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No Associative PI-Algebra Coincides with Its Commutant

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Abstract

We prove that each (possibly not finitely generated) associative PI-algebra does not coincide with its commutant. We thus solve I. V. L'vov's problem in the Dniester Notebook. The result follows from the fact (also established in this article) that, in every T-prime variety, some weak identity holds and there exists a central polynomial (the existence of a central polynomial was earlier proved by A. R. Kemer). Moreover, we prove stability of T-prime varieties (in the case of characteristic zero, this was done by S. V. Okhitin who used A. R. Kemer's classification of T-prime varieties).

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References

  1. Dnestrovskaya Tetrad ' [in Russian], Inst. Mat., Novosibirsk (1993).

  2. Kemer A. R., Nonmatrix Varieties, Varieties with Power Growth and Finitely Generated PI-Matrices, Dis. Kand. Fiz.-Mat. Nauk, Novosibirsk (1981).

    Google Scholar 

  3. Kemer A. R., Ideals of Identities of Associative Algebras [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Barnaul (1988). English transl.: Ideals of Identities of Associative Algebras, Amer. Math. Soc., Providence RI (1991). (Transl. Math. Monogr.; 87.)

    Google Scholar 

  4. Kemer A. R., “On the multilinear components of the regular prime varieties,” in: Methods in Ring Theory, M. Dekker, New York, 1998, pp. 171–183. (Lecture Notes Pure Appl. Math.; 138).

    Google Scholar 

  5. Latyshev V. N., Nonmatrix Varieties of Associative Algebras [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Moscow (1977).

    Google Scholar 

  6. Latyshev V. N., “On some varieties of associative algebras,” Izv. Akad. Nauk SSSR Ser. Mat., 37, No. 5, 1010–1037 (1973).

    Google Scholar 

  7. Latyshev V. N., “Stable ideals of identities,” Algebra i Logika, 20, No. 5, 563–570, 600 (1981).

    Google Scholar 

  8. Razmyslov Yu. P., Identities of Algebras and Their Representations [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  9. Razmyslov Yu. P., “On one Kaplansky's problem,” Izv. Akad. Nauk SSSR Ser. Mat., 37, No. 3, 483–501 (1973).

    Google Scholar 

  10. Formanek E., “Central polynomials for matrix rings,” J. Algebra, 23, No., 129–133 (1972).

    Google Scholar 

  11. Okhitin S. V., “On stable T-ideals and central polynomials,” Vestnik MGU Ser. Mat. Mekh., No. 3, 85–89 (1986).

  12. Kemer A. R., “Multilinear identities of the algebras over a field of characteristic p,” Internat. J. Algebra Comput., 5, No. 2, 189–197 (1995).

    Google Scholar 

  13. Belov A. Ya., “On non-Specht varieties,” Fund. i Prikl. Mat., 5, No. 1, 47–66 (1999).

    Google Scholar 

  14. Zubkov A. N., “A generalization of the Razmyslov-Procesi theorem,” Algebra i Logika, 35, No. 4, 433–457 (1996).

    Google Scholar 

  15. Zubrilin K. A., “Algebras satisfying Capelli identities,” Mat. Sb., 186, No. 3, 53–64 (1995).

    Google Scholar 

  16. Razmyslov Yu. P., “Algebras satisfying identity relations of Capelli type,” Izv. Akad. Nauk SSSR Ser. Mat., 45, No. 1, 143–166 (1981).

    Google Scholar 

  17. Razmyslov Yu. P., “On the Jacobson radical in PI-algebras,” Algebra i Logika, 13, No. 3, 337–360 (1974).

    Google Scholar 

  18. Razmyslov Yu. P., “Trace identities of full matrix algebras over a field of characteristic zero,” Izv. Akad. Nauk SSSR Ser. Mat., 38, No. 4, 723–756 (1974).

    Google Scholar 

  19. Braun A., “The radical in a finitely generated PI-algebra,” Bull. Amer. Math. Soc., 7, No. 2, 385–386 (1982).

    Google Scholar 

  20. Procesi C., Rings with Polynomial Identities, Acad. Press, New York (1973).

    Google Scholar 

  21. Rowen L. H., Polynomial Identities in Ring Theory, Acad. Press, New York (1980).

    Google Scholar 

  22. Procesi C., “The invariant theory of n × n matrices,” Adv. Math., 19, No. 3, 306–381 (1976).

    Google Scholar 

  23. Donkin S., “On tilting modules for algebraic groups,” Math. Z., 212, No. 1, 39–60 (1993).

    Google Scholar 

  24. Donkin S., “Invariant functions on matrices,” Math. Proc. Cambridge Philos. Soc., 113, No. 1, 23–43 (1993).

    Google Scholar 

  25. Donkin S., “Invariants of several matrices,” Invent. Math., 110, No. 2, 389–401 (1992).

    Google Scholar 

  26. Zubrilin K. A, “On class of nilpotency of obstruction to embeddability of algebras satisfying Capelli identities,” Fund. Prikl. Mat., 1, No. 2, 409–430 (1995).

    Google Scholar 

  27. Kemer A. R., “Identities of finitely generated algebras over the infinite field,” Izv. Akad. Nauk SSSR Ser. Mat., 54, No. 4, 726–753 (1990).

    Google Scholar 

  28. Belov A. Ya., Algebras with Polynomial Identities: Representations and Combinatorial Methods [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Moscow (2002).

    Google Scholar 

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Authors and Affiliations

  1. International University of Bremen; Moscow Open Education Institute, Russia

    A. Ya. Belov

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Belov, A.Y. No Associative PI-Algebra Coincides with Its Commutant. Siberian Mathematical Journal 44, 969–980 (2003). https://doi.org/10.1023/B:SIMJ.0000007472.85188.56

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  • DOI: https://doi.org/10.1023/B:SIMJ.0000007472.85188.56

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