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Classification and structure of generalized Legendrian racks
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We study algebraic aspects of generalized Legendrian racks, which are nonassociative structures based on the Legendrian Reidemeister moves. We answer an open question characterizing the group of GL-structures on a given rack. As applications, we classify several infinite families of GL-racks. We also compute automorphism groups of dihedral GL-quandles. Then we compute the centers of the category of GL-racks and several of its full subcategories. We also construct an equivalence of categories between racks and GL-quandles. We also study tensor products of racks and GL-racks coming from universal algebra. Surprisingly, the categories of racks and GL-racks have tensor units. The induced symmetric monoidal structure on medial racks is closed, and similarly for medial GL-racks.
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On medial Latin quandles and affine modules
As categories, Latin medial quandles are equivalent to affine modules over Z[t^{±1},(1−t)^{−1}], and medial commutative quandles to affine modules over Z[1/2].
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