# T-ideal

## Quasi-polynomials of Capelli. III

In this paper polynomials of Capelli type (double and quasi-polynomials of Capelli) belonging to a free associative algebra $F\{X\cup Y\}$ considering over an arbitrary field $F$ and generated by two disjoint countable sets $X, Y$ are investigated. It is shown that double Capelli's polynomials $C_{4k,\{1\}}$, $C_{4k,\{2\}}$ are consequences of the standard polynomial $S^-_{2k}$.

## To Chang Theorem

Multilinear polynomials H (¯x, ¯y) and R(¯x, ¯y), the sum of which is the Chang polynomial F(¯x, ¯y) have been introduced in this paper. It has been proved by mathematical induction method that each of them is a consequence of the standard polynomial S−(¯x). In particular it has been shown that the double Capelli polynomial of add degree C2m−1(¯x, ¯y) is also a consequence of the polynomial S−m(¯x, ¯y).

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## Quasi-polynomials of Capelli

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.

## To Chang Theorem. II

Multilinear polynomials H +(¯x, ¯y| ¯ w), H −(¯x, ¯y| ¯ w) ∈ F{X ∪ Y }, the sum of which is a polynomial H (¯x, ¯y| ¯ w) Chang (where F{X∪Y } is a free associative algebra over an arbitrary field F of characteristic not equal two, generated by a countable set X ∪ Y ) have been introduced in this paper. It has been proved that each of them is a consequence of the standard polynomial S−(¯x).

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## To Chang Theorem. III

Various multilinear polynomials of Capelli type belonging to a free associative algebra F {X ∪ Y } over an arbitrary ﬁeld F generated by a countable set X ∪ Y are considered. The formulas expressing coefﬁcients of polynomial Chang R(¯x, ¯y|¯w) are found. It is proved that if the characteristic of ﬁeld F is not equal two then polynomial R(¯x, ¯y| ¯w) may be represented by different ways in the form of sum of two consequencesof standard polynomial S− (¯x).

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## 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).