Skip to main content
SpringerLink
Log in

On the structure of a relatively free Grassmann algebra

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We investigate the multiplicative and T-space structure of the relatively free algebra F (3) with a unity corresponding to the identity [[x 1 , x 2], x 3] = 0 over an infinite field of characteristic p > 0. The highest emphasis is placed on unitary closed T-spaces over a field of characteristic p > 2. We construct a diagram containing all basic T-spaces of the algebra F (3), which form infinite chains of the inclusions. One of the main results is the decomposition of quotient T-spaces connected with F (3) into a direct sum of simple components. Also, the studied T-spaces are commutative subalgebras of F (3); thus, the structure of F(3) and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics of the case p = 2. In the Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. V. Aladova, A. V. Grishin, and E. A. Kireeva, “T-spaces. The history of the question, applications and the last results,” Chebyshevskii Sb., 5, 4 (12), 39–57 (2004).

    Google Scholar 

  2. Yu. A. Bakhturin and A. Yu. Ol’shanskii, “Identities,” Itogi Nauki Tekh. Sovr. Probl. Mat. Fund. Napr., 18, 117–240 (1988).

    MATH  MathSciNet  Google Scholar 

  3. A. Ya. Belov, Algebras with Polynomial Identities: Representations and Combinatorial Methods [in Russian], Doctoral Dissertation in Physics and Mathematics, Moscow (2002).

  4. A. Ya. Belov, “On non-Spechtian varieties,” Fundam. Prikl. Mat., 5, No. 1, 47–66 (1999).

    MATH  MathSciNet  Google Scholar 

  5. A. Ya. Belov, “On rings asymptotically close to associative rings,” Mat. Tr., 10, No. 1, 29–96 (2007).

    MathSciNet  Google Scholar 

  6. A. Ya. Belov, “Hilbert series for T-spaces,” Usp. Mat. Nauk, to appear.

  7. P. J. Chiripov and P. N. Siderov, “On bases for identities of some varieties of associative algebras,” Pliska, 2, 103–115 (1981).

    MathSciNet  Google Scholar 

  8. A. V. Grishin, “On the finite basis property for systems of generalized polynomials,” Math. USSR-Izv., 37, No. 2, 243–272 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. V. Grishin, “On the finite basis property of T-spaces over a field of finite characteristic,” in: Proc. of Moscow–Tainan Algebraic Workshop (1994), pp. 225–227.

  10. A. V. Grishin, “On the finite basis property of abstract T-spaces,” Fundam. Prikl. Mat., 1, No. 3, 669–700 (1995).

    MATH  MathSciNet  Google Scholar 

  11. A. V. Grishin, “Examples of T-spaces and T-ideals over a field of characteristic 2 without the finite basis property,” Fundam. Prikl. Mat., 5, No. 1, 101–118 (1999).

    MATH  MathSciNet  Google Scholar 

  12. A. V. Grishin, “On non-Spechtianness of the variety of associative rings that satisfy the identity x 32 = 0,” Electron. Res. Announc. Am. Math. Soc., No. 6, 50–51 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. V. Grishin, “The variety of associative rings which satisfy the identity x 32 = 0 is not Specht,” in: 12th Int. Conf. FPSAC’00. Proceedings, pp. 686–691.

  14. A. V. Grishin, “Structural and algorithmic problems in T-spaces over a field of characteristic p > 0,” Russ. Math. Surv., 60, No. 3, 568–569 (2005).

    Article  MathSciNet  Google Scholar 

  15. A. V. Grishin, “On independent systems in unitary relatively free algebras,” Fundam. Prikl. Mat., 14, No. 8, 69–71 (2008).

    MathSciNet  Google Scholar 

  16. A. V. Grishin, “On center of a relatively free Grassmann algebra,” in: Modern Problems of Mathematics, Mechanics and Their Applications (on the 70th birthday of V. A. Sadovnichy). Abstracts of papers [in Russian], Universitetskaya Kniga, Moscow (2009), pp. 406–407.

    Google Scholar 

  17. A. V. Grishin, “On T-spaces and related concepts and results,” J. Math. Sci., 163, No. 6, 677–681 (2009).

    Article  Google Scholar 

  18. A. V. Grishin, “On the centre of a relatively free Grassmann algebra,” Usp. Mat. Nauk, 65, No. 4, 191–192 (2010).

    Google Scholar 

  19. A. V. Grishin and V. V. Shchigolev, “On T-spaces and their applications,” J. Math. Sci., 134, No. 1, 1799–1878 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  20. A. V. Grishin and L. M. Surmina (Tsybulya), “On the structure of some T-spaces in the relatively free Grassmann algebra,” in: Actual Problems of Mathematics, Informatics and Education [in Russian], Publ. of MPSU, Moscow (2007), pp. 73–82.

    Google Scholar 

  21. A. V. Grishin and L. M. Surmina (Tsybulya), “On the T-algebra of n-words,” in: Int. Conf. on Algebra and Number Theory Dedicated to the 80th Birthday of V. E. Voskresensky (Samara 2007 ). Abstracts of Papers [in Russian], Univers Grupp, Samara (2007), pp. 15–16.

    Google Scholar 

  22. A. V. Grishin and L. M. Surmina (Tsybulya), “On T-spaces of n-words over a field of characteristic p > 0,” Russ. Math. Surv., 62, No. 4, 802–803 (2007).

    Article  MATH  Google Scholar 

  23. A. V. Grishin and L. M. Tsybulya, “Two theorems on the structure of a relatively free Grassmann algebra,” Russ. Math. Surv., 63, No. 4, 782–783 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  24. A. V. Grishin and L. M. Tsybulya, “On the (p, n)-problem,” Vestn. Samar. Gos. Univ. Estestv. Ser. Mat., 57, No. 7, 35–55 (2007).

    Google Scholar 

  25. A. V. Grishin and L. M. Tsybulya, “Decomposition of the commutator quotient T-spaces into the direct sum of simple components,” in: Int. Algebraic Conf. Dedicated to the 100th Birthday of A. G. Kurosh. Abstracts of Papers (Moscow 2008 ) [in Russian], Publ. of Faculty of Mech. and Math. of MSU, Moscow (2008), pp. 302–303.

    Google Scholar 

  26. A. V. Grishin and L. M. Tsybulya, “On the multiplicative and T-space structure of the relatively free Grassmann algebra,” Sb. Math., 200, No. 9, 1299–1338 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  27. A. V. Grishin and S. V. Urbakhanov, “On codimensions for the sapces of 2-monomials over a field of characteristic 2,” Chebyshevskii Sb., 3, No. 2, 34–42 (2002).

    MATH  MathSciNet  Google Scholar 

  28. C. K. Gupta and A. N. Krasilnikov, “A non-finitely based system of polynomial identities which contains the identity x 6 = 0,” Quart. J. Math., 53, 173–183 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  29. A. R. Kemer, “Varieties and ℤ2-graded algebras,” Izv. Akad. Nauk SSSR, Ser. Mat., 48, No. 5, 1042–1059 (1984).

    MathSciNet  Google Scholar 

  30. A. R. Kemer, “The finite basis property of identities of associative algebras,” Algebra Logika, 26, 597–641 (1987).

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  32. E. A. Kireeva, “Fully invariant submodules of polynomial algebras are finitely generated,” Chebyshevskii Sb., 2, 54–60 (2001).

    MATH  MathSciNet  Google Scholar 

  33. E. A. Kireeva, “Limit T-spaces,” J. Math. Sci., 152, No. 4, 540–557 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  34. E. A. Kireeva and A. N. Krasilnikov, “On some extremal varieties of associative algebras,” Math. Notes, 78, No. 4, 503–517 (2005).

    Article  MathSciNet  Google Scholar 

  35. V. N. Latyshev, “On complexity of nonmatrix varieties of associative algebras,” Algebra Logika, 16, No. 2, 149–183 (1977).

    Google Scholar 

  36. S. V. Okhitin, “Central polynomials of an algebra of second-order matrices,” Moscow Univ. Math. Bull., 43, No. 4, 49–51 (1988).

    MATH  MathSciNet  Google Scholar 

  37. Ju. P. Razmyslov, “On finite basis property for identities of matrix algebra of second order over a field of characteristic zero,” Algebra Logika, 12, No. 1, 83–113 (1973).

    MathSciNet  Google Scholar 

  38. V. V. Shchigolev, “Examples of infinitely based T-ideals,” Fundam. Prikl. Mat., 5, No. 1, 307–312 (1999).

    MATH  MathSciNet  Google Scholar 

  39. V. V. Shchigolev, “Examples of T-spaces with an infinite basis,” Sb. Math., 191, No. 3, 459–476 (2000).

    Article  MathSciNet  Google Scholar 

  40. V. V. Shchigolev, “Finite basis property of T-spaces over fields of characteristic zero,” Izv. Ross. Akad. Nauk, Ser. Mat., 65, No. 5, 1041–1071 (2001).

    MATH  MathSciNet  Google Scholar 

  41. W. Specht, “Gesetze in Ringen,” Math. Z., 52, 557–589 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  42. L. M. Surmina (Tsybulya), “On some T-spaces of n-words,” in: Int. Conf. on Algebra and Number Theory Dedicated to the 80th Birthday of V. E. Voskresensky (Samara 2007). Abstracts of Papers [in Russian], Univers Grupp, Samara (2007), pp. 47–48.

    Google Scholar 

  43. The Kourovka Notebook (Unsolved Problems in Group Theory) [in Russian], Inst. Mat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1967).

  44. L. M. Tsybulya, “Theorem on the independence,” Int. Sci. Conf. “Modern Problems of Differential Geometry and General Algebra” Dedicated to the 100th Birthday of V. V. Vagner (Saratov 2008 ). Abstracts of Papers, Publ. of Saratov Univ., Saratov (2008), pp. 138–140.

    Google Scholar 

  45. L. M. Tsybulya, “The T ∗-space of n-words in a relatively free Grassmann algebra without a unity,” Deposited at VINITI, No. 975-V2008 (2008).

  46. L. M. Tsybulya, “Theorems on equalization and monomiality in a relatively free Grassmann algebra,” J. Math. Sci., 163, No. 6, 759–773 (2009).

    Article  Google Scholar 

  47. L. M. Tsybulya, “Reduction to monomial substitutions,” in: Int. Algebraic Conf. Dedicated to the 100th Birthday of A. G. Kurosh. Abstracts of Papers (Moscow 2008) [in Russian], Publ. of Faculty of Mech. and Math. of MSU, Moscow (2008), pp. 367–368.

    Google Scholar 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Pedagogical State University, Krasnoprudnaya str., 14, 107140, Moscow, Russia

    A. V. Grishin & L. M. Tsybulya

Authors
  1. A. V. Grishin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. M. Tsybulya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Grishin.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 15, No. 8, pp. 3–93, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grishin, A.V., Tsybulya, L.M. On the structure of a relatively free Grassmann algebra. J Math Sci 171, 149–212 (2010). https://doi.org/10.1007/s10958-010-0131-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-010-0131-7

Keywords

Search

Navigation