Abstract
By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely generated and projective \(\Bbbk \)-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich’s reconstruction theorem for Hopf algebras. When \(\Bbbk \) is a field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.
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Acknowledgments
This paper was written while the author was member of the ”National Group for Algebraic and Geometric Structures and their Applications” (GNSAGA-INdAM). The author is sincerely grateful to Alessandro Ardizzoni and Claudia Menini for their contribution and to the referee for her/his useful suggestions.
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Appendix A: A relation for the preantipode of a quasi-bialgebra
Appendix A: A relation for the preantipode of a quasi-bialgebra
Recall from [18] that a preantipode for a quasi-bialgebra (A, Δ, ε, m, u, Φ) is a \(\Bbbk \)-linear map S : A → A that satisfies
for all a,b ∈ A, where \(\sum {\Phi }^{1}\otimes {\Phi }^{2}\otimes {\Phi }^{3}={\Phi }\). Let us introduce also the following extended notation for the reassociator and its inverse:
Let (A,m,u,Δ,ε,Φ,S) be a quasi-bialgebra with preantipode and consider the A-actions on End(A) = Hom(A,A) defined by \(\left (f\leftharpoonup a\right )(b) = f(ab)\) and \(\left (a\rightharpoonup f\right )(b) = f(ba)\) for all a,b ∈ A and for all f ∈EndA. Define the elements
where \(\left (x\left (y\rightharpoonup f\right )\right )(a) = xf(ay)\) and \(\left (\left (f\leftharpoonup x\right )y\right )(a) = f(ax)y\) for all a,x,y ∈ A and for all f ∈EndA. Let us introduce the following notation for shortness:
Lemma.
In the foregoing notation we have that for every a ∈ A
Moreover, the following relations hold for every a,b ∈ A
Proof
The reassociator Φ satisfies the dual relation to Eq. 3, i.e.
In particular, it satisfies
Applying \(\left (A\otimes m\right ) \left (A\otimes A\otimes m\right ) \left (A\otimes A\otimes (S\leftharpoonup a)\otimes A\right ) \) to both sides we get
which is the first identity in Eq. 33. The second one is proved analogously. Let us check that Eq. 34 holds as well ((35) is proved similarly). We compute
where in (∗) we used the quasi-coassociativity Φ ⋅ (Δ ⊗ A)Δ = (A ⊗Δ)Δ ⋅Φ. □
Lemma.
Let (A,m,u,Δ,ε,Φ,S) be a quasi-bialgebra with preantipode and let p,q be defined as above. For all a ∈ A we have that
Proof
Keeping in mind that Φ− 1 is counital, i.e. that it satisfies
we may compute directly
□
Proposition.
Let (A,m,u,Δ,ε,Φ,S) be a quasi-bialgebra with preantipode. For all a,b ∈ A we have
Proof
We know from Lemma A.2 that \(S(a)=\sum q^{1}\left (1_{A}\right )S\left (p^{1}aq^{2}\right ) p^{2}\left (1_{A}\right )\). Relation (36) is proved directly by applying it to \(S\left (ab\right ) \):
□
Formula (36) can be viewed as an anti-multiplicativity of the preantipode.
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Saracco, P. Coquasi-Bialgebras with Preantipode and Rigid Monoidal Categories. Algebr Represent Theor 24, 55–80 (2021). https://doi.org/10.1007/s10468-019-09931-2
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DOI: https://doi.org/10.1007/s10468-019-09931-2