BiFrobenius algebras (Q2708909)

BiFrobenius algebras (bF algebras in short) are introduced and studied in the paper. Let \(A\) be a finite-dimensional algebra and \(\phi\in A^*\). The pair \((A,\phi)\) is called a Frobenius algebra if \(A^*=\phi\leftharpoonup A\) (or equivalently, \(A^*=A\rightharpoonup\phi\)). Dually, let \(C\) be a finite-dimensional coalgebra and \(t\in C\). The pair \((C,t)\) is called a Frobenius coalgebra if \(C=t\leftharpoonup C^*\) (or equivalently, \(C=C^*\rightharpoonup t\)). Let \(H\) be a finite-dimensional algebra and coalgebra, and \(\phi\in H^*\) and \(t\in H\). Define the map \(S\colon H\ni h\mapsto\sum\phi(t_1h)t_2\in H\). The 4-tuple \((H,\phi,t,S)\) is called a bF algebra if \((H,\phi)\) is a Frobenius algebra and \((H,t)\) is a Frobenius coalgebra such that the counit \(\varepsilon_H\) is an algebra map and the identity \(1_H\) is a group-like element, and such that \(S\) is both an anti-algebra map and an anti-coalgebra map.NEWLINENEWLINENEWLINELet \((H,\phi,t,S)\) be a bF algebra. Then \(\phi\leftharpoonup t=\varepsilon\) and \(t\leftharpoonup\phi=1\), and hence \(\phi\) and \(t\) are right integrals, i.e., \(th=\varepsilon(h)\) and \(\sum\phi(h_1)h_2=\phi(h)1\), \(\forall h\in H\). And there is an algebra map \(\alpha\colon H\to k\) (the right modular function) and a group-like element \(a\in H\) (the right modular element) such that \(ht=\alpha(h)t\) and \(\sum h_1\phi(h_2)=\phi(h)a\), \(\forall h\in H\). The paper gives Radford's formula for \(S^4\), i.e., \(S^4=a(\alpha^{-1}\rightharpoonup h\leftharpoonup\alpha)a^{-1}=\alpha^{-1}\rightharpoonup(aha^{-1})\leftharpoonup\alpha\), \(\forall h\in H\). It is proved that the bF algebras are equivalent to Koppinen's double Frobenius algebras. The authors also deal with the braided version of bF algebras and prove Radford's \(S^4\)-formula for this case.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].