Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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ISSN 2541-9005 (Online)

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Antonov S. Y., Antonova A. V. Quasi-polynomials of Capelli. III. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 2, pp. 142-150. DOI: 10.18500/1816-9791-2021-21-2-142-150, EDN: HMVRSQ

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Quasi-polynomials of Capelli. III

Antonova Alina Vladimirovna, Kazan State Power Engineering University, Russia

In this paper polynomials of Capelli type (double and quasi-polynomials of Capelli) belonging to a free associative algebra $F\{X\cup Y\}$ considering over an arbitrary field $F$ and generated by two disjoint  countable  sets $X, Y$  are investigated.  It  is shown  that  double Capelli's  polynomials $C_{4k,\{1\}}$, $C_{4k,\{2\}}$ are consequences of the standard polynomial $S^-_{2k}$. Moreover, it  is  proved that  these  polynomials equal to zero both for square and for rectangular matrices of corresponding  sizes. In this paper it is also shown that all Capelli's quasi-polynomials of the $(4k+1)$ degree are minimal identities of odd component of $Z_2$-graded matrix algebra $M^{(m, k)}(F)$ for any  $F$ and $m\ne k$.

  1. Antonov S. Yu., Antonova A. V. Quasi-polynomials of Capelli. II. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 1, pp. 4–16 (in Russian). https://doi.org/10.18500/1816-9791-2020-20-1-4-16
  2. Vincenzo O. M. On the graded identities of M1,1(E). Israel Journal of Mathematics, 1992, vol. 80, no. 3, pp. 323–335. https://doi.org/10.1007/BF02808074
  3. Mattina D. On the graded identities and cocharacters of the algebra of 3×3 matrices. Linear Algebra and its Applications, 2004, vol. 384, pp. 55–75. https://doi.org/10.1016/S0024-3795(04)00034-5
  4. Aver’yanov I. V. Basis of graded identities of the superalgebra M1,2(F). Mathematical Notes, 2009, vol. 85, pp. 467–483. https://doi.org/10.1134/S0001434609030195
  5. Vincenzo O. M. Z2-graded polynomial identities for superalgebras of block-triangular matrices. Serdica Mathematical Journal, 2004, vol. 30, no. 2–3, pp. 111–134.
  6. Di Vincenzo O. M., Nardozza V. Z2-graded cocharacters for superalgebras of triangular matrices. Journal of Pure and Applied Algebra, 2004, vol. 194, iss. 1–2, pp. 193–211. https://doi.org/10.1016/j.jpaa.2004.04.004
  7. Amitsur S. A., Levitzki J. Minimal identities for algebras. Proceedings of the American Mathematical Society, 1950, vol. 1, no. 4, pp. 449–463. https://doi.org/10.1090/S0002-9939-1950-0036751-9
  8. Razmyslov Yu. P On the Jacobson radical in PI algebras. Algebra and Logic, 1974, vol. 13, no. 3, pp. 337–360.
  9. Gateva T. V. The complexity of a bundle of varieties of associative algebras. Russian Mathematical Surveys, 1981, vol. 36, no. 1, pp. 233.
  10. Chang Q. Some consequences of the standard polynomial. Proceedings of the American Mathematical Society, 1988, vol. 104, no. 3, pp. 707–710. https://doi.org/10.1090/S0002- 9939-1988-0964846-8
  11. Giambruno A., Sehgal S. K. On a polynomial identity for n×n matrices. Journal of Algebra, 1989, vol. 126, no. 2, pp. 451–453.
  12. Antonov S. Yu., Antonova A. V. On multiple polynomials of Capelli type Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2016, vol. 158, no. 1, pp. 5–25 (in Russian).
  13. Antonov S. Y. The least degree identities subspace M1(m,k) (F) of matrix superalgebra M(m,k) (F). Russian Mathematical (Izvestiya VUZ. Matematika), 2012, no. 56, pp. 1–16. https://doi.org/10.3103/S1066369X12110011
  14. Latyshev V. N. Combinatorial generators of the multilinear polynomial identities. Journal of Mathematical Sciences, 2008, vol. 149, iss. 2, pp. 1107–1112. https://doi.org/10.1007/s10958-008-0049-5
  15. Belov A. Ya. The local finite basis property and local representability of varieties of associative rings. Izvestiya: Mathematics, 2010, vol. 74, iss. 1, pp. 1–126. http://dx.doi.org/10.1070/IM2010v074n01ABEH002481