Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Axioms for Weak Bialgebras   Let A be a finite dimensional unital associative algebra over a field K,
which is also equipped with a coassociative counital coalgebra structure
(\Delta,\eps). A is called a Weak Bialgebra if the coproduct \Delta is
multiplicative. We do not require \Delta(1) = 1 \otimes 1 nor multiplicativity
of the counit \eps. Instead, we propose a new set of counit axioms, which are
modelled so as to guarantee that \Rep\A becomes a monoidal category with unit
object given by the cyclic A-submodule \E := (A --> \eps) \subset \hat A (\hat
A denoting the dual weak bialgebra). Under these monoidality axioms \E and
\bar\E := (\eps <-- A) become commuting unital subalgebras of \hat A which are
trivial if and only if the counit \eps is multiplicative. We also propose
axioms for an antipode S such that the category \Rep\A becomes rigid. S is
uniquely determined, provided it exists. If a monoidal weak bialgebra A has an
antipode S, then its dual \hat A is monoidal if and only if S is a bialgebra
anti-homomorphism, in which case S is also invertible. In this way we obtain a
definition of weak Hopf algebras which in Appendix A will be shown to be
equivalent to the one given independently by G. B\"ohm and K. Szlach\'anyi.
Special examples are given by the face algebras of T. Hayashi and the
generalised Kac algebras of T. Yamanouchi.

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The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc