On Nakayama automorphisms of double Frobenius algebras
Here the author continues his study of double Frobenius (dF-) algebras [from J. Algebra 182, No. 1, 256-273 (1996; Zbl 0897.16022) and Commun. Algebra 26, No. 11, 3669-3690 (1998; Zbl 0914.16014)], and refers back to these papers for the relevant definitions and basic results. A dF-algebra over a field \(k\) is a finite-dimensional \(k\)-space \(A\) with two binary operations \(\cdot\) and \(*\) such that \((A,\cdot)\) and \((A,*)\) are associative algebras with (different) identity. Also there are algebra maps \(\varepsilon\colon(A,\cdot)\to k\) and \(\omega\colon(A,*)\to k\) so that the bilinear forms \(a\times b\mapsto\varepsilon(a\cdot b)\) or \(\omega(a*b)\) are nondegenerate. Finally, if \(a\mapsto a^\sigma\) is the unique bijection such that \(\varepsilon(a^\sigma*b)=\omega(a\cdot b)\), then there are \(s,t\in A\) such that for all \(a,b\in A\), \[ (a*b)^\sigma=b^\sigma*s*a^\sigma\quad\text{and}\quad(a\cdot b)^{\sigma^{-1}}=b^{\sigma^{-1}}\cdot t\cdot a^{\sigma^{-1}}. \] Examples of dF-algebras include finite-dimensional Hopf algebras (the second multiplication here comes from that on \(H^*\), so in fact from the comultiplication) and Bose-Mesner algebras (algebras of \(n\times n\) matrices with the usual matrix product and the trace map or with the Hadamard product and the map taking a matrix to the sum of its elements). For \(A\) a Frobenius algebra, the map \(N\colon A\to A\) defined by \(\langle\omega,a\cdot b\rangle=\langle\omega,b\cdot N(a)\rangle\) is called the Nakayama automorphism. In this paper, the author studies Nakayama automorphisms of dF-algebras. As an application, Radford's result that the antipode of a finite-dimensional Hopf algebra is of finite order is generalized to dF-algebras along with the formula for \(S^{4n}\) in terms of group-likes.
scientific article; zbMATH DE number 1275252
scientific article