Modules over geometric quandles and representations of Lie-Yamaguti algebras. (English) Zbl 1492.17005
The notion of modules over racks or quandles was introduced by N. Andruskiewitsch and M. Graña [Adv. Math. 178, No. 2, 177–243 (2003; Zbl 1032.16028)].
It is known that there is a correspondence between the category of rack (quandle) modules and the category of abelian group objects in the category of racks (quandles). In particular, modules over topological racks (quandles) (topological spaces with a quandle (rack) structure) were investigated in [M. Elhamdadi and E.-K. M. Moutuou, J. Knot Theory Ramifications 25, No. 3, Article ID 1640002, 17 p. (2016; Zbl 1345.57022)].
The author of the paper gives a characterization of modules over different classes of geometric racks (quandles) (racks or quandles with additional geometric structures) such as: transitive smooth quandles, complex analytic quandles, quandles variety, regular \(s\)-manifolds. In particular, the author shows that there is an equivalence of the category of linear quandle modules over \(Q\), where \(Q\) is a smoooth quandle (complex analytic quandle or a quandle variety) and the category of vectors bundles in the category of geometric quandles over \(Q\).
Finally, he shows that there is a faithful functor between the category of regular quandle modules over a connected regular \(s\)-manifold \(Q\) and the category of regular representations of the infinitesimal \(s\)-manifold.
It is known that there is a correspondence between the category of rack (quandle) modules and the category of abelian group objects in the category of racks (quandles). In particular, modules over topological racks (quandles) (topological spaces with a quandle (rack) structure) were investigated in [M. Elhamdadi and E.-K. M. Moutuou, J. Knot Theory Ramifications 25, No. 3, Article ID 1640002, 17 p. (2016; Zbl 1345.57022)].
The author of the paper gives a characterization of modules over different classes of geometric racks (quandles) (racks or quandles with additional geometric structures) such as: transitive smooth quandles, complex analytic quandles, quandles variety, regular \(s\)-manifolds. In particular, the author shows that there is an equivalence of the category of linear quandle modules over \(Q\), where \(Q\) is a smoooth quandle (complex analytic quandle or a quandle variety) and the category of vectors bundles in the category of geometric quandles over \(Q\).
Finally, he shows that there is a faithful functor between the category of regular quandle modules over a connected regular \(s\)-manifold \(Q\) and the category of regular representations of the infinitesimal \(s\)-manifold.
Reviewer: Agata Pilitowska (Warszawa)
MSC:
| 17A40 | Ternary compositions |
| 14M17 | Homogeneous spaces and generalizations |
| 17D99 | Other nonassociative rings and algebras |
| 22F30 | Homogeneous spaces |
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