On cotriangular Hopf algebras (Q2747078)

In an earlier paper [Math. Res. Lett. 5, No. 1-2, 191-197 (1998; Zbl 0907.16016)] the authors proved that any semisimple triangular Hopf algebra over an algebraically closed field \(k\) of characteristic zero can be obtained from a finite group \(G\) by twisting the comultiplication of the group algebra \(k[G]\). Since semisimple Hopf algebras are finite-dimensional and cosemisimple, duality gives that any cotriangular semisimple Hopf algebra over \(k\) is obtained from \(k[G]^*\) by twisting its multiplication.NEWLINENEWLINENEWLINEThe paper under review extends this result on finite-dimensional cotriangular Hopf algebras to the infinite-dimensional case. The precise result is: let \((A,R)\) be a cotriangular Hopf algebra, \(R\) in \((A\otimes A)^*\). Then \(A\) is obtained from the function algebra of a proalgebraic group \(G\) by twisting its multiplication (possibly changing its \(R\)-form by a central group-like element of \(A^*\) of order \(\leq 2\)) if and only if the trace of \(S^2\) (\(S\) is the antipode of \(A\)) restricted to any finite-dimensional subcoalgebra \(C\) of \(A\) equals the dimension of \(C\). As with the earlier paper [op. cit.], the key step is the use of a theorem of Deligne on Tannakian categories. Examples are given of such twisted function algebras. In one of these infinite-dimensional examples (\(G\) is the group of affine transformations of the line), \(S^2\) is not the identity map. However, the authors conjecture that in the setting of the theorem, \(S^2\) must always be unipotent. They prove this in some special cases.