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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2303.13208 |
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| _version_ | 1866913577472884736 |
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| author | Barbier, Sigiswald De Medts, Tom Smet, Michiel |
| author_facet | Barbier, Sigiswald De Medts, Tom Smet, Michiel |
| contents | Kantor pairs, (quadratic) Jordan pairs, and similar structures have been instrumental in the study of $\mathbb{Z}$-graded Lie algebras and algebraic groups. We introduce the notion of an operator Kantor pair, a generalization of Kantor pairs to arbitrary (commutative, unital) rings, similar in spirit as to how quadratic Jordan pairs and algebras generalize linear Jordan pairs and algebras. Such an operator Kantor pair is formed by a pair of $Φ$-groups $(G^+,G^-)$ of a specific kind, equipped with certain homogeneous operators. For each such a pair $(G^+,G^-)$, we construct a $5$-graded Lie algebra $L$ together with actions of $G^\pm$ on $L$ as automorphisms. Moreover, we can associate a group $G(G^+,G^-) \subset \operatorname{Aut}(L)$ to this pair generalizing the projective elementary group of Jordan pairs. If the non-$0$-graded part of $L$ is projective, we can uniquely recover $G^+,G^-$ from $G(G^+,G^-)$ and the grading on $L$ alone. We establish, over rings $Φ$ with $1/30 \in Φ$, a one to one correspondence between Kantor pairs and operator Kantor pairs. Finally, we construct operator Kantor pairs for the different families of central simple structurable algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13208 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Operator Kantor Pairs Barbier, Sigiswald De Medts, Tom Smet, Michiel Rings and Algebras Group Theory 17B60, 17B70, 17A30, 17C99, 16T05 Kantor pairs, (quadratic) Jordan pairs, and similar structures have been instrumental in the study of $\mathbb{Z}$-graded Lie algebras and algebraic groups. We introduce the notion of an operator Kantor pair, a generalization of Kantor pairs to arbitrary (commutative, unital) rings, similar in spirit as to how quadratic Jordan pairs and algebras generalize linear Jordan pairs and algebras. Such an operator Kantor pair is formed by a pair of $Φ$-groups $(G^+,G^-)$ of a specific kind, equipped with certain homogeneous operators. For each such a pair $(G^+,G^-)$, we construct a $5$-graded Lie algebra $L$ together with actions of $G^\pm$ on $L$ as automorphisms. Moreover, we can associate a group $G(G^+,G^-) \subset \operatorname{Aut}(L)$ to this pair generalizing the projective elementary group of Jordan pairs. If the non-$0$-graded part of $L$ is projective, we can uniquely recover $G^+,G^-$ from $G(G^+,G^-)$ and the grading on $L$ alone. We establish, over rings $Φ$ with $1/30 \in Φ$, a one to one correspondence between Kantor pairs and operator Kantor pairs. Finally, we construct operator Kantor pairs for the different families of central simple structurable algebras. |
| title | Operator Kantor Pairs |
| topic | Rings and Algebras Group Theory 17B60, 17B70, 17A30, 17C99, 16T05 |
| url | https://arxiv.org/abs/2303.13208 |