Lie and Jordan products in interchange algebras
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2014
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| _version_ | 1866912490880761856 |
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| author | Bremner, Murray Madariaga, Sara |
| author_facet | Bremner, Murray Madariaga, Sara |
| contents | We study Lie brackets and Jordan products derived from associative operations $\circ, \bullet$ satisfying the interchange identity $(w \bullet x ) \circ ( y \bullet z ) \equiv (w \circ y ) \bullet ( x \circ z )$. We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree $\le 7$ relating (i) the two Lie brackets, (ii) one Lie bracket and one Jordan product, and (iii) the two Jordan products. For the Lie-Lie case, there are two new identities in degree 6 and another two in degree 7. For the Lie-Jordan case, there are no new identities in degree $\le 6$ and a complex set of new identities in degree 7. For the Jordan-Jordan case, there is one new identity in degree 4, two in degree 5, and complex sets of new identities in degrees 6 and 7. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1408_3069 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Lie and Jordan products in interchange algebras Bremner, Murray Madariaga, Sara Rings and Algebras Representation Theory Primary 17A30, Secondary 17A50, 17B60, 17C50, 18D50 We study Lie brackets and Jordan products derived from associative operations $\circ, \bullet$ satisfying the interchange identity $(w \bullet x ) \circ ( y \bullet z ) \equiv (w \circ y ) \bullet ( x \circ z )$. We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree $\le 7$ relating (i) the two Lie brackets, (ii) one Lie bracket and one Jordan product, and (iii) the two Jordan products. For the Lie-Lie case, there are two new identities in degree 6 and another two in degree 7. For the Lie-Jordan case, there are no new identities in degree $\le 6$ and a complex set of new identities in degree 7. For the Jordan-Jordan case, there is one new identity in degree 4, two in degree 5, and complex sets of new identities in degrees 6 and 7. |
| title | Lie and Jordan products in interchange algebras |
| topic | Rings and Algebras Representation Theory Primary 17A30, Secondary 17A50, 17B60, 17C50, 18D50 |
| url | https://arxiv.org/abs/1408.3069 |