Geodesic
Quasi-polynomials of Capelli.~III
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 2, pp. 142-150.

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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 $m\ne k$. 
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S. Yu. Antonov; A. V. Antonova. Quasi-polynomials of Capelli.~III. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 2, pp. 142-150. http://geodesic.mathdoc.fr/item/ISU_2021_21_2_a0/

[1] Antonov S. Yu., Antonova A. V., “Quasi-polynomials of Capelli. II”, Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 20:1 (2020), 4–16 (in Russian) | DOI | DOI | MR | Zbl

[2] Vincenzo O. M., “On the graded identities of $M_{1,1}(E)$”, Israel Journal of Mathematics, 80:3 (1992), 323–335 | DOI | MR | Zbl

[3] Mattina D., “On the graded identities and cocharacters of the algebra of $3\times3$ matrices”, Linear Algebra and its Applications, 384 (2004), 55–75 | DOI | MR | Zbl

[4] Aver'yanov I. V., “Basis of graded identities of the superalgebra $M_{1,2}(F)$”, Mathematical Notes, 85 (2009), 467–483 | DOI | DOI | MR

[5] Vincenzo O. M., “$Z_2$-graded polynomial identities for superalgebras of block-triangular matrices”, Serdica Mathematical Journal, 30:2–3 (2004), 111–134 | MR | Zbl

[6] Di Vincenzo O. M., Nardozza V., “$Z_2$-graded cocharacters for superalgebras of triangular matrices”, Journal of Pure and Applied Algebra, 194:1–2 (2004), 193–211 | DOI | MR | Zbl

[7] Amitsur S. A., Levitzki J., “Minimal identities for algebras”, Proceedings of the American Mathematical Society, 1:4 (1950), 449–463 | DOI | MR | Zbl

[8] Razmyslov Yu. P., “On the Jacobson radical in PI algebras”, Algebra and Logic, 13:3 (1974), 337–360 | MR

[9] Gateva T. V., “The complexity of a bundle of varieties of associative algebras”, Russian Mathematical Surveys, 36:1 (1981), 233 | DOI | MR | Zbl | Zbl

[10] Chang Q., “Some consequences of the standard polynomial”, Proceedings of the American Mathematical Society, 104:3 (1988), 707–710 | DOI | MR | Zbl

[11] Giambruno A., Sehgal S. K., “On a polynomial identity for $n\times n$ matrices”, Journal of Algebra, 126:2 (1989), 451–453 | DOI | MR | Zbl

[12] Antonov S. Yu., Antonova A. V., “On multiple polynomials of Capelli type”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 158, no. 1, 2016, 5–25 (in Russian) | MR

[13] Antonov S. Y., “The least degree identities subspace $M _ 1 ^ {(m, k)} (F)$ of matrix superalgebra $ M ^ {(m, k)} (F)$”, Russian Mathematical, 2012, no. 56, 1–16 | DOI | MR | Zbl

[14] Latyshev V. N., “Combinatorial generators of the multilinear polynomial identities”, Journal of Mathematical Sciences, 149:2 (2008), 1107–1112 | DOI | MR | MR | Zbl

[15] Belov A. Ya., “The local finite basis property and local representability of varieties of associative rings”, Izvestiya: Mathematics, 74:1 (2010), 1–126 | DOI | DOI | MR | Zbl