Special identities for the pre-Jordan product in the free dendriform algebra
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| Format: | Preprint |
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2012
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| author | Bremner, Murray R. Madariaga, Sara |
| author_facet | Bremner, Murray R. Madariaga, Sara |
| contents | Pre-Jordan algebras were introduced recently in analogy with pre-Lie algebras. A pre-Jordan algebra is a vector space $A$ with a bilinear multiplication $x \cdot y$ such that the product $x \circ y = x \cdot y + y \cdot x$ endows $A$ with the structure of a Jordan algebra, and the left multiplications $L_\cdot(x)\colon y \mapsto x \cdot y$ define a representation of this Jordan algebra on $A$. Equivalently, $x \cdot y$ satisfies these multilinear identities: [see PDF]. The pre-Jordan product $x \cdot y = x \succ y + y \prec x$ in any dendriform algebra also satisfies these identities. We use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree $\le 7$ for this product is implied by the identities of degree 4, but that there exist new identities of degree 8 which do not follow from those of lower degree. There is an isomorphism of $S_8$-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1212_5631 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Special identities for the pre-Jordan product in the free dendriform algebra Bremner, Murray R. Madariaga, Sara Rings and Algebras Mathematical Physics Primary 17C05. Secondary 17-04, 17A30, 17A50, 17C50, 18D50, 20C30 Pre-Jordan algebras were introduced recently in analogy with pre-Lie algebras. A pre-Jordan algebra is a vector space $A$ with a bilinear multiplication $x \cdot y$ such that the product $x \circ y = x \cdot y + y \cdot x$ endows $A$ with the structure of a Jordan algebra, and the left multiplications $L_\cdot(x)\colon y \mapsto x \cdot y$ define a representation of this Jordan algebra on $A$. Equivalently, $x \cdot y$ satisfies these multilinear identities: [see PDF]. The pre-Jordan product $x \cdot y = x \succ y + y \prec x$ in any dendriform algebra also satisfies these identities. We use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree $\le 7$ for this product is implied by the identities of degree 4, but that there exist new identities of degree 8 which do not follow from those of lower degree. There is an isomorphism of $S_8$-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra. |
| title | Special identities for the pre-Jordan product in the free dendriform algebra |
| topic | Rings and Algebras Mathematical Physics Primary 17C05. Secondary 17-04, 17A30, 17A50, 17C50, 18D50, 20C30 |
| url | https://arxiv.org/abs/1212.5631 |