Jordan matrix algebras defined by generators and relations

Yingyu Luo; Yu Wang; Junjie Gu; Huihui Wang; Yingyu Luo; Yu Wang; Junjie Gu; Huihui Wang

doi:10.3934/math.2022168

AIMS Mathematics

2022, Volume 7, Issue 2: 3047-3055. doi: 10.3934/math.2022168

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Research article

Jordan matrix algebras defined by generators and relations

1.
College of Mathematics, Changchun Normal University, Changchun 130032, China
2.
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received: 07 October 2021 Accepted: 22 November 2021 Published: 24 November 2021
MSC : 17C10, 16S10, 16S50

In the present paper we describe Jordan matrix algebras over a field by generators and relations. We prove that the minimun number of generators of some special Jordan matrix algebras over a field is $2$ .

Keywords:

Citation: Yingyu Luo, Yu Wang, Junjie Gu, Huihui Wang. Jordan matrix algebras defined by generators and relations[J]. AIMS Mathematics, 2022, 7(2): 3047-3055. doi: 10.3934/math.2022168

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Abstract

1. Introduction

Let $F$ be a field of characteristic not $2$ . Let $A$ be an $F$ -algebra. By $A^{(+)}$ we denote the Jordan algebra induced by $A$ in the usual manner:

$a\circ b = \frac{1}{2}(ab+ba).$

We call $A^{(+)}$ the special Jordan algebra. There exist Jordan algebras that are not special, they are called exceptional. Let $S$ be a subset of $A^{(+)}$ . By $< S >$ we denote the subalgebra of $A^{(+)}$ generated by $S$ . For more detailed introduction of Jordan algebras we refer the reader a book of Jacobson ^[7].

Let $X$ be a nonempty set. The free algebra on $X$ over $F$ will be denoted by $F(X)$ . Let $F(X)^{(+)}$ be the Jordan algebra induced by $F(X)$ in the usual manner. Let $R$ be a nonempty subset of $F(X)^{(+)}$ . By $(R)$ we denote the ideal of $F(X)^{(+)}$ generated by $R$ . This forms the Jordan factor algebra $F(X)^{(+)}/(R)$ . By the universal property of $F(X)$ , a function $f:X\rightarrow A$ can be extended to an algebra homomorphism $\bar{f}$ from $F(X)$ into $A$ . It is clear that $\bar{f}$ induces an algebra homomorphism from $F(X)^{(+)}$ into $A^{(+)}$ . Suppose that $\bar{f}(R) = 0$ . We note that there exists an algebra homomorphism $\hat{f}$ from $F(X)^{(+)}/(R)$ into $A^{(+)}$ .

Set $X = \{\xi_i\; |\; i\in I\}$ and $R = \{f_j = f_j(\xi_{i_1}, \ldots, \xi_{i_{n(j)}})\; |\; j\in J\}$ . Note that every element in $R$ is a Jordan polynomial. For example,

$2\xi_1+\xi_2\circ \xi_3-\xi_3^4.$

Denote the coset $\xi_i+(R)$ in $F(X)^{(+)}/(R)$ by $x_i$ . Note that

$f_j(x_{i_1}, \ldots, x_{i_{n(j)}}) = 0$

for every $j\in J$ . Following the case of algebras in [,Section 6.2], we write $F(X)^{(+)}/(R)$ as $L(X; R)^{(+)}$ . We say that this Jordan algebra is defined by the generators $x_i$ and relations $f_j$ . We always hope that the number of generators of $L(X, R)^{(+)}$ is the minimum. For more detailed introduction of generators and relations of algebras we refer the reader to [1,Section 6.2].

As we know, both matrix algebras and Jordan matrix algebras are important algebras that we often come across. We remand the reader to the papers ^{[5,6,7,8,9,10,11,12,13]} for a general theory of matrix Jordan algebras and the papers ^[2,3,4] that focus on polynomial identities of Jordan matrix algebras. It is easy to check that the minimun number of generators of matrix algebras over a field is $2$ (see Proposition 1). However, it is not easy to determine the minimun number of generators of Jordan matrix algebras over a field.

In the present paper we shall describe Jordan matrix algebras over a field by generators and relations (see Theorem 2.1). We prove that the minimun number of generators of some special Jordan matrix algebras over a field is $2$ .

2. Jordan matrix algebras defined by generators and relations

Let $n\geq 2$ be an integer. By $M_n(F)$ we denote the algebra of all $n\times n$ matrices over $F$ . By $e_{ij}$ we denote the standard matrix unit of $M_n(F)$ . By $\delta_{ij}$ we denote the symbol of Kronecker delta.

We begin with the following simple result.

Theorem 2.1. Let $F$ be a field. Set

$X = \{\xi, \eta, \rho\}.$

Set

$\xi_{11} = \xi, \quad \xi_{12} = 2\xi_{11}\circ\eta.$

We set inductively

$\xi_{1, i+1} = 2\xi_{1i}\circ \eta$

for all $i = 2, \ldots, n-1$ . Similarly, we set inductively

$\xi_{i+1, 1} = 2\xi_{i1}\circ \rho$

for all $i = 1, \ldots, n-1$ . Furthemore, we set

$\xi_{ij} = 2\xi_{i1}\circ \xi_{1j}-\delta_{ij}\xi_{11}$

for all $i, j = 2, \ldots, n$ . Let $R$ be the following subset of $L(X)^{(+)}$ :

$\begin{eqnarray*} \begin{split} &\xi-\xi_{11}; \; \; \eta-\sum\limits_{i = 1}^{n-1}\xi_{i, i+1}; \; \; \rho-\sum\limits_{i = 1}^{n-1}\xi_{i+1, i};\\ &2\xi_{ij}\circ \xi_{st}-\delta_{js}\xi_{it}-\delta_{it}\xi_{sj}, \quad i, j, s, t = 1, \ldots, n;\\ &\sum\limits_{i = 1}^n\xi_{ii}-1. \end{split} \end{eqnarray*}$

We have that $L(X, R)^{(+)}\cong M_n(F)^{(+)}$ .

Proof. Set

$\begin{eqnarray*} \begin{split} x& = \xi+(R);\\ y& = \eta+(R);\\ z& = \rho+(R);\\ x_{ij}& = \xi_{ij}+(R), \quad i, j = 1, \ldots, n. \end{split} \end{eqnarray*}$

It follows from the elements in $R$ that

$\begin{eqnarray*} \begin{split} x& = x_{11}; \; \; y = \sum\limits_{i = 1}^{n-1}x_{i, i+1};\; \; z = \sum\limits_{i = 1}^{n-1}x_{i+1, i};\\ 2x_{ij}\circ x_{st}& = \delta_{js}x_{it}+\delta_{it}x_{sj}, i, j, s, t = 1, \ldots, n;\\ \sum\limits_{i = 1}^nx_{ii}& = 1. \end{split} \end{eqnarray*}$

It follows from the relation above that every element of $L(X; R)^{(+)}$ is a linear combination of the following set

$T = \{x_{ij}\; |\; i, j = 1, 2, \ldots, n\}.$

We claim that $T$ is an independent subset of $L(X, R)^{(+)}$ . Suppose that

$\begin{equation} \sum\limits_{i, j = 1}^{n}\lambda_{ij}x_{ij} = 0 \end{equation}$

(2.1)

for some $\lambda_{ij}\in F$ . We define a function $f:X\rightarrow M_n(F)^{(+)}$ as follows:

$f(\xi) = e_{11}, \quad f(\eta) = \sum\limits_{i = 1}^{n-1}e_{i, i+1}, \quad f(\rho) = \sum\limits_{j = 1}^{n-1}e_{j+1, j}.$

By the universal property of $F(X)$ we have that there exists an algebra homomorphism $\bar{f}:F(X)^{(+)}\rightarrow M_n(F)^{(+)}$ such that

$\bar{f}(\xi) = e_{11}, \quad\bar{f}(\eta) = \sum\limits_{i = 1}^{n-1}e_{i, i+1}, \quad\bar{f}(\rho) = \sum\limits_{j = 1}^{n-1}e_{j+1, j}$

and

$\begin{eqnarray*} \begin{split} \bar{f}(\xi_{ij}) = 2\bar{f}(\xi_{i1}\circ \xi_{1j})-\delta_{ij}\bar{f}(\xi_{11} = 2\bar{f}(\xi_{i1})\circ\bar{f}(\xi_{1j})-\delta_{ij}\bar{f}(\xi_{11}) = 2e_{i1}\circ e_{1j}-\delta_{ij}e_{11} = e_{ij} \end{split} \end{eqnarray*}$

for all $i, j = 1, \ldots, n$ . Note that $\{e_{ij}\; |\; i, j = 1, \ldots, n\}$ is a basis of $M_n(F)^{(+)}$ . This implies that $\bar{f}$ is surjective. It is easy to check that $\bar{f}(R) = 0$ . Hence there exists a surjective algebra homomorphism $\hat{f}:L(X; R)^{(+)}\rightarrow M_n(F)^{(+)}$ such that

$\hat{f}(x) = e_{11}, \quad \hat{f}(y) = \sum\limits_{i = 1}^{n-1}e_{i, i+1}, \quad \hat{f}(z) = \sum\limits_{j = 1}^{n-1}e_{j+1, j}$

and

$\hat{f}(x_{ij}) = e_{ij}$

for all $i, j = 1, \ldots, n$ . We get from (2.1) that

$\begin{eqnarray*} \begin{split} \sum\limits_{i = 1}^n\lambda_{ij}e_{ij} = 0. \end{split} \end{eqnarray*}$

It implies that $\lambda_{ij} = 0$ for all $i, j = 1, \ldots, n$ . Consequently, $T$ is a basis of $L(X, R)^{(+)}$ . In view of the above relations we get that the Jordan operation table of $T$ is the same as that of $\{e_{ij}\; |\; i, j = 1, \ldots, n\}$ , a standard basis of $M_n(F)^{(+)}$ . Therefore $L(X, R)^{(+)}\cong M_n(F)^{(+)}$ . The proof of the result is complete.

We remark that Theorem 2.1 implies the following result:

Corollary 1. Let $F$ be a field. We have that

$M_n(F)^{(+)} = \left < e_{11}, \sum\limits_{i = 1}^{n-1}e_{i, i+1}, \sum\limits_{j = 1}^{n-1}e_{j+1, j}\right > .$

In view of Corollary 2.1 we see that every Jordan matrix algebra over a field can be generated by three elements.

We remark that the minimun number of generators of matrix algebras over a field is $2$ . For example, we easily prove the following result. We give its proof for completeness.

Proposition 2.1. Let $F$ be a field. We have that

$M_n(F) = \left < \sum\limits_{i = 1}^{n-1}e_{i, i+1}, \sum\limits_{j = 1}^{n-1}e_{j+1, j}\right > .$

Proof. Set

$E = \sum\limits_{i = 1}^{n-1}e_{i, i+1}, \quad Q = \sum\limits_{j = 1}^{n-1}e_{j+1, j}.$

We have that

$E^{n-1}Q^{n-1} = e_{11}\in < E, Q > .$

We get that $< e_{11}, E, Q > = < E, Q >$ . In view of Corollary 2.1 we note that $M_n(F)^{(+)} = < e_{11}, E, Q >$ . This implies that

$M_n(F) = < e_{11}, E, Q > = < E, Q > .$

This proves the result.

We now give the main result of the paper, which shows that some special Jordan matrix algebras over a field can be generated by two elements.

Theorem 2.2. Let $F$ be a field. Let $n\geq 3$ be an integer. Suppose that $char(F) = 0$ or $char(F) > [\frac{3(n-1)}{2}]$ , the integer part of $\frac{3(n-1)}{2}$ . Set

$X = \{\xi, \eta\}.$

We set

$\begin{eqnarray*} \begin{split} &a_i = i, \quad i = 1, \ldots, n-2;\\ &a_{n-1} = [\frac{3(n-1)}{2}];\\ &a_n = n-1 \end{split} \end{eqnarray*}$

and

$\begin{eqnarray} \left( \begin{array}{cc} \lambda_1\\ \lambda_2\\ \vdots\\ \lambda_n \end{array} \right) = \left( \begin{array}{cc} a_1, a_1^2, \ldots, a_1^{n}\\ a_2, a_2^2, \ldots, a_2^{n}\\ \vdots\\ a_n, a_n^2, \ldots, a_n^{n} \end{array} \right)^{-1}\left( \begin{array}{cc} 1\\ 0\\ \vdots\\ 0 \end{array} \right) \end{eqnarray}$

(2.2)

Set

$\xi_{11} = \lambda_1\xi\circ \eta+\lambda_2 (\xi\circ \eta)^2+\cdots +\lambda_n(\xi\circ \eta)^{n}.$

We set

$\xi_{12} = \xi_{11}\circ \xi.$

Furthermore, we set inductively

$\xi_{1, i+1} = [\frac{i+1}{2}]^{-1}\xi_{1i}\circ \xi$

for all $i = 1, \ldots, n-2$ and

$\xi_{1n} = (n-1)^{-1}\xi_{1, n-1}\circ \xi.$

Similarly, we set

$\xi_{21} = 2\xi_{11}\circ \eta.$

We set inductively

$\xi_{i+1, 1} = 2\xi_{i1}\circ \eta$

for all $i = 1, \ldots, n-1$ . Moreover, we set

$\xi_{ij} = 2(\xi_{i1}\circ \xi_{1j})-\delta_{ij}\xi_{11}$

for all $i, j = 2, \ldots, n$ . Let $R$ be the following subset of $L(X)^{(+)}$ :

$\begin{eqnarray*} \begin{split} &\xi-\sum\limits_{i = 1}^{n-2}2[\frac{i+1}{2}]\xi_{i, i+1}-2(n-1)\xi_{n-1, n};\\ &\eta-\sum\limits_{i = 1}^{n-1}\xi_{i+1, i};\\ &2\xi_{ij}\circ \xi_{st}-\delta_{js}\xi_{it}-\delta_{it}\xi_{sj}, \quad i, j, s, t = 1, \ldots, n;\\ &\sum\limits_{i = 1}^n\xi_{ii}-1. \end{split} \end{eqnarray*}$

We have that $L(X, R)^{(+)}\cong M_n(F)^{(+)}$ . Moreover, the minimun number of generators of $M_n(F)^{(+)}$ is $2$ .

Proof. Set

$\begin{eqnarray*} \begin{split} x& = \xi+(R);\\ y& = \eta+(R);\\ x_{ij}& = \xi_{ij}+(R), \quad i, j = 1, \ldots, n. \end{split} \end{eqnarray*}$

It follows from the elements in $R$ that

$\begin{eqnarray*} \begin{split} x& = \sum\limits_{i = 1}^{n-2}2[\frac{i+1}{2}]x_{i, i+1}+2(n-1)x_{n-1, n};\\ y& = \sum\limits_{i = 1}^{n-1}x_{i+1, i};\\ 2x_{ij}\circ x_{st}& = \delta_{js}x_{it}+\delta_{it}x_{sj}, i, j, s, t = 1, \ldots, n;\\ \sum\limits_{i = 1}^nx_{ii}& = 1. \end{split} \end{eqnarray*}$

It follows from the relation above that every element of $L(X; R)^{(+)}$ is a linear combination of the following set

$T = \{x_{ij}\; |\; i, j = 1, 2, \ldots, n\}.$

We claim that $T$ is an independent subset of $L(X, R)^{(+)}$ . Suppose that

$\begin{equation} \sum\limits_{i, j = 1}^{n}\lambda_{ij}x_{ij} = 0 \end{equation}$

(2.3)

for some $\lambda_{ij}\in F$ . We define a function $f:X\rightarrow M_n(F)^{(+)}$ as follows:

$f(\xi) = 2\left(\sum\limits_{i = 1}^{n-2}[\frac{i+1}{2}]e_{i, i+1}+(n-1)e_{n-1, n}\right), \quad f(\eta) = \sum\limits_{j = 1}^{n-1}e_{j+1, j}.$

By the universal property of $F(X)$ we have that there exists an algebra homomorphism $\bar{f}:F(X)^{(+)}\rightarrow M_n(F)^{(+)}$ such that

$\bar{f}(\xi) = 2\left(\sum\limits_{i = 1}^{n-2}[\frac{i+1}{2}]e_{i, i+1}+(n-1)e_{n-1, n}\right), \quad\bar{f}(\eta) = \sum\limits_{j = 1}^{n-1}e_{j+1, j}$

and

$\begin{eqnarray*} \begin{split} \bar{f}(\xi\circ \eta)& = \bar{f}(\xi)\circ \bar{f}(\eta)\\ & = 2\left(\sum\limits_{i = 1}^{n-2}[\frac{i+1}{2}]e_{i, i+1}+(n-1)e_{n-1, n}\right)\circ\left(\sum\limits_{j = 1}^{n-1}e_{j+1, j}\right)\\ & = \sum\limits_{i = 1}^{n-1}ie_{ii}+[\frac{3(n-1)}{2}]e_{n-1, n-1}+(n-1)e_{nn}\\ & = \sum\limits_{i = 1}^na_ie_{ii}. \end{split} \end{eqnarray*}$

We get from (2.2) that

$\begin{eqnarray*} \begin{split} \bar{f}(\xi_{11})& = \bar{f}(\lambda_1\xi\circ \eta+\lambda_2 (\xi\circ \eta)^2+\cdots +\lambda_n(\xi\circ \eta)^{n})\\ & = \lambda_1\bar{f}(\xi\circ \eta)+\lambda_2 \bar{f}\left(\xi\circ \eta\right)^{2}+\cdots +\lambda_n\bar{f}(\xi\circ \eta)^{n}\\ & = \lambda_1\left(\sum\limits_{i = 1}^na_ie_{ii}\right)+\lambda_2 \left(\sum\limits_{i = 1}^na_ie_{ii}\right)^2\\ &\ \ \ +\cdots+\lambda_n\left( \sum\limits_{i = 1}^na_ie_{ii}\right)^{n}\\ & = (\lambda_1a_1+\lambda_2 a_1^2+\cdots +\lambda_na_1^{n})e_{11}+\cdots+\\ &\ \ \ (\lambda_1a_n+\lambda_2 a_n^2+\cdots +\lambda_na_n^{n})e_{nn}\\ & = e_{11}. \end{split} \end{eqnarray*}$

We have that

$\begin{eqnarray*} \begin{split} \bar{f}(\xi_{12})& = \bar{f}(\xi_{11}\circ \xi)\\ & = \bar{f}(\xi_{11})\circ \bar{f}(\xi)\\ & = e_{11}\circ \left(\sum\limits_{i = 1}^{n-2}2[\frac{i+1}{2}]e_{i, i+1}+2(n-1)e_{n-1, n}\right)\\ & = e_{12} \end{split} \end{eqnarray*}$

and

$\bar{f}(\xi_{1, i+1}) = [\frac{i+1}{2}]^{-1}\bar{f}(\xi_{1i}\circ \xi) = e_{1, i+1}$

for all $i = 1, \ldots, n-2$ . Moreover, we have that

$\bar{f}(\xi_{1n}) = (n-1)^{-1}\bar{f}(\xi_{1, n-1}\circ \xi) = e_{1n}.$

Similarly, we have that

$\bar{f}(\xi_{i1}) = e_{i1}$

for all $i = 2, \ldots, n-1$ . Moreover, we have that

$\begin{eqnarray*} \begin{split} \bar{f}(\xi_{ij}) = 2\bar{f}(\xi_{i1}\circ\xi_{1j})-\delta_{ij}\bar{f}(\xi_{11}) = 2e_{i1}\circ e_{1j}-\delta_{ij}e_{11} = e_{ij} \end{split} \end{eqnarray*}$

for all $i, j = 2, \ldots, n$ . Note that $\{e_{ij}\; |\; i, j = 1, \ldots, n\}$ is a basis of $M_n(F)^{(+)}$ . This implies that $\bar{f}$ is surjective. It is easy to check that $\bar{f}(R) = 0$ . Hence there exists a surjective algebra homomorphism $\hat{f}:L(X; R)^{(+)}\rightarrow M_n(F)^{(+)}$ such that

$\hat{f}(x) = \sum\limits_{i = 1}^{n-2}2[\frac{i+1}{2}]e_{i, i+1}+2(n-1)e_{n-1, n}, \quad\hat{f}(y) = \sum\limits_{j = 1}^{n-1}e_{j+1, j}.$

Moreover, we have that

$\hat{f}(x_{1j}) = e_{1j}, \quad \hat{f}(x_{i1}) = e_{i1}$

for all $i, j = 1, \ldots, n$ and so

$\hat{f}(x_{ij}) = e_{ij}$

for all $i, j = 1, \ldots, n$ . It follows from (2.3) that

$\begin{eqnarray*} \begin{split} \sum\limits_{i, j = 1}^n\lambda_{ij}e_{ij} = 0. \end{split} \end{eqnarray*}$

This implies that $\lambda_{ij} = 0$ for all $i, j = 1, \ldots, n$ . Consequently, $T$ is a basis of $L(X, R)^{(+)}$ . In view of the above relations we see that the Jordan operation table of $T$ is the same as that of $\{e_{ij}\; |\; i, j = 1, 2, \ldots, n\}$ , a standard basis of $M_n(F)^{(+)}$ . Therefore $L(X, R)^{(+)}\cong M_n(F)^{(+)}$ .

Suppose that $M_n(F)^{(+)} = < A >$ for some $A\in M_n(F)^{(+)}$ . It is clear that $M_n(F) = < A >$ . This implies that $M_n(F)$ is commutative, a contradiction. We get that the minimun number of generators of $M_n(F)^{(+)}$ is $2$ . The proof of the result is complete.

We remark that Theorem 2.2 implies the following result:

Corollary 2.2. Let $F$ be a field. Let $n\geq 3$ be an integer. Suppose that $char(F) = 0$ or $char(F) > [\frac{3(n-1)}{2}]$ . We have that

$M_n(F)^{(+)} = \left < \sum\limits_{i = 1}^{n-2}2[\frac{i+1}{2}]e_{i, i+1}+2(n-1)e_{n-1, n}, \sum\limits_{j = 1}^{n-1}e_{j+1, j}\right > .$

3. Conclusions

The main conclusion of the paper is to show that the minimun number of generators of $M_n(F)^{(+)}$ (where $n\geq 3$ and $char(F) = 0$ or $char(F) > [\frac{3(n-1)}{2}]$ ) is $2$ .

Acknowledgements

The authors are grateful to the referees for useful comments.

Conflict of interest

The authors declare no conflicts of interest in this paper.

References

[1]	M. Brešar, Introduction to nocommutative algebras, Springer, 2014. doi: 10.1007/978-3-319-08693-4.
[2]	L. Centrone, F. Martino, A note on cocharacter sequence of Jordan upper triangular matrix algebra, Commun. Algebra, 45 (2017), 1687–1695. doi: 10.1080/00927872.2016.1222408. doi: 10.1080/00927872.2016.1222408
[3]	L. Centrone, F. Martino, M. da Silva Souza, Specht property for some varieties of Jordan algebras of almost polynomial growth, J. Algebra, 521 (2019), 137–165. doi: 10.1016/j.jalgebra.2018.11.017. doi: 10.1016/j.jalgebra.2018.11.017
[4]	A. Cirrito, F. Martino, Ordinary and graded cocharacter of the Jordan algebra of $2\times 2$ upper triangular matrices, Linear Algebra Appl., 451 (2004), 246–259. doi: 10.1016/j.laa.2014.03.011. doi: 10.1016/j.laa.2014.03.011
[5]	C. Costara, Nonlinear invertibility preserving maps on matrix algebras, Linear Algebra Appl., 602 (2020), 216–222. doi: 10.1016/j.laa.2020.05.010. doi: 10.1016/j.laa.2020.05.010
[6]	M. Grašič, Zero product determined Jordan algebra, Linear Multilinear Algebra, 59 (2011), 671–685. doi: 10.1080/03081087.2010.485199. doi: 10.1080/03081087.2010.485199
[7]	N. Jacobson, Structure and representations of Jordan algebras, Colloquium Publications, Vol. 39, American Mathematical Society, 1968.
[8]	M. Kochetov, F. Y. Yasumura, Group grading on the Lie and Jordan algebras of block-triangular matrices, J. Algebra, 537 (2019), 147–172. doi: 10.1016/j.jalgebra.2019.07.020. doi: 10.1016/j.jalgebra.2019.07.020
[9]	X. B. Ma, G. H. Ding, L. Wang, Square-zero determined matrix algebras, Linear Multilinear Algebra, 59 (2011), 1311–1317. doi: 10.1080/03081087.2010.539606. doi: 10.1080/03081087.2010.539606
[10]	K. McCrimmon, A taste of Jordan algebras, Springer, 2004. doi: 10.1007/b97489.
[11]	H. P. Pertersson, The isotopy problem for Jordan matrix algebras, Trans. Amer. Math. Soc., 244 (1978), 185–197. doi: 10.1090/S0002-9947-1978-0506615-7. doi: 10.1090/S0002-9947-1978-0506615-7
[12]	C. C. Xi, J. B. Zhang, Structure of centralizer matrix algebras, Linear Algebra Appl., 622 (2021), 215–249. doi: 10.1016/j.laa.2021.03.034. doi: 10.1016/j.laa.2021.03.034
[13]	D. Y. Wang, Q. Hu C. G. Xia, Jordan derivation of certain Jordan matrix algebras, Linear Multilinear Algebra, 56 (2008), 581–588. doi: 10.1080/03081080701492165. doi: 10.1080/03081080701492165

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AIMS Mathematics

Jordan matrix algebras defined by generators and relations

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Abstract

1. Introduction

2. Jordan matrix algebras defined by generators and relations

3. Conclusions

Acknowledgements

Conflict of interest

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AIMS Mathematics

Jordan matrix algebras defined by generators and relations

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1. Introduction

2. Jordan matrix algebras defined by generators and relations

3. Conclusions

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