Radford's formula for generalized weak bi-Frobenius algebras.
Let \(H\) be a finite dimensional Hopf algebra over a field \(\mathbf K\), and let \(H^*\) be its linear dual. Then \(H\) has the structure of a left and right \(H^*\)-module by \(f\rightharpoonup h=\text{mult}(f\otimes 1)\Delta(h)\) and \(h\leftharpoonup f=\text{mult}(f\otimes 1)\Delta(h)\), respectively, where \(h\in H\), \(f\in H^*\), and \(\Delta\) is the comultiplication on \(H\). Radford's formula states that \(S^4(h)=g(\alpha\rightharpoonup h\leftharpoonup\alpha^{-1})g^{-1}\) for some distinguished grouplike elements \(g\in H\), \(\alpha\in H^*\).NEWLINENEWLINE Recently, adaptations of Radford's formula have appeared in other categories of generalized Hopf algebras, such as weak Hopf algebras, biFrobenius Hopf algebras, co-Frobenius Hopf algebras, algebraic quantum groups, and bornological quantum groups.NEWLINENEWLINE The work under review marks the introduction of generalized weak biFrobenius algebra (hereafter, GWBF-algebra). Let \(H\) be finite dimensional algebra with unit \(1\) which also has a coalgebra structure with counit \(\varepsilon\). Pick \(\Psi\in H^*\), \(\Psi\neq 0\). Then there is a bijective map \(S\colon H\to H\) such that \(\text{mult}(\Psi\otimes S)(h\otimes 1)\Delta(g)=\text{mult}(\Psi\otimes 1)\Delta(h)(g\otimes 1)\) for all \(g,h\in H\). If \(S\) satisfies a sequence of axioms, including \(S\) is an anti-algebra map and anti-coalgebra map; and \((H,\Psi)\) is a Frobenius algebra, then \((H,\Psi,S)\) is said to be a GWBF-algebra. As an example, a weak Hopf algebra \(H\) gives a GWBF-algebra \((H,\Psi,S)\) where \(S\) is the antipode and \(\Psi\) is a right integral. Examples of GWBF-algebras which are ``new'' (in the sense that they are not weak Hopf algebras nor biFrobenius Hopf algebras) are also given.NEWLINENEWLINE The main result of the paper is to establish Radford's formula, which is done quite explicitly.
scientific article; zbMATH DE number 6339709
scientific article