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. II. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 2, pp. 127-137. DOI: 10.18500/1816-9791-2017-17-2-127-137

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: 50)

To Chang Theorem. II

Antonov Stepan Yuryevich, Kazan State Power Engineering University, Russia
Antonova Alina Vladimirovna, Kazan State Power Engineering University, Russia

Multilinear polynomials H +(¯x, ¯y| ¯ w), H −(¯x, ¯y| ¯ w) ∈ F{X ∪ Y }, the sum of which is a polynomial H (¯x, ¯y| ¯ w) Chang (where F{X∪Y } is a free associative algebra over an arbitrary field F of characteristic not equal two, generated by a countable set X ∪ Y ) have been introduced in this paper. It has been proved that each of them is a consequence of the standard polynomial S−(¯x). In particular it has been shown that the Capelli quasi-polynomials b2m−1(¯xm, ¯y) and h2m−1(¯xm, ¯y) are also consequences of the polynomial S−m (¯x). The minimal degree of the polynomials b2m−1(¯xm, ¯y), h2m−1(¯xm, ¯y) in which they are a polynomial identity of matrix algebra Mn(F) has been also found in the paper. The obtained results are the translation of Chang results to some Capelli quasi polynomials of odd degree.

  1. Chang Q. Some consequences of the standard polynomial. Proc. Amer. Math. Soc., 1988, vol. 104, no. 3, pp. 707–710. DOI: https://doi.org/10.2307/2046778.
  2. Antonov S. Yu., Antonova A. V. To Chang theorem. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2015, vol. 15, iss. 3, pp. 247–251 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2015-15-3-247-251.
  3. Gateva T. V. The complexity of a bundle of varieties of associative algebras. Russian Math. Surveys, 1981, vol. 36, no. 1, pp. 233. DOI: https://doi.org/10.1070/RM1981v036n01ABEH002548.
  4. 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.
  5. Benanti F., Drensky V. On the consequences of the standard polynomial. Comm. Algebra, 1998, vol. 26, pp. 4243–4275.
  6. Leron U. Multilinear identities of the matrix ring. Trans. Amer. Math. Soc., 1973. vol. 183, pp. 175–202.
  7. Amitsur S. A., Levitzki J. Minimal identities for algebras. Proc. Amer. Math. Soc., 1950, vol. 1, no. 4, pp. 449–463.
  8. Owens F. W. Applications of graph theory to matrix theory. Amer. Math. Soc., 1975, vol. 51, no. 1, pp. 242–249.
  9. Rosset S. A new proof of the Amitsur–Levitzki identity. Israel J. Math., 1976, vol. 23, pp. 187–188.
  10. Szigeti J., Tuza Z., Revesz G. Eulerian polynomial identities on matrix rings. J. of Algebra, 1993, vol. 161, iss. 1, pp. 90–101. DOI: https://doi.org/10.1006/jabr.1993.1207.
  11. Lee A., Revesz G., Szigeti J., Tuza Z. Capelli polynomials, almost-permutation matrices and sparse Eulerian graphs. Descrete Math., 2001, vol. 230, no. 1–3, pp. 49–61.
  12. Antonov S. Yu. The least degree identities subspace M1(m,k)(F) of matrix superalgebra M(m,k) (F). Russian Math. (Iz. VUZ), 2012, vol. 56, no. 11, pp. 1–16. DOI: https://doi.org/10.3103/S1066369X12110011.