Geodesic
Quasi-polynomials of Capelli
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 371-382 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper deals with the class of Capelli polynomials in free associative algebra $F\{Z\}$ where $F$ is an arbitrary field and $Z$ is a countable set. The interest to these objects is initiated by assumption that the polynomials (Capelli quasi-polynomials) of some odd degree introduced will be contained in the basis ideal $Z_2$-graded identities of $Z_2$-graded matrix algebra $M^{(m,k)}(F)$ when $\mathrm{char}\,F=0$. In connection with this assumption the fundamental properties of Capelli quasi-polynomials have been given in the paper. In particularly, the decomposition of Capelli type polynomials have been given by the polynomials of the same type and some betweeness of their $T$-ideals have been shown. Besides, taking into account some properties of Capelli quasi-polynomials obtained and also the Chang theorem we show that all Capelli quasi-polynomials of even degree $2n$ $(n>1)$ are consequence of standard polynomial $S_n^-$ in case when the characteristic of field $F$ is not equal to two. At last we find the least $n \in N$ at which any of Capelli quasi-polynomials of even degree $2n$ belongs to ideal of matrix algebra $M_m(F)$ identities. 
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S. Yu. Antonov; A. V. Antonova. Quasi-polynomials of Capelli. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 371-382. http://geodesic.mathdoc.fr/item/ISU_2015_15_4_a1/

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

[2] Antonov S. Yu., “The least degree identities subspace $M_1^{(m,k)}(F)$ of matrix superalgebra $M^{(m,k)}(F)$”, Izvestiya VUZ. Matematika, 2012, no. 11, 1–16 | MR | Zbl

[3] Antonov S. Yu., “Some types of identities of subspaces $M_0^{(m,k)}(F)$, $M_1^{(m,k)}(F)$ of matrix superalgebra $M^{(m,k)}(F)$”, Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki, 154:1 (2012), 189–201 (in Russian)

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

[5] Antonov S. Yu., Antonova A. V., “To Chang Theorem”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 15:3 (2015), 247–251 (in Russian)

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