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Antonov S. Y., Antonova A. V. Quasi-Polynomials of Capelli. II. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 1, pp. 4-16. DOI: 10.18500/1816-9791-2020-20-1-4-16, EDN: AXSHCX
Quasi-Polynomials of Capelli. II
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 S 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 Mk(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 Z2-graded matrix algebra Mk+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 Z2-graded matrix algebra Mm+k(F) for m > k. The conditions under which Capelli quasi-polynomials of degree 4k + 1 being identities of the subspace M (m,k) 1 (F) are given.
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