Geodesic
Quasi-polynomials of Capelli.~II
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 1, pp. 4-16.

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This paper observes the continuation of the study of a certain kind of polynomials of type Capelli (Capelli quasi-polynomials) belonging to the free associative algebra $F\{X\bigcup Y\}$ considered over an arbitrary field $F$ and generated by two disjoint countable sets $X$ and $Y$. It is proved that if $char F=0$ then among the Capelli quasi-polynomials of degree $4k-1$ there are those that are neither consequences of the standard polynomial $S^-_{2k}$ nor identities of the matrix algebra $M_k(F)$. It is shown that if $char F=0$ then only two of the six Capelli quasi-polynomials of degree $4k-1$ are identities of the odd component of the $Z_2$-graded matrix algebra $M_{k+k}(F)$. It is also proved that all Capelli quasi-polynomials of degree $4k+1$ are identities of certain subspaces of the odd component of the $Z_2$-graded matrix algebra $M_{m+k}(F)$ for $m>k$. The conditions under which Capelli quasi-polynomials of degree $4k+1$ being identities of the subspace $M_1^{(m,k)}(F)$ are given. 
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S. Yu. Antonov; A. V. Antonova. Quasi-polynomials of Capelli.~II. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 1, pp. 4-16. http://geodesic.mathdoc.fr/item/ISU_2020_20_1_a0/

[1] Antonov S. Yu., “Some types of identities of subspaces $M^{(m,k)}_0(F), M^{(m,k)}_1(F)$ of matrix superalgebra $M^{(m,k)}(F)$”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 154, no. 1 (2012), 189–201 (in Russian)

[2] Antonov S. Yu., Antonova A. V., “Quasi-polynomials of Capelli”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 15:4 (2015), 371–382 (in Russian) | DOI

[3] Birmajer D., “Polynomial detection of matrix subalgebras”, Proc. Amer. Math. Soc., 133:4 (2004), 1007–1012

[4] Chang Q., “Some consequences of the standard polynomial”, Proc. Amer. Math. Soc., 104:3 (1988), 707–710

[5] Kostant B., “A theorem of Frobenius, a theorem of Amitsur-Levitzki and cohomology theory”, J. Math. Mech., 7 (1958), 237–264

[6] Rowen L. H., “Standard polynomials in matrix algebras”, Proc. Amer. Math. Soc., 190 (1974), 253–284

[7] Wenxin M., Racine M., “Minimal identities of symmetric matrices”, Proc. Amer. Math. Soc., 320:1 (1990), 171–192

[8] Vincenzo O. M., “On the graded identities of $M_{1,1}(E)$”, Israel J. Math., 80:3 (1992), 323–335

[9] Mattina D., “On the graded identities and cocharacters of the algebra of 3x3 matrices”, J. Linear Algebra App., 384 (2004), 55–75 | DOI

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

[11] Vincenzo O. M., “$Z_2$-graded polynomial identities for superalgebras of block-triangular matrices”, Serdica Math. J., 30 (2004), 111–134

[12] Vincenzo O. M., “$Z_2$-graded cocharacters for superalgebras of triangular matrices”, J. of Pure and Applied Algebra, 194:1–2 (2004), 193–211 | DOI

[13] Amitsur S. A., Levitzki J., “Minimal identities for algebras”, Proc. Amer. Math. Soc., 1:4 (1950), 449–463

[14] Antonov S. Yu., Antonova A. V., “To Chang Theorem. II”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 17:2 (2017), 127–137 (in Russian) | DOI