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Images of multilinear graded polynomials on upper triangular matrix algebras

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

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.

References

Albert, A. A. and Muckenhoupt, B., On matrices of trace zero . Michigan Math. J. 4(1957), 1–3.CrossRefGoogle Scholar
Amitsur, A. S. and Levitzki, J., Minimal identities for algebras . Proc. Amer. Math. Soc. 1(1950), 449–463.CrossRefGoogle Scholar
Amitsur, S. and Rowen, L., Elements of reduce trace 0 . Israel J. Math. 87(1994), 161–179.CrossRefGoogle Scholar
Brešar, M., Introduction to noncommutative algebra, Springer, Cham, 2014.CrossRefGoogle Scholar
Brešar, M., Commutators and images of noncommutative polynomials , Adv. Math.. 374 (2020), Article ID 107346, 21 pages.CrossRefGoogle Scholar
Brešar, M. and Klep, I., Values of noncommutative polynomials, Lie skew-ideals and tracial Nullstellensätze . Math. Res. Lett. 16(2009), no. 4, 605–626.CrossRefGoogle Scholar
Brešar, M. and Klep, I., A note on values of noncommutative polynomials . Proc. Amer. Math. Soc. 138(2010), no. 7, 2375–2379.CrossRefGoogle Scholar
Chuang, C.-L., On ranges of polynomials in finite matrix rings . Proc. Amer. Math. Soc. 140(1990), 293–302.CrossRefGoogle Scholar
Colombo, J. and Koshlukov, P., Central polynomials in the matrix algebra of order two . Linear Algebra Appl. 377(2004), 53–67.CrossRefGoogle Scholar
de Mello, T. C., The image of multilinear polynomials evaluated on $3\times 3$ upper triangular matrices . Commun. Math. 29(2021), 183–186.CrossRefGoogle Scholar
Di Vincenzo, O. M., Koshlukov, P., Valenti, A., Gradings on the algebra of upper triangular matrices and their graded identities . J. Algebra 275(2004), no. 2, 550–566.CrossRefGoogle Scholar
Drensky, V., A minimal basis of identities for a second-order matrix algebra over a field of characteristic 0 . Algebra Logika 20(1981), 282–290 (in Russian); in English translation Algebra Logic 20(1982), 188–194.Google Scholar
Drensky, V., Free algebras and PI-algebras: Graduate course in algebra, Springer, Singapore, 1999.Google Scholar
Elduque, A. and Kochetov, M., Gradings on simple Lie algebras, Mathematical Surveys and Monographs, 189, American Mathematical Society, Providence, RI; Atlantic Association for Research in the Mathematical Sciences, Halifax, NS, 2013.CrossRefGoogle Scholar
Fagundes, P., The images of multilinear polynomials on strictly upper triangular matrices . Linear Algebra Appl. 563(2019), no. 1, 287–301.CrossRefGoogle Scholar
Fagundes, P. and de Mello, T. C., Images of multilinear polynomials of degree up to four on upper triangular matrices . Oper. Matrices. 13(2019), 283–292.CrossRefGoogle Scholar
Filippov, V. T., Kharchenko, V. K., and Shestakov, I. P. (eds.), Dniester notebook: Unsolved problems in the theory of rings and modules. 4th ed., Mathematics Institute, Russian Academy of Sciences, Siberian Branch, Novosibirsk, 1993.Google Scholar
Formanek, E., Central polynomials for matrix rings . J. Algebra. 23(1972), 129–132.CrossRefGoogle Scholar
Gargate, I. and de Mello, T. C., Images of multilinear polynomials on $n\times n$ upper triangular matrices over infinite fields, Israel J. Math., 2022, to appear.CrossRefGoogle Scholar
Giambruno, A. and Zaicev, M., Central polynomials and growth functions . Israel J. Math. 226(2018), 15–28.CrossRefGoogle Scholar
Gonçalves, D. and Riva, E., Graded polynomial identities for the upper triangular matrix algebra over a finite field . J. Algebra. 559(2020), 625–645.CrossRefGoogle Scholar
Gonçalves, D. and Salomão, M., ${\mathbb{Z}}_2$ -graded polynomial identities for the Jordan algebra of $2\times 2$ upper triangular matrices. Preprint, 2020.Google Scholar
Jacobson, N., Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, 39, American Mathematical Society, Providence, RI, 1968.CrossRefGoogle Scholar
Kanel-Belov, A., Malev, S., and Rowen, L., The images of non-commutative polynomials evaluated on $2\times 2$ matrices . Proc. Amer. Math. Soc. 140(2012), 465–478.CrossRefGoogle Scholar
Kanel-Belov, A., Malev, S., and Rowen, L., The images of multilinear polynomials evaluated on $3\times 3$ matrices . Proc. Amer. Math. Soc. 144(2016), 7–19.CrossRefGoogle Scholar
Kanel-Belov, A., Malev, S., and Rowen, L., Power-central polynomials on matrices . J. Pure Appl. Algebra 220(2016), 2164–2176.CrossRefGoogle Scholar
Kemer, A. R., Ideals of identities of associative algebras, Translations of Mathematical Monographs, 87, American Mathematical Society, Providence, RI, 1991.CrossRefGoogle Scholar
Koshlukov, P., Basis of the identities of the matrix algebra of order two over a field of characteristic $\mathrm{p}\ne 2$ . J. Algebra. 241(2001), 410–434.CrossRefGoogle Scholar
Koshlukov, P. and Martino, F., Polynomial identities for the Jordan algebra of upper triangular matrices of order $2$ . J. Pure Appl. Algebra. 216(2012), 2524–2532.CrossRefGoogle Scholar
Koshlukov, P. and Yasumura, F., Group gradings on the Jordan algebra of upper triangular matrices . Linear Algebra Appl. 534(2017), 1–12.CrossRefGoogle Scholar
Kulyamin, V., Images of graded polynomials in matrix rings over finite group algebras . Usp. Mat. Nauk 55(2000), no. 2, 141–142 (in Russian); in English translation Russ. Math. Surv. 55(2000), 345–346.Google Scholar
Luo, Y. Y. and Wang, Y., On Fagundes–Mello conjecture . J. Algebra. 592(2022), 118–152.CrossRefGoogle Scholar
Malev, S., The images of non-commutative polynomials evaluated on $2\times 2$ matrices over an arbitrary field , J. Algebra Appl. 13(2014), no. 6, Article ID 1450004, 12 pages.CrossRefGoogle Scholar
Malev, S., The images of noncommutative polynomials evaluated on the quaternion algebra , J. Algebra Appl. 20(2021), no. 5, Article ID 2150074, 8 pages.CrossRefGoogle Scholar
Okhitin, S., Central polynomials of the algebra of second order matrices , Vestn. Mosk. Univ., Ser. I 43(1988), no. 4, 61–63 (in Russian); in English translation Moscow Univ. Math. Bull. 43(1988), no. 4, 49–51.Google Scholar
Razmyslov, Y. P., Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero , Algebra Logika 12(1973), 83–113 (in Russian); English translation Algebra Logic 12(1974), 47–63.CrossRefGoogle Scholar
Razmyslov, Y. P., On a problem of Kaplansky , Izv. Akad. Nauk SSSR, Ser. Mat. 37(1973), 483–501 (in Russian); in English translation. Math USSR. Izv. 7(1973), 479–496.Google Scholar
Shoda, K., Einige Sätze über Matrizen . Japan J. Math. 13(1937), 361–365 (in German).CrossRefGoogle Scholar
Slinko, A., Special varieties of Jordan algebras , Mat. Zametki 26(1979), no. 3, 337–344 (in Russian); in English translation. Math. Notes 26(1979/80), nos. 3–4, 661–665.Google Scholar
Špenko, Š., On the image of a noncommutative polynomial . J. Algebra. 377(2013), 298–311.CrossRefGoogle Scholar
Valenti, A. and Zaicev, M., Group gradings on upper triangular matrices . Arch. Math. 89(2007), 33–40.CrossRefGoogle Scholar
Wall, C. T. C., Graded Brauer groups . J. Reine Angew. Math. 213(1964), 187–199.CrossRefGoogle Scholar
Wang, Y., The images of multilinear polynomials on $2\times 2$ upper triangular matrix algebras . Linear Multilinear Algebra 67(2019), 2366–2372.CrossRefGoogle Scholar
Wang, Y., Zhou, J., and Luo, Y. Y., The image of polynomials on $2\times 2$ upper triangular matrix algebras . Linear Algebra Appl. 610(2021), 560–570.CrossRefGoogle Scholar
Zhou, J. and Wang, Y., The images of completely homogeneous polynomials on $2\times 2$ upper triangular matrix algebras . Algebr. Represent. Theory 24(2021), no. 5, 1221–1229.CrossRefGoogle Scholar