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This article is cited in 3 scientific papers (total in 3 papers)
Mathematics
Quasi-polynomials of Capelli
S. Yu. Antonov, A. V. Antonova Kazan State Power Engineering University, 51, Krasnosel'skaya st., 420066, Kazan, Russia
Abstract:
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.
Key words:
$T$-ideal, standard polynomial, Capelli polynomial.
Citation:
S. Yu. Antonov, A. V. Antonova, “Quasi-polynomials of Capelli”, Izv. Saratov Univ. Math. Mech. Inform., 15:4 (2015), 371–382
Linking options:
https://www.mathnet.ru/eng/isu605 https://www.mathnet.ru/eng/isu/v15/i4/p371
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