Research
My research interests lie in the interplay between representation theory and mathematical physics. In particular I am interested in Lie algebras, (Poisson) vertex algebras and integrable systems.
All my papers are available on arXiv.
Preprints
Valeri D., Yang D., Remarks on intersection numbers and integrable hierarchies. II. Tau-structure.
De Sole A., Kac V.G., Valeri D., Poisson vertex algebras and Hamiltonian PDE.
Publications
De Sole A., Kac V.G., Valeri D., Adler-Oevel-Ragnisco type operators and Poisson vertex algebras, Pure Appl. Math. Q. 20, no. 3 (2024), 1181-1249.
https://dx.doi.org/10.4310/PAMQ.2024.v20.n3.a5
Feigin M., Valeri D., Wright J., Flat coordinates of algebraic Frobenius manifolds in small dimensions, J. Geom. Phys. 200 (2024), 105151.
https://doi.org/10.1016/j.geomphys.2024.105151
Fairon M., Valeri D., Double multiplicative Poisson vertex algebras, Int. Math. Res. Not. 2023 (2023), no. 17, 14991-15072.
https://doi.org/10.1093/imrn/rnac245 (free version)
Dubrovin B., Valeri D., Yang D., Affine Kac--Moody algebras and tau-functions for the Drinfeld--Sokolov hierarchies: The matrix-resolvent method, SIGMA 18 (2022), 077.
https://doi.org/10.3842/SIGMA.2022.077
De Sole A., Kac V.G., Valeri D., On Lax operators, Jpn. J. Math. 17 (2022), no. 1, 63-116.
https://doi.org/10.1007/s11537-021-2134-1
Casati M., Lorenzoni P., Valeri D., Vitolo R., Weakly nonlocal Poisson brackets: tools, examples, computations, Comput. Phys. Commun. 274 (2022), Paper No. 108284.
https://doi.org/10.1016/j.cpc.2022.108284
De Sole A., Jibladze M., Kac V.G., Valeri D., Integrable triples in semisimple Lie algebras, Lett. Math. Phys. 111 (2021), 117.
https://doi.org/10.1007/s11005-021-01456-4
De Sole A., Jibladze M., Kac V.G., Valeri D., Integrability of classical affine W-algebras, Transform. Groups 26 (2021), no. 2, 479-500.
https://doi.org/10.1007/s00031-021-09645-0
Valeri D., W-algebras via Lax type operators, in: Paranjape M.B., MacKenzie R., Thomova Z., Winternitz P., Witczak-Krempa W. (eds) Quantum Theory and Symmetries, CRM Series in Mathematical Physics, Springer, Cham.
https://doi.org/10.1007/978-3-030-55777-5_17
Carpentier S., De Sole A., Kac V.G., Valeri D., van de Leur J., p-reduced multicomponent KP hierarchy and classical W-algebras W(gl_N,p), Comm. Math. Phys. 380 (2020), no. 2, 655-722.
https://doi.org/10.1007/s00220-020-03817-x
De Sole A., Fedele L., Valeri D., Generators of the quantum finite W-algebras in type A, J. Algebra Appl. 19 (2020), n. 9.
https://doi.org/10.1142/S0219498820501753
De Sole A., Kac V.G., Valeri D., Wakimoto M., Poisson λ-brackets for differential-difference equations, Int. Math. Res. Not. 2020 (2020), n.13, 4144-4190.
https://doi.org/10.1093/imrn/rny242
De Sole A., Kac V.G., Valeri D., Wakimoto M., Local and non-local multiplicative Poisson vertex algebras and differential-difference equations, Comm. Math. Phys. 370 (2019), no. 3, 1019-1068.
https://doi.org/10.1007/s00220-019-03416-5
Genovese G., Lucà R., Valeri D., Invariant measures for the periodic derivative nonlinear Schrödinger equation, Math. Ann. 374 (2019), no. 3-4, 1075-1138.
https://doi.org/10.1007/s00208-018-1754-0
De Sole A., Kac V. G., Valeri D., A Lax type operator for quantum finite W-algebras, Selecta Math. 24 (2018), no. 5, 4617-4657.
https://doi.org/10.1007/s00029-018-0439-6
Casati M., Valeri D., MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine W-algebras, Boll. Unione Mat. Ital. 11 (2018), no. 4, 503-531.
https://doi.org/10.1007/s40574-017-0146-9
De Sole A., Kac V. G., Valeri D., Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras, Comm. Math. Phys. 360 (2018), n.3, 851-918.
https://doi.org/10.1007/s00220-018-3142-8
De Sole A., Kac V. G., Valeri D., Finite W-algebras for gl_N, Adv. Math. 327 (2018), 173-224.
https://doi.org/10.1016/j.aim.2017.06.016
Masoero D., Raimondo A., Valeri D., Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections II. The non simply-laced case, Comm. Math. Phys. 349 (2017), n. 3, 1063-1105.
https://doi.org/10.1007/s00220-016-2744-2
De Sole A., Kac V. G., Valeri D., Classical affine W-algebras for gl_N and associated integrable Hamiltonian hierarchies, Comm. Math. Phys. 348 (2016), n. 1, 265-319.
https://doi.org/10.1007/s00220-016-2632-9
De Sole A., Kac V. G., Valeri D., A new scheme of integrability for (bi)Hamiltonian PDE, Comm. Math. Phys. 347 (2016), n. 2, 449-488.
https://doi.org/10.1007/s00220-016-2684-x
De Sole A., Kac V. G., Valeri D., Structure of classical (finite and affine) W-algebras, J. Eur. Math. Soc. 18 (2016), n. 9, 1873-1908.
https://doi.org/10.4171/jems/632
Genovese G., Lucà R., Valeri D., Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation, Selecta Math. 22 (2016), no. 3, 1663-1702 (also Oberwolfach Preprints (OWP) 2015-04).
https://doi.org/10.1007/s00029-016-0225-2
Masoero D., Raimondo A., Valeri D., Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case, Comm. Math. Phys. 344 (2016), n. 3, 719-750.
https://doi.org/10.1007/s00220-016-2643-6
De Sole A., Kac V. G., Valeri D., Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras, Int. Math. Res. Not. 2015 (2015), n.21, 11186-11235.
https://doi.org/10.1093/imrn/rnv017
De Sole A., Kac V. G., Valeri D., Double Poisson vertex algebras and non-commutative Hamiltonian equations, Adv. Math. 281 (2015), 1025-1099.
https://doi.org/10.1016/j.aim.2015.05.011
De Sole A., Kac V. G., Valeri D., Integrability of Dirac reduced bi-Hamiltonian equations, Trends in Contemporary Mathematics, Springer INDAM series, Vol. 8 (2014), 13-32.
https://doi.org/10.1007/978-3-319-05254-0_2
De Sole A., Kac V. G., Valeri D., Dirac reduction for Poisson vertex algebras, Comm. Math. Phys. 331 (2014), n. 3, 1155-1190.
https://doi.org/10.1007/s00220-014-2103-0
De Sole A., Kac V. G., Valeri D., Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents, Comm. Math. Phys. 331 (2014), n. 2, 623-676. Erratum in: Comm. Math. Phys. 333 (2015), n. 3, 1617-1619.
https://doi.org/10.1007/s00220-014-2049-2
Valeri D., Classical W-algebras within the theory of Poisson vertex algebras, Advances in Lie superalgebras, Springer INDAM series, Vol. 7 (2014), 203-221 (pdf).
https://doi.org/10.1007/978-3-319-02952-8_12
De Sole A., Kac V. G., Valeri D., Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras, Comm. Math. Phys. 323 (2013), n. 2, 663-711.
Notes
If you are curious about what a vertex algebra is you may have a look at these notes from the course "An introduction to vertex algebras" given by professor Victor Kac at La Sapienza (Roma) during December 2008 and January 2009. I was not able to solve all the exercises assigned, so if you have any suggestions I would be grateful. Obviously everything there is written there has to be taken with the tongs (especially my proposed solutions of the exercises!). Comments and corrections are welcome.
Software
With Matteo Casati we developed a Mathematica package to perform computations in Poisson vertex algebras. If you experience troubles in performing the computations by hand, you can download the package and ask the computer for help.
With Matteo Casati, Paolo Lorenzoni and Raffaele Vitolo we developed a package (for Maple, Mathematica and Reduce) to perform computations in weakly non-local Poisson vertex algebras. All details here.
The Mathematica code used in the joint paper with Misha Feigin and Johan Wright about the flat coordinates of algebraic Frobenius manifolds is available here.