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Images of multilinear graded polynomials on upper triangular matrix algebras

Part of: Rings with polynomial identity Basic linear algebra Jordan algebras

Published online by Cambridge University Press:  19 September 2022

Pedro Fagundes  and
Plamen Koshlukov
Pedro Fagundes
Affiliation:
IMECC, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária “Zeferino Vaz,” Distr. Barão Geraldo, Campinas, São Paulo CEP 13083-859, Brazil e-mail: pedro.fagundes@ime.unicamp.br
Plamen Koshlukov*
Affiliation:
IMECC, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária “Zeferino Vaz,” Distr. Barão Geraldo, Campinas, São Paulo CEP 13083-859, Brazil e-mail: pedro.fagundes@ime.unicamp.br
*
e-mail: plamen@unicamp.br
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Abstract

In this paper, we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices $UT_n$. For positive integers $q\leq n$, we classify these images on $UT_{n}$ endowed with a particular elementary ${\mathbb {Z}}_{q}$-grading. As a consequence, we obtain the images of multilinear graded polynomials on $UT_{n}$ with the natural ${\mathbb {Z}}_{n}$-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras $UT_{2}$ and $UT_{3}$, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra $UJ_{2}$, and also for $UJ_{3}$ endowed with the natural elementary ${\mathbb {Z}}_{3}$-grading.

Keywords

Images of polynomialsLvov–Kaplansky conjectureupper triangular matricesgraded algebrasJordan algebras

MSC classification

Primary: 15A54: Matrices over function rings in one or more variables
Secondary: 16R50: Other kinds of identities (generalized polynomial, rational, involution) 17C05: Identities and free Jordan structures
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 5 , October 2023 , pp. 1540 - 1565
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

P. Fagundes was supported by São Paulo Research Foundation (FAPESP), Grant No. 2019/16994-1. P. Koshlukov was partially supported by São Paulo Research Foundation (FAPESP), Grant No. 2018/23690-6 and by CNPq Grant No. 302238/2019-0.

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