# Graded central polynomials for the matrix algebra of order two

Daniela La Mattina, Plamen Koshlukov, Alexei Krasilnikov, Antônio Pereira Brandão Jr.

Research output: Other contribution

5 Citations (Scopus)

### Abstract

Let K be an infinite integral domain and $A=M_2(K)$ the algebra of $2\times 2$ matrices over $K$. The authors consider the natural $\mathbb{Z}_2$-grading of $A$ obtained by requiring that the diagonal matrices and the off-diagonal matrices are of homogeneous degree $0$ and $1$, respectively. When $K$ is a field, a basis of the graded identities of $A$ was described in [O. M. Di Vincenzo, On the graded identities of $M_{1,1}(E).$ Israel J. Math. 80 (1992), no. 3, 323-–335] in case $\mbox{char}\, K = 0$ and in [P. E. Koshlukov and S. S. de Azevedo, Graded identities for T-prime algebras over fields of positive characteristic. Israel J. Math. 128 (2002), 157-–176] when $K$ is infinite and $\mbox{char}\, K >2$. Here the authors remark that the same basis holds in case K is an infinite integral domain. They also study the $T_2$-space of central polynomials of $A$ and find a finite set of generators. Their proof does not depend on the characteristic, hence holds also for infinite fields of characteristic $2.$
Original language English Published - 2009

Matrix Algebra
Integral domain
Polynomial
Diagonal matrix
Finite Set
Generator
Algebra

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cite this

La Mattina, D., Koshlukov, P., Krasilnikov, A., & Brandão Jr., A. P. (2009). Graded central polynomials for the matrix algebra of order two.

Graded central polynomials for the matrix algebra of order two. / La Mattina, Daniela; Koshlukov, Plamen; Krasilnikov, Alexei; Brandão Jr., Antônio Pereira.

2009, .

Research output: Other contribution

La Mattina, D, Koshlukov, P, Krasilnikov, A & Brandão Jr., AP 2009, Graded central polynomials for the matrix algebra of order two..
La Mattina D, Koshlukov P, Krasilnikov A, Brandão Jr. AP. Graded central polynomials for the matrix algebra of order two. 2009.
La Mattina, Daniela ; Koshlukov, Plamen ; Krasilnikov, Alexei ; Brandão Jr., Antônio Pereira. / Graded central polynomials for the matrix algebra of order two. 2009.
@misc{0e1ba5b897624e50b1386d39bb05d85c,
title = "Graded central polynomials for the matrix algebra of order two",
abstract = "Let K be an infinite integral domain and $A=M_2(K)$ the algebra of $2\times 2$ matrices over $K$. The authors consider the natural $\mathbb{Z}_2$-grading of $A$ obtained by requiring that the diagonal matrices and the off-diagonal matrices are of homogeneous degree $0$ and $1$, respectively. When $K$ is a field, a basis of the graded identities of $A$ was described in [O. M. Di Vincenzo, On the graded identities of $M_{1,1}(E).$ Israel J. Math. 80 (1992), no. 3, 323-–335] in case $\mbox{char}\, K = 0$ and in [P. E. Koshlukov and S. S. de Azevedo, Graded identities for T-prime algebras over fields of positive characteristic. Israel J. Math. 128 (2002), 157-–176] when $K$ is infinite and $\mbox{char}\, K >2$. Here the authors remark that the same basis holds in case K is an infinite integral domain. They also study the $T_2$-space of central polynomials of $A$ and find a finite set of generators. Their proof does not depend on the characteristic, hence holds also for infinite fields of characteristic $2.$",
keywords = "central polynomials",
author = "{La Mattina}, Daniela and Plamen Koshlukov and Alexei Krasilnikov and {Brand{\~a}o Jr.}, {Ant{\^o}nio Pereira}",
year = "2009",
language = "English",
type = "Other",

}

TY - GEN

T1 - Graded central polynomials for the matrix algebra of order two

AU - La Mattina, Daniela

AU - Koshlukov, Plamen

AU - Krasilnikov, Alexei

AU - Brandão Jr., Antônio Pereira

PY - 2009

Y1 - 2009

N2 - Let K be an infinite integral domain and $A=M_2(K)$ the algebra of $2\times 2$ matrices over $K$. The authors consider the natural $\mathbb{Z}_2$-grading of $A$ obtained by requiring that the diagonal matrices and the off-diagonal matrices are of homogeneous degree $0$ and $1$, respectively. When $K$ is a field, a basis of the graded identities of $A$ was described in [O. M. Di Vincenzo, On the graded identities of $M_{1,1}(E).$ Israel J. Math. 80 (1992), no. 3, 323-–335] in case $\mbox{char}\, K = 0$ and in [P. E. Koshlukov and S. S. de Azevedo, Graded identities for T-prime algebras over fields of positive characteristic. Israel J. Math. 128 (2002), 157-–176] when $K$ is infinite and $\mbox{char}\, K >2$. Here the authors remark that the same basis holds in case K is an infinite integral domain. They also study the $T_2$-space of central polynomials of $A$ and find a finite set of generators. Their proof does not depend on the characteristic, hence holds also for infinite fields of characteristic $2.$

AB - Let K be an infinite integral domain and $A=M_2(K)$ the algebra of $2\times 2$ matrices over $K$. The authors consider the natural $\mathbb{Z}_2$-grading of $A$ obtained by requiring that the diagonal matrices and the off-diagonal matrices are of homogeneous degree $0$ and $1$, respectively. When $K$ is a field, a basis of the graded identities of $A$ was described in [O. M. Di Vincenzo, On the graded identities of $M_{1,1}(E).$ Israel J. Math. 80 (1992), no. 3, 323-–335] in case $\mbox{char}\, K = 0$ and in [P. E. Koshlukov and S. S. de Azevedo, Graded identities for T-prime algebras over fields of positive characteristic. Israel J. Math. 128 (2002), 157-–176] when $K$ is infinite and $\mbox{char}\, K >2$. Here the authors remark that the same basis holds in case K is an infinite integral domain. They also study the $T_2$-space of central polynomials of $A$ and find a finite set of generators. Their proof does not depend on the characteristic, hence holds also for infinite fields of characteristic $2.$

KW - central polynomials

UR - http://hdl.handle.net/10447/75679

UR - http://www.ams.org/mathscinet/pdf/2520727.pdf?pg1=RVRI&pg3=authreviews&s1=734661&vfpref=html&r=4

M3 - Other contribution

ER -