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Operads and varieties of algebras defined by polylinear identities

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Abstract

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.

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References

  1. Tronin S. N., “Abstract clones and operads,” Siberian Math. J., 43, No. 4, 746–755 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  2. Tronin S. N., “Operads in the category of convexors. I,” Russian Math., 46, No. 3, 38–46 (2002).

    MATH  MathSciNet  Google Scholar 

  3. Tronin S. N., “Operads in the category of convexors. II,” Russian Math., 46, No. 5, 59–67 (2002).

    MATH  MathSciNet  Google Scholar 

  4. Tronin S. N., “On characterization of varieties of algebras over W-operads,” in: Abstracts: International Algebraic Conference Dedicated to the 250th Anniversary of Moscow State University and 75th Anniversary of the Chair of Higher Algebra, Izdat. Mekhmata MGU, 2004, pp. 127–128.

  5. Tronin S. N., “Operad theory and a universal algebra,” in: Algebra and Analysis-2004. Proceedings of the International Conference Dedicated to the 200th Anniversary of Kazan’ State University, Kazan’, July 2–9, 2004, Izdat. Kazan. Mat. Obshch., Kazan’, 2004, pp. 20–21 (Proceedings of the Lobachevskiĭ Center in Mathematics; 23).

    Google Scholar 

  6. Kurosh A. G., “Multioperator rings and algebras,” Uspekhi Mat. Nauk, 24, No. 1, 3–15 (1969).

    MATH  Google Scholar 

  7. Baranovich T. M. and Burgin M. S., “Linear Ω-algebras,” Uspekhi Mat. Nauk, 30, No. 4, 61–106 (1975).

    Google Scholar 

  8. Smirnov V. A., Simplicial and Operad Methods in Homotopy Theory [in Russian], Faktorial-Press, Moscow (2002).

    Google Scholar 

  9. Markl M., Shnider S., and Stasheff J., Operads in Algebra, Topology and Physics, Amer. Math. Soc., Providence, RI (2002) (Math. Surveys and Monographs; V. 96).

    Google Scholar 

  10. Leinster T., Higher Operads, Higher Categories, Cambridge Univ. Press, Cambridge (2003) (London Math. Soc. Lecture Notes Ser.).

    Google Scholar 

  11. Lawvere F. W., “Functorial semantics of algebraic theories,” Proc. Nat. Acad. Sci. USA, 50, No. 5, 869–872 (1963).

    MATH  MathSciNet  Google Scholar 

  12. Katsov E. B., “Tensor product of functors,” Siberian Math. J., 19, No. 2, 222–229 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  13. Pinus A. G., Conditional Terms and Their Applications to Algebra and the Theory of Computations [in Russian], Izdat. NGTU, Novosibirsk (2002).

    Google Scholar 

  14. Movsisyan Yu. M., Hyperidentities and Hypervarieties in Algebras [in Russian], Izdat. Erevan Univ., Erevan (1990).

    Google Scholar 

  15. Tronin S. N. and Kopp O. A., “Matrix linear operads,” Russian Math., 44, No. 6, 50–59 (2000).

    MathSciNet  Google Scholar 

  16. MacLane S., Categories for the Working Mathematician [Russian translation], Fizmatlit, Moscow (2004).

    Google Scholar 

  17. Smirnov D. M., Varieties of Algebras [in Russian], VO Nauka. Sibirsk. Izdat. Firma, Novosibirsk (1992).

    Google Scholar 

  18. Kizner F. I., “Two theorems on identities in multioperator algebras,” Uspekhi Mat. Nauk, 24, No. 1, 39–42 (1969).

    MATH  MathSciNet  Google Scholar 

  19. Shirshov A. I., Zhevlakov K. A., Slin’ko A. M., and Shestakov I. P., Rings That Are Nearly Associative [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  20. Klyachko A. A., “Lie elements in the tensor algebra,” Siberian Math. J., 15, No. 6, 914–920 (1974).

    Article  Google Scholar 

  21. Mishchenko S. P., “Growth in varieties of Lie algebras,” Uspekhi Mat. Nauk, 45, No. 6, 25–45 (1990).

    MATH  MathSciNet  Google Scholar 

  22. Boardman J. M. and Vogt R. M., Homotopy Invariant Algebraic Structures on Topological Spaces [Russian translation], Mir, Moscow (1977).

    Google Scholar 

  23. Artamonov V. A., “Clones of polylinear operations,” Uspekhi Mat. Nauk, 24, No. 1, 47–59 (1969).

    MATH  MathSciNet  Google Scholar 

  24. Mal’tsev A. I., “Structure characterization of some classes of algebras,” Dokl. Akad. Nauk SSSR, 120, No. 1, 29–32 (1958).

    MathSciNet  Google Scholar 

  25. Tronin S. N., “On varieties prescribed by polylinear identities,” in: Abstracts: XIX All-Union Algebraic Conference, 9–11 September 1987. Part 2, L’vov, 1987, p. 280.

  26. Tronin S. N., “On some properties of finitary algebraic theories,” in: Abstracts: V Siberian School on Varieties of Algebraic Systems, 1–5 July 1988, Barnaul, 1988, pp. 68–70.

  27. Tronin S. N., “On some properties of algebraic theories of varieties of linear algebras. I. Varieties prescribed by polylinear identities,” submitted to VINITI August 11, 1988, No. 6511-B88.

  28. Tronin S. N., On Retractions of Free Algebras and Modules [in Russian], Dis. Kand. Fiz.-Mat. Nauk, Kishinëv (1989).

    Google Scholar 

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Authors and Affiliations

  1. Kazan’ State University, Kazan’, Russia

    S. N. Tronin

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  1. S. N. Tronin
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Original Russian Text Copyright © 2006 Tronin S. N.

The author was supported by the Academy of Sciences of the Republic of Tatarstan (Grant 05-5.1-284).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 3, pp. 670–694, May–June, 2006.

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Tronin, S.N. Operads and varieties of algebras defined by polylinear identities. Sib Math J 47, 555–573 (2006). https://doi.org/10.1007/s11202-006-0067-9

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