Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Antonov S. Y., Antonova A. V. Quasi-polynomials of Capelli. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2015, vol. 15, iss. 4, pp. 371-382. DOI: 10.18500/1816-9791-2015-15-4-371-382, EDN: VIZDPH

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Quasi-polynomials of Capelli

Antonova Alina Vladimirovna, Kazan State Power Engineering University, Russia

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 Z2 -graded identities of Z2 -graded matrix algebra M(m,k)(F) when 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 ∈ N at which any of Capelli quasi-polynomials of even degree 2n belongs to ideal of matrix algebra Mm(F) identities.

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