is worth noting that such convention allows one to define a map \(\mathcal C\rightarrow \mathcal C_\natural ,\;f\mapsto {\overline{f}}\) which is commonly referred to as “universal trace”.

  3. More details can be found in [Art18, AR].

  4. For our purpose we actually need only one or two connected components.

  5. Note that this double bracket is closely related but different from the double bracket on a conjugate surface corresponding to the torus network. The main difference is that we no longer think of objects of \(\mathcal C\) as boundary points, but rather as points on the fixed cut of a conjugate surface. This records an additional information about the cut of the conjugate surface.

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Acknowledgements

S.A. is grateful to A. Alexeev, A. Berenstein, and M. Kontsevich for many fruitful discussions and to V. Retakh and V. Roubtsov for thesis advising [Art18] and long collaboration [AR], which resulted in starting formulas of Sect. 2.2. S.A. is also grateful to IHES, Michigan State University, University of Angers, and University of Geneva for hospitality during my visits. M.S. is grateful to the following institutions and programs: Research Institute for Mathematical Sciences, Kyoto (Spring 2019), Research in Pairs Program at the Mathematisches Forschungsinstitut Oberwolfach (Summer 2019), Mathematical Science Research Institute, Berkeley (Fall 2019) for their hospitality and outstanding working conditions they provided. S.A. was partially supported by RFBR-18-01-00926 and RFBR-19-51-50008-YaF. N.O. was partially supported by the NSF grant DMS-1745638. M.S. was partially supported by the NSF grant DMS-1702115 and partially supported by International Laboratory of Cluster Geometry NRU HSE, RF Government grant, ag. No 075-15-2021-608 from 08.06.2021.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    S. Arthamonov

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    N. Ovenhouse

  3. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    M. Shapiro

  4. National Research University Higher School of Economics, Moscow, Russia

    M. Shapiro

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  1. S. Arthamonov
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Correspondence to S. Arthamonov.

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Arthamonov, S., Ovenhouse, N. & Shapiro, M. Noncommutative Networks on a Cylinder. Commun. Math. Phys. 405, 129 (2024). https://doi.org/10.1007/s00220-023-04873-9

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