On the Hochschild Cohomologies of Associative Conformal Algebras with a Finite Faithful Representation

  • P. S. KolesnikovEmail author
  • R. A. Kozlov


Associative conformal algebras of conformal endomorphisms are of essential importance for the study of finite representations of conformal Lie algebras (Lie vertex algebras). We describe all semisimple algebras of conformal endomorphisms which have the trivial second Hochschild cohomology group with coefficients in every conformal bimodule. As a consequence, we state a complete solution of the radical splitting problem in the class of associative conformal algebras with a finite faithful representation.


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The work was supported by the Program of fundamental scientific researches of the Siberian Branch of Russian Academy of Sciences, I.1.1, Project 0314-2016-0001. The authors are grateful to the referees for useful comments that helped to improve the exposition.


  1. 1.
    Bakalov B., D’Andrea A., Kac V.G.: Theory of finite pseudoalgebras. Adv. Math. 162, 1–140 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bakalov B., Kac V.G., Voronov A.: Cohomology of conformal algebras. Commun. Math. Phys. 200, 561–589 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beilinson, A.A., Drinfeld, V.G.: Chiral Algebras. American Mathematical Society, Colloquium Publications, vol. 51. American Mathematical Society, Providence, RI (2004)Google Scholar
  4. 4.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bokut L.A., Fong Y., Ke W.-F.: Gröbner–Shirshov bases and composition lemma for associative conformal algebras: an example. Contemp. Math. 264, 63–90 (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bokut L.A., Fong Y., Ke W.-F.: Composition-diamond lemma for associative conformal algebras. J. Algebra 272, 739–774 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borcherds R.E.: Vertex algebras, Kac–Moody algebras, and the monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boyallian C., Kac V.G., Liberati J.-I.: On the classification of subalgebras of CendN and gcN. J. Algebra 260, 32–63 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boyallian C., Kac V.G., Liberati J.-I.: Classification of finite irreducible modules over the Lie conformal superalgebra CK 6. Commun. Math. Phys. 317, 503–546 (2013)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Boyallian, C., Liberati, J.-I.: Classification of irreducible representations over finite simple Lie conformal superalgebras. In: Polcino Milies, C. (ed.) Groups, Algebras and Applications, XVIII Latin American Algebra Colloquium (São Pedro, Brazil, August 3–8, 2009). Contemporary Mathematics, vol. 537, pp. 85–121. American Mathematical Society, Providence (2011)Google Scholar
  11. 11.
    Cheng S.-J., Kac V.G.: Conformal modules. Asian J. Math. 1, 181–193 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cheng S.-J., Kac V.G.: A new N = 6 superconformal algebra. Commun. Math. Phys. 186, 219–231 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cheng, S.-J., Kac, V.G., Wakimoto, M.: Extensions of conformal modules. In: Kashiwara, M et al. (eds.) Topological Field Theory, Primitive Forms and Related Topics. Proceedings of the 38th Taniguchi Symposium (Kyoto, Japan, December 9–13, 1996). Progress in Mathematics, vol. 160, pp. 79–129. Birkhauser, Boston (1998).Google Scholar
  14. 14.
    Cheng S.-J., Kac V.G., Wakimoto M.: Extensions of Neveu-Schwarz conformal modules. J. Math. Phys. 41, 2271–2294 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    D’Andrea A, Kac V.G.: Structure theory of finite conformal algebras. Sel. Math. New Ser. 4, 377–418 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dolguntseva I.A.: The Hochschild cohomology for associative conformal algebras. Algebra Logic 46, 373–384 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dolguntseva I.A.: Triviality of the second cohomology group of the conformal algebras Cendn and Curn. St. Petersburg Math. J. 21, 53–63 (2010)Google Scholar
  18. 18.
    Fattori D., Kac V.G.: Classification of finite simple Lie conformal superalgebras. J. Algebra 258, 23–59 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fattori, D., Kac, V.G., Retakh, A.: Structure theory of finite Lie conformal superalgebras. In: Lie Theory and its Applications in Physics V, 27–63. World Scientific Publication, River Edge (2004)Google Scholar
  20. 20.
    Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. Mathematical Surveys and Monograps, vol. 88. American Mathematical Society, Providence (2001)Google Scholar
  21. 21.
    Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Applied Mathematics, vol. 134. Academic, Boston, MA (1998)Google Scholar
  22. 22.
    Kac, V.G.: Vertex Algebras for Beginners. University Lecture Series, 10, 2nd edn. American Mathematical Society, Providence, 1996 (1998)Google Scholar
  23. 23.
    Kac, V.G.: Formal distribution algebras and conformal algebras. In: De Wit, D. et al. (eds.) 12th International Congress of Mathematical Physics (ICMP 97), pp. 80–97. International Press, Cambridge (1999)Google Scholar
  24. 24.
    Kolesnikov P.S.: Associative conformal algebras with finite faithful representation. Adv. Math. 202, 602–637 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kolesnikov P.S.: On the Wedderburn principal theorem in conformal algebras. J. Algebra Appl. 6, 119–134 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kolesnikov P.S.: On finite representations of conformal algebras. J. Algebra 331, 169–193 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kolesnikov P.S.: The Ado theorem for finite Lie conformal algebras with Levi decomposition. J. Algebra Appl. 15, 1650130 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kozlov, R.A.: Hochshild Cohomology of the Associative Conformal Algebra Cend1,x. arXiv:1710.02969
  29. 29.
    Martinez C., Zelmanov E.: Irreducible representations of the exceptional Cheng–Kac superalgebra. Trans. Am. Math. Soc. 366, 5853–5876 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mostovoy J., Perez-Izquierdo J.M., Shestakov I.P.: Hopf algebras in non-associative Lie theory. Bull. Math. Sci. 4, 129–173 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Retakh A.: Associative conformal algebras of linear growth. J. Algebra 237, 769–788 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Retakh A.: Unital associative pseudoalgebras and their representations. J. Algebra 277, 769–805 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Roitman M.: On free conformal and vertex algebras. J. Algebra 217, 496–527 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zelmanov, E.: On the structure of conformal algebras. In: Chan, K.Y. et al. (eds). International Conference on Combinatorial and Computational Algebra (May 24–29, 1999, Hong Kong, China). Contempory Mathematical, vol. 264, pp. 139–153. American Mathematical Society, Providence (2000)Google Scholar
  35. 35.
    Zelmanov, E.: Idempotents in conformal algebras. In: Fong, Y. et al. (eds) Proceedings of the 3rd International Algebra Conference (Tainan, Taiwan, June 16–July 1, 2002), pp. 257–266. Kluwer Academic Publishers, Dordrecht (2003)Google Scholar

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Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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