Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Antonov S. Y., Antonova A. V. To Chang Theorem. III. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 2, pp. 128-143. DOI: 10.18500/1816-9791-2018-18-2-128-143, EDN: XQFNQD

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 252)
Article type: 

To Chang Theorem. III

Antonova Alina Vladimirovna, Kazan State Power Engineering University, Russia

Various multilinear polynomials of Capelli type belonging to a free associative algebra F {X ∪ Y } over an arbitrary field F generated by a countable set X ∪ Y are considered. The formulas expressing coefficients of polynomial Chang R(¯x, ¯y|¯w) are found. It is proved that if the characteristic of field F is not equal two then polynomial R(¯x, ¯y| ¯w) may be represented by different ways in the form of sum of two consequencesof standard polynomial S− (¯x). The decomposition of Chang polynomial H (¯x, ¯y|¯w) different from already known is given. Besides, the connection between polynomials R(¯x, ¯y|¯w) and H (¯x, ¯y|¯w) is found. Some consequences of standard polynomial being of great interest for algebras with polynomial identities are obtained. In particular, a new identity of min imal degree for odd component of Z2 -graded matrix algebra M(m,m) (F) is given.

  1.  Antonov S. Yu., Antonova A. V. To Chang theorem. II. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2017, vol. 17, no. 2, pp. 127–137 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2017-17-2-127-137
  2. Antonov S. Yu., Antonova A. V. To Chang theorem. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2015, vol. 15, no. 3, pp. 247–251 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2015-15-3-247-251
  3. Antonov S. Yu., Antonova A. V. Quasi-polynomials of Capelli. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2015, vol. 15, no. 4, pp. 371–382 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2015-15-4-371-382
  4. Chang Q. Some consequences of the standard polynomial. Proc. Amer. Math. Soc., 1988, vol. 104, no. 3, pp. 707–710.
  5. Antonov S. Yu., Antonova A. V. On multiple polynomials of Capelli type. Physics and mathematics, Uchenye Zapiski Kazanskogo Universiteta. Ser. Fiziko-Matematicheskie Nauki, 2016, vol. 158, no. 1, pp. 5–25 (in Russian).
  6. Gateva T. V. The complexity of a bundle of varieties of associative algebras. Russian Math. Surveys, 1981, vol. 36, iss. 1, pp. 233.
  7. Kemer A. R. Remark on the standard identity. Math. Notes, 1978, vol. 23, no. 5, pp. 414–416. DOI: https://doi.org/10.1007/BF01789011
  8. Leron U. Multilinear identities of the matrix ring. Trans. Amer. Math. Soc., 1973. vol. 183, pp. 175–202.
  9. Amitsur S. A., Levitzki J. Minimal identities for algebras. Proc. Amer. Math. Soc., 1950,vol. 1, no. 4, pp. 449–463. DOI: https://doi.org/10.1090/S0002-9939-1950-0036751-9
  10. Mal’tsev Y. N. Basis for identities of the algebra of upper triangular matrices. Algebra and Logic, 1971, vol. 10, iss. 4, pp. 242–247. DOI: https://doi.org/10.1007/BF02219811
  11. Siderov P. N. A basis for the identities of an algebra of triangular matrices over an arbitrary field. PLISKA Studia Math. Bulgar., 1981, vol. 2, pp. 143–152 (in Russian).
  12. Kostant B. A theorem of Frobenius, a theorem of Amitsu r – Levitzki, and cohomology theory. J. Math. Mech., 1958, vol. 7, pp. 237–264. DOI: https://doi.org/10.1007/b94535_8
  13. Rowen L. H. Standard polynomials in matrix algebras. Trans. Amer. Math. Soc., 1974, vol. 190, pp. 253–284.
  14. Wenxin M., Racine M. Minimal identities of symmetric matrices. Trans. Amer. Math. Soc., 1990, vol. 320, no. 1, pp. 171–192. DOI: https://doi.org/10.1090/S0002-9947-1990-0961598-6
  15. Aver’yanov I. V. Basis of graded identities of the superalgebra M1,2(F). Math. Notes, 2009, vol. 85, no. 4, pp. 467–483. DOI: https://doi.org/10.1134/S0001434609030195
Short text (in English):
(downloads: 70)