To Chang theorem. III

S. Yu. Antonov ; A. V. Antonova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 2, pp. 128-143.

Résumé

Various multilinear polynomials of Capelli type belonging to a free associative algebra $F\{X\cup Y\}$ over an arbitrary field $F$ generated by a countable set $X \cup Y$ are considered. The formulas expressing coefficients of polynomial Chang ${\mathcal R}(\bar x, \bar y \vert \bar w)$ are found. It is proved that if the characteristic of field $F$ is not equal two then polynomial ${\mathcal R}(\bar x, \bar y \vert \bar w)$ may be represented by different ways in the form of sum of two consequences of standard polynomial $S^-(\bar x)$. The decomposition of Chang polynomial ${\mathcal H}(\bar x, \bar y \vert \bar w)$ different from already known is given. Besides, the connection between polynomials ${\mathcal R}(\bar x, \bar y \vert \bar w)$ and ${\mathcal H}(\bar x, \bar y \vert \bar w)$ is found. Some consequences of standard polynomial being of great interest for algebras with polynomial identities are obtained. In particular, a new identity of minimal degree for odd component of $Z_2$-graded matrix algebra $M^{(m,m)}(F)$ is given.

Bibliographie

[1] Antonov S. Yu., Antonova A. V., “To Chang theorem II”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 17:2 (2017), 127–137 (in Russian) | DOI | MR

[2] 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) | DOI | MR

[3] Antonov S. Yu., Antonova A. V., “Quasi-polynomials of Capelli”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 15:4 (2015), 371–382 (in Russian) | DOI

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

[5] Antonov S. Yu., Antonova A. V., “On multiple polynomials of Capelli type”, Physics and mathematics, Uchenye Zapiski Kazanskogo Universiteta. Ser. Fiziko-Matematicheskie Nauki, 158:1 (2016), 5–25 (in Russian) | MR

[6] Gateva T. V., “The complexity of a bundle of varieties of associative algebras”, Russian Math. Surveys, 36:1 (1981), 233 | DOI | MR

[7] Kemer A. R., “Remark on the standard identity”, Math. Notes, 23:5 (1978), 414–416 | DOI | MR

[8] Leron U., “Multilinear identities of the matrix ring”, Trans. Amer. Math. Soc., 183 (1973), 175–202 | DOI | MR

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

[10] Mal'tsev Y. N., “Basis for identities of the algebra of upper triangular matrices”, Algebra and Logic, 10:4 (1971), 242–247 | DOI | MR

[11] Siderov P. N., “A basis for the identities of an algebra of triangular matrices over an arbitrary field”, PLISKA Studia Math. Bulgar., 2 (1981), 143–152 | MR

[12] Kostant B., “A theorem of Frobenius, a theorem of Amitsur–Levitzki, and cohomology theory”, J. Math. Mech., 7 (1958), 237–264 | DOI | MR

[13] Rowen L. H., “Standard polynomials in matrix algebras”, Trans. Amer. Math. Soc., 190 (1974), 253–284 | DOI | MR

[14] Wenxin M., Racine M., “Minimal identities of symmetric matrices”, Trans. Amer. Math. Soc., 320:1 (1990), 171–192 | DOI | MR

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