A Coordinatization Theorem for the Jordan algebra of symmetric 2x2 matrices
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| Format: | Preprint |
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2024
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| _version_ | 1866911779991322624 |
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| author | Laliena, Jesús Solís, Victor López Shestakov, Ivan |
| author_facet | Laliena, Jesús Solís, Victor López Shestakov, Ivan |
| contents | The Jacobson Coordinatization Theorem describes the structure of unitary Jordan algebras containing the algebra $H_n(F)$ of symmetric nxn matrices over a field F with the same identity element, for $n\geq 3$. In this paper we extend the Jacobson Coordinatization Theorem for n=2. Specifically, we prove that if J is a unitary Jordan algebra containing the Jordan matrix algebra $H_2(F)$ with the same identity element, then J has a form $J=H_2(F)\otimes A_0+k\otimes A_1$, where $A=A_0+A_1$ is a $Z_2$-graded Jordan algebra with a partial odd Leibniz bracket {,} an $k=e_{12}-e_{21}\in M_2(F)$ with the multiplication given by $(a\otimes b)(c\otimes d)=ac\otimes bd + [a,c]\otimes \{b,d\},$ the commutator [a,c] is taken in $M_2(F)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_10556 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Coordinatization Theorem for the Jordan algebra of symmetric 2x2 matrices Laliena, Jesús Solís, Victor López Shestakov, Ivan Rings and Algebras 17Cxx, 17C10 The Jacobson Coordinatization Theorem describes the structure of unitary Jordan algebras containing the algebra $H_n(F)$ of symmetric nxn matrices over a field F with the same identity element, for $n\geq 3$. In this paper we extend the Jacobson Coordinatization Theorem for n=2. Specifically, we prove that if J is a unitary Jordan algebra containing the Jordan matrix algebra $H_2(F)$ with the same identity element, then J has a form $J=H_2(F)\otimes A_0+k\otimes A_1$, where $A=A_0+A_1$ is a $Z_2$-graded Jordan algebra with a partial odd Leibniz bracket {,} an $k=e_{12}-e_{21}\in M_2(F)$ with the multiplication given by $(a\otimes b)(c\otimes d)=ac\otimes bd + [a,c]\otimes \{b,d\},$ the commutator [a,c] is taken in $M_2(F)$. |
| title | A Coordinatization Theorem for the Jordan algebra of symmetric 2x2 matrices |
| topic | Rings and Algebras 17Cxx, 17C10 |
| url | https://arxiv.org/abs/2402.10556 |