Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polyadic Hopf algebras and quantum groups   This article continues the study of concrete algebra-like structures in our
polyadic approach, where the arities of all operations are initially taken as
arbitrary, but the relations between them, the arity shapes, are to be found
from some natural conditions ("arity freedom principle"). In this way,
generalized associative algebras, coassociative coalgebras, bialgebras and Hopf
algebras are defined and investigated. They have many unusual features in
comparison with the binary case. For instance, both the algebra and its
underlying field can be zeroless and nonunital, the existence of the unit and
counit is not obligatory, and the dimension of the algebra is not arbitrary,
but "quantized". The polyadic convolution product and bialgebra can be defined,
and when the algebra and coalgebra have unequal arities, the polyadic version
of the antipode, the querantipode, has different properties. As a possible
application to quantum group theory, we introduce the polyadic version of
braidings, almost co-commutativity, quasitriangularity and the equations for
the $R$-matrix (which can be treated as a polyadic analog of the Yang-Baxter
equation). Finally, we propose another concept of deformation which is governed
not by the twist map, but by the medial map, where only the latter is unique in
the polyadic case. We present the corresponding braidings, almost co-mediality
and $M$-matrix, for which the compatibility equations are found.

3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
1
The categories of this paper are: cs
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
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The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci