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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 50

On the Locality of Formal Distributions Over Right-Symmetric and Novikov Algebras

Author(s)
Leonid A. Bokut1, Pavel S. Kolesnikov1

1Sobolev Institute of Mathematics, Novosibirsk, Russian Federation

Abstract
The Dong Lemma in the theory of vertex algebras states that the locality property of formal distributions over a Lie algebra is preserved under the action of a vertex operator. A similar statement is known for associative algebras. We study local formal distributions over pre-Lie (right-symmetric), pre-associative (dendriform), and Novikov algebras to show that the analogue of the Dong Lemma holds for Novikov algebras but does not hold for pre-Lie and pre-associative ones.
About the Authors

Leonid A. Bokut, Dr. Sci. (Phys.–Math.), Prof., Sobolev Institute of Mathematics, Novosibirsk, 630090, Russian Federation, bokut@math.nsc.ru

Pavel S. Kolesnikov, Dr. Sci. (Phys.–Math.), Sobolev Institute of Mathematics, Novosibirsk, 630090, Russian Federation, pavelsk77@gmail.com

For citation

Bokut L. A., Kolesnikov P. S. On the Locality of Formal Distributions Over Right-Symmetric and Novikov Algebras. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 83–100. (in Russian)

https://doi.org/10.26516/1997-7670.2024.50.83

Keywords
conformal algebra, locality function, pre-Lie algebra, Novikov algebra
UDC
512.554
MSC
17D25, 17B69
DOI
https://doi.org/10.26516/1997-7670.2024.50.83
References
  1. Balinskii A.A., Novikov S.P. Poisson brackets of hydrodynamics type, Frobenius algebras and Lie algebras. Soviet Math. Dokl., 1985, vol. 32, no. 1, pp. 228–231.
  2. Bokut L.A., Chen Y., Li Y. Gr¨obner–Shirshov bases for Vinberg–Koszul– Gerstenhaber right-symmetric algebras. J. Math. Sci. (N. Y.), 2010, vol. 166, no. 5, pp. 603–612. https://doi.org/10.1007/s10958-010-9875-3
  3. Gel’fand I.M., Dorfman I.Ya. Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl., 1979, vol. 13, no. 4, pp. 248–262.
  4. Kolesnikov P.S., Nesterenko A.A. Conformal envelopes of Novikov–Poisson algebras. Sib. Math. J., 2023, vol. 64, no. 3, pp. 598–610.
  5. Bai C., Bellier O., Guo L., Ni X. Splitting of operations, Manin products, and Rota–Baxter operators. Int. Math. Res. Not. IMRN, 2013, no. 3, pp. 485–524. https://doi.org/10.1093/imrn/rnr266
  6. Bakalov B., D’ Andrea A., Kac V. G. Theory of finite pseudoalgebras. Adv. Math., 2001, vol. 162, pp. 1–140. https://doi.org/10.1006/aima.2001.1993
  7. Bokut L.A., Chen Y., Zhang Z. Gr¨obner–Shirshov bases method for Gelfand– Dorfman–Novikov algebras. J. Algebra Appl., 2017, vol. 16, no. 1, art. no. 1750001, 22 pp. https://doi.org/10.1142/S0219498817500013
  8. D’Andrea A., Kac V.G. Structure theory of finite conformal algebras. Selecta Math. New Ser., 1998, vol. 4, pp. 377–418. https://doi.org/10.1007/s000290050036
  9. Dotsenko V., Tamaroff P. Endofunctors and Poincar´e–Birkhoff–Witt theorems. Int. Math. Res. Not. IMRN, 2021, no. 16, pp. 12670–12690. https://doi.org/10.1093/imrn/rnz369
  10. Dzhumadil’daev A.S., L¨ofwall C. Trees, free right-symmetric algebras, free Novikov algebras and identities. Homology, Homotopy Appl., 2002, vol. 4, no. 2, pp. 165–190. https://doi.org/10.4310/hha.2002.v4.n2.a8
  11. Fattori D., Kac V.G. Classification of finite simple Lie conformal superalgebras. J. Algebra, 2002, vol. 258, no. 1, pp. 23–59. https://doi.org/10.1016/S0021- 8693(02)00504-5
  12. Frenkel E., Ben-Zvi D. Vertex Algebras and Algebraic Curves. 2nd ed. Mathematical Surveys and Monographs. Providence, RI: AMS, 2004, vol. 88. https://doi.org/10.1090/surv/088
  13. Gubarev V. Poincar´e–Birkhoff–Witt theorem for pre-Lie and post-Lie algebras. J. Lie Theory, 2020, vol. 30, no. 1, pp. 223–238.
  14. Gubarev V.Yu., Kolesnikov P.S. Embedding of dendriform algebras into RotaBaxter algebras. Cent. Eur. J. Math., 2013, vol. 11, no. 2, pp. 226–245. https://doi.org/10.2478/s11533-012-0138-z
  15. Hong Y., Bai C. On antisymmetric infinitesimal conformal bialgebras. J. Algebra, 2021, vol. 586, pp. 325–356. https://doi.org/10.1016/j.jalgebra.2021.06.029
  16. Hong Y., Li F. Left-symmetric conformal algebras and vertex algebras. J. Pure and Appl. Algebra, 2015, vol. 219, no. 8, pp. 3543–3567. https://doi.org/10.1016/j.jpaa.2014.12.012
  17. Kac V.G. Vertex Algebras for Beginners. Univ. Lect. Ser. Providence, RI: AMS, 1997, vol. 10. https://doi.org/10.1090/ulect/010
  18. Cantarini N., Kac V. G. Classification of linearly compact simple Jordan and generalized Poisson superalgebras. J. Algebra, 2007, vol. 313, no. 1, pp. 100–124. https://doi.org/10.1016/j.jalgebra.2006.10.040
  19. Kac V.G. Formal distribution algebras and conformal algebras. 12th international congress of mathematical physics (ICMP97). De Wit, D. et al. (eds.). Cambridge, MA: Internat. Press, 1999, pp. 80–97.
  20. Kolesnikov P.S. Associative conformal algebras with finite faithful representation. Adv. Math., 2006, vol. 202, no. 2, pp. 602–637. https://doi.org/10.1016/j.aim.2005.04.001
  21. Kolesnikov P. Gr¨obner–Shirshov bases for pre-associative algebras // Comm. Algebra, 2017, vol. 45, no. 12, pp. 5283–5296. https://doi.org/10.1080/00927872.2017.1304552
  22. Kolesnikov P.S. Universal enveloping Poisson conformal algebras. Internat. J. Algebra Comput., 2020, vol. 30, no. 5, pp. 1015–1034. https://doi.org/10.1142/S0218196720500289
  23. Loday J.-L. Dialgebras. Dialgebras and related operads. Lectures Notes in Math. Loday J.-L. , Frabetti A., Chapoton F., Goichot F. (eds.). Berlin, Springer-Verl., 2001, vol. 1763, pp. 7–66. https://doi.org/10.1007/3-540-45328-8_2
  24. Loday J.-L. Une version non commutative des alg´ebres de Lie: les alg´ebres de Leibniz. Enseign. Math., 1993, vol. 39, pp. 269–293.
  25. Loday J.-L., Pirashvili T. Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann., 1993, vol. 296, pp. 139–158. https://doi.org/10.1007/BF01445099
  26. Pei J., Bai C., Guo L., Ni X. Replicators, Manin white product of binary operads and average operators. New trends in algebras and combinatorics. Hackensack, NJ, World Scientific Publishing Co., 2020, pp. 317–353. https://doi.org/10.1142/9789811215476_0019
  27. Roitman M. On free conformal and vertex algebras. J. Algebra, 1999, vol. 217, pp. 496–527. https://doi.org/10.1006/jabr.1998.7834
  28. Yuan L. O-operators and Nijenhuis operators of associative conformal algebras. J. Algebra, 2022, vol. 609, pp. 245–291. https://doi.org/10.1016/j.jalgebra.2022.07.003
  29. Zelmanov E.I. On the structure of conformal algebras. Proc. Intern. Conf. on Combinatorial and Computational Algebra. Contemp. Math., Providence, RI: AMS, 2000, vol. 264, pp. 139–153. https://doi.org/10.1090/conm/264/04216

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