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MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine \({\mathcal {W}}\)-algebras

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Abstract

We give an introduction to the Mathematica packages MasterPVA and MasterPVAmulti used to compute \(\lambda \)-brackets in Poisson vertex algebras, which play an important role in the theory of infinite-dimensional Hamiltonian systems. As an application, we give an introduction to the Mathematica package WAlg aimed to compute the \(\lambda \)-brackets among the generators of classical affine \({\mathcal {W}}\)-algebras. The use of these packages is shown by providing some explicit examples.

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Acknowledgements

We wish to thank Alberto De Sole, Boris Dubrovin and Victor Kac for introducing us to the fascinating theory of integrable systems. Part of this work was done during the visit of the authors to the Department of Mathematics of the University of Rome La Sapienza in January and February 2016, and to the Department of Mathematics and Applications of the University of Milan-Bicocca in January 2017. We wish to thank this institutions for the kind hospitality. We also wish to thank theatrOn.it for hosting the packages files. M. Casati is supported by the INdAM-Cofund-2012 grant “MPoisCoho—Poisson cohomology of multidimensional Hamiltonian structures”. D. Valeri is supported by an NSFC “Research Fund for International Young Scientists” grant (11550110178).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

    Matteo Casati

  2. Istituto Nazionale d’Alta Matematica, 00185, Rome, Italy

    Matteo Casati

  3. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Daniele Valeri

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  1. Matteo Casati
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  2. Daniele Valeri
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Correspondence to Daniele Valeri.

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Casati, M., Valeri, D. MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine \({\mathcal {W}}\)-algebras. Boll Unione Mat Ital 11, 503–531 (2018). https://doi.org/10.1007/s40574-017-0146-9

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  • DOI: https://doi.org/10.1007/s40574-017-0146-9

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