Images of multilinear graded polynomials on upper triangular matrix algebras

. In this paper we study the images of multilinear graded polyno- mials on the graded algebra of upper triangular matrices UT n . For positive integers q ≤ n , we classify these images on UT n endowed with a particular elementary Z q -grading. As a consequence, we obtain the images of multilinear graded polynomials on UT n with the natural 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 Z 3 -grading.

[1]  Yingyu Luo,et al.  On Fagundes-Mello conjecture , 2021, Journal of Algebra.

[2]  I. Gargate,et al.  Images of multilinear polynomials on n × n upper triangular matrices over infinite fields , 2021, Israel Journal of Mathematics.

[3]  Thiago Castilho de Mello.  The image of multilinear polynomials evaluated on 3 × 3 upper triangular matrices , 2021, Communications in Mathematics.

[4]  J. Zhou,et al.  The image of polynomials on 2 × 2 upper triangular matrix algebras , 2021 .

[5]  Dimas José Gonçalves,et al.  Graded polynomial identities for the upper triangular matrix algebra over a finite field , 2020 .

[6]  J. Zhou,et al.  The Images of Completely Homogeneous Polynomials on 2 × 2 Upper Triangular Matrix Algebras , 2020, Algebras and Representation Theory.

[7]  M. Brevsar.  Commutators and images of noncommutative polynomials , 2020, Advances in Mathematics.

[8]  S. Malev.  The images of noncommutative polynomials evaluated on the quaternion algebra , 2019, 1906.04973.

[9]  Yu Wang.  The images of multilinear polynomials on 2 × 2 upper triangular matrix algebras , 2019, Linear and Multilinear Algebra.

[10]  P. Fagundes,et al.  Images of multilinear polynomials of degree up to four on upper triangular matrices , 2018, Operators and Matrices.

[11]  P. Fagundes.  The images of multilinear polynomials on strictly upper triangular matrices , 2018, Linear Algebra and its Applications.

[12]  A. Giambruno,et al.  Central polynomials and growth functions , 2018, Israel Journal of Mathematics.

[13]  P. Koshlukov,et al.  Group gradings on the Jordan algebra of upper triangular matrices , 2017, 1708.03032.

[14]  S. Malev,et al.  The images of multilinear polynomials evaluated on 3×3 matrices , 2015 .

[15]  M. Brešar.  Introduction to Noncommutative Algebra , 2014 .

[16]  S. Malev.  The images of non-commutative polynomials evaluated on $2\times 2$ matrices over an arbitrary field , 2013 .

[17]  S. Malev,et al.  Power-central polynomials on matrices , 2013, 1310.1598.

[18]  A. Elduque,et al.  Gradings on Simple Lie Algebras , 2013 .

[19]  vSpela vSpenko.  On the image of a noncommutative polynomial , 2012, 1212.4600.

[20]  P. Koshlukov,et al.  Polynomial identities for the Jordan algebra of upper triangular matrices of order 2 , 2012 .

[21]  S. Malev,et al.  The images of non-commutative polynomials evaluated on 2 x 2 matrices , 2010, 1310.8563.

[22]  I. Klep,et al.  A note on values of noncommutative polynomials , 2009, 0909.2640.

[23]  Igor Klep,et al.  Values of noncommutative polynomials, Lie SkewIdeals and tracial Nullstellensätze , 2008, 0810.1774.

[24]  A. Valenti,et al.  Group gradings on upper triangular matrices , 2007 .

[25]  A. Valenti,et al.  Gradings on the algebra of upper triangular matrices and their graded identities , 2004 .

[26]  P. Koshlukov,et al.  Central polynomials in the matrix algebra of order two , 2004 .

[27]  P. Koshlukov.  Basis of the Identities of the Matrix Algebra of Order Two over a Field of Characteristic p ≠ 2☆ , 2001 .

[28]  Aleksandr Robertovich Kemer,et al.  Ideals of Identities of Associative Algebras , 1991 .

[29]  C. Chuang.  On ranges of polynomials in finite matrix rings , 1990 .

[30]  V. S. Drenski,et al.  A minimal basis of identities for a second-order matrix algebra over a field of characteristic o , 1981 .

[31]  A. Slinko.  Special varieties of Jordan algebras , 1979 .

[32]  Ju P Razmyslov.  ON A PROBLEM OF KAPLANSKY , 1973 .

[33]  E. Formanek.  Central polynomials for matrix rings , 1972 .

[34]  N. Jacobson.  Structure and Representations of Jordan Algebras , 1968 .

[35]  A. S. Amitsur,et al.  Minimal identities for algebras , 1950 .

[36]  V. Drensky.  Free algebras and PI-algebras : graduate course in algebra , 2000 .

[37]  V V Kulyamin.  Images of graded polynomials in matrix rings over finite group algebras , 2000 .

[38]  Yu. P. Razmyslov.  Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero , 1973 .

[39]  C. Wall.  Graded Brauer Groups. , 1964 .

[40]  O. Taussky.  Matrices with Trace Zero , 1962 .

[41]  Von Kenjiro Shoda.  Einige Sätze über Matrizen , 1936 .