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TMF, 2024 Volume 220, Number 1, Pages 25–43 (Mi tmf10656)

$n$-valued quandles and associated bialgebras

V. G. Bardakovabc, T. A. Kozlovskayac, D. V. Talalaevde

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State Agrarian University, Novosibirsk, Russia
c Regional Scientific and Educational Mathematical Center, Tomsk State University, Tomsk, Russia
d Lomonosov Moscow State University, Moscow, Russia
e Demidov Yaroslavl State University, Yaroslavl, Russia

Abstract: We study $n$-valued quandles and $n$-corack bialgebras. These structures are closely related to topological field theories in dimensions $2$ and $3$, to the set-theoretic Yang–Baxter equation, and to the $n$-valued groups, which have attracted considerable attention or researchers. We elaborate the basic methods of this theory, find an analogue of the so-called coset construction known in the theory of $n$-valued groups, and construct $n$-valued quandles using $n$-multiquandles. In contrast to the case of $n$-valued groups, this construction turns out to be quite rich in algebraic and topological applications. We study the properties of $n$-corack bialgebras, which play a role similar to that of bialgebras in group theory.

Keywords: multiset, multivalued group, multigroup, rack, quandle, $n$-valued quandle, bialgebra, rack bialgebra.

MSC: 20N20, 16S34, 05E30

Received: 12.12.2023
Revised: 24.02.2024

DOI: 10.4213/tmf10656


 English version:
Theoretical and Mathematical Physics, 2024, 220:1, 1080–1096

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© Steklov Math. Inst. of RAS, 2024