Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Antipodes Of Hom-Hopf algebras   In the recent definition of Hom-Hopf algebras the antipode S is the relative
Hominverse of the identity map with respect to the convolution product. We
observe that some fundamental properties of the antipode of Hopf algebras and
Hom-Hopf algebras, with the original definition, do not hold generally in the
new setting. We show that the antipode is a relative Hom-anti algebra and a
relative anti-coalgebra morphism. It is also relative Hom-unital, and relative
Hom-counital. Furthermore if the twisting maps of multiplications and
comultiplications are invertible then S is an anti-algebra and an
anti-coalgebra map. We show that any Hom-bialgebra map between two Hom-Hopf
algebras is a relative Hom-morphism of Hom-Hopf alegbras. Specially if the
corresponding twisting maps are all invertible then it is a Hom-Hopf algebra
map. If the Hom-Hopf algebra is commutative or cocommutative we observe that
S^2 is equal to the identity map in some sense. At the end we study the images
of primitive and group-like elements under the antipode.

0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
16
The categories of this paper are: stat