Symmetry for finite dimensional Hopf algebras

This note refines criteria given by R. G. Larson and M. E. Sweedler for a finite dimensional Hopf algebra to be a symmetric algebra, with applications to restricted universal enveloping algebras and to certain finite dimensional subalgebras of the hyperalgebra of a semisimple algebraic group in characteristic p. Let A be a finite dimensional associative algebra over a field K. Then A is called Frobenius if there exists a nondegenerate bilinear form f: A x A -* K which is associative in the sense that f (ab, c) = f (a, bc) for all a, b, c E A [3, Chapter IX]. A is called symmetric if there exists a symmetric form of this type [3, ?66]. For example, semisimple algebras and group algebras of finite groups are synumetric. We investigate here the extent to which finite dimensional Hopf algebras (with antipode) are symmetric; they are always Frobenius, thanks to the main theorem of [8]. 1. Hopf algebras. In this section H denotes a finite dimensional Hopf algebra over an arbitrary field K, with antipode s and augmentation e: H -* K. According to the main theorem of [8], existence of the antipode implies (and is implied by) the existence of a (nonsingular) left integral A E H, which is unique up to scalar multiples. By definition, A satisfies: hA = e(h)A, for all h E H. Equally well, H has a right integral A', unique up to scalar multiples. If A' is proportional to A, H is called unimodular. With a left integral A is associated a nondegenerate bilinear associative form b on H [8, ?7]. As a result, H is a Frobenius algebra. From the second corollary of Proposition 8 in [8], applied to the dual Hopf algebra (whose antipode has the same order as s), we obtain immediately: THEOREM 1. With notation as above, b is symmetric if and only if H is unimodular and s2= 1. In particular, if the latter conditions hold, then H is symmetric. We can apply this to the algebras un (n = 1, 2, .. .) defined in [6, Appendix U], [7]. These are finite dimensional Hopf subalgebras of the hyperalgebra UK of a simply connected, semisimple algebraic group G over an algebraically Reccived by the editors February 28, 1977. AMS (MOS) subject classifications (1970). Primary 16A24, 16A36, 17B50; Secondary 17B45.