On antipodes and integrals in Hopf algebras over rings and the quantum Yang-Baxter equation
In a previous paper [Trans. Am. Math. Soc. 349, No. 9, 3823-3836 (1997)] the authors showed that for every Frobenius algebra \(A\) over a commutative ring a pair of dual bases with respect to a Frobenius homomorphism of \(A\) defines a canonical solution of the quantum Yang-Baxter equation (QYBE). Let \(H\) be a Hopf algebra over a commutative ring \(K\) which is a Frobenius algebra over \(K\) such that the Frobenius homomorphism is an integral for the dual of \(H\). (By a result of \textit{B. Pareigis} [J. Algebra 18, No. 4, 588-596 (1971; Zbl 0225.16008)] the latter property is satisfied if \(H\) is a finitely generated projective \(K\)-module and the Picard group of \(K\) is trivial.) The first main result of the paper under review is a formula for a norm of \(H\) and its image under the comultiplication by using a pair of dual bases with respect to a Frobenius homomorphism of \(H\). From these formulas the authors obtain an explicit description of the canonical solution of the QYBE mentioned above in terms of the antipode and an arbitrary integral of \(H\). In particular, they prove a trace formula for the square of the antipode, which in the case that \(K\) is a field is due to \textit{R. G. Larson} and \textit{D. E. Radford} and was used in an essential way in their proof of a conjecture of Kaplansky [J. Algebra 117, No. 2, 267-289 (1988; Zbl 0649.16005), Am. J. Math. 110, No. 1, 187-195 (1988; Zbl 0637.16006)]. If in addition \(H\) is assumed to be unimodular (i.e., left and right integrals of \(H\) coincide), then they show that the antipode is the identity on the subspace of integrals of \(H\) and that the Nakayama automorphism (which measures the symmetry of the bilinear form associated to the Frobenius homomorphism) coincides with the square of the antipode. As a consequence, the fourth power of the antipode is the identity if \(H\) and its dual are unimodular, and the Drinfel'd double of \(H\) is symmetric if \(K\) is a field. The first result has been proved by \textit{R. G. Larson} [in J. Algebra 17, No. 3, 352-368 (1971; Zbl 0217.33801)] for \(K\) a field, and the second result can be found in a paper of \textit{M. Lorenz} [J. Algebra 188, No. 2, 476-505 (1997; Zbl 0873.16023)]. The second main result of this paper is a characterization of the separability of \(H\) by several equivalent conditions some of which in the case that \(K\) is a field are due to \textit{D. G. Higman} [Can. J. Math. 7, 490-508 (1955; Zbl 0065.26001)] or \textit{R. G. Larson} and \textit{M. E. Sweedler} [Am. J. Math. 91, 75-94 (1969; Zbl 0179.05803)], respectively. Moreover, the authors obtain an explicit formula for the separability idempotent in terms of a pair of dual bases with respect to a Frobenius homomorphism which is closely related to their canonical solution of the QYBE. This in conjunction with the trace formula for the square of the antipode leads to a generalization of several results in the two papers of R. G. Larson and D. E. Radford cited above. Namely, let \(H\) be a Hopf algebra over a commutative ring \(K\) which is a finitely generated projective \(K\)-module. Then \(H\) and its dual are separable if and only if the trace of the square of the antipode of \(H\) is a unit in \(K\). If \(K\) contains the rational numbers, then \(H\) is separable if and only if its dual is separable, and in this case \(H\) (and its dual) are involutory.
scientific article; zbMATH DE number 1053875
scientific article