In algebra, a Taft Hopf algebra is a Hopf algebra introduced by that is neither commutative nor cocommutative and has an antipode of large even order.
Suppose that k is a field with a primitive nth root of unity ö for some positive integer n. The Taft algebra is the n<sup>2</sup>-dimensional associative algebra generated over k by c and x with the relations c<sup>n</sup>=1, x<sup>n</sup>=0, xc=öcx. The coproduct takes c to câÂÂc and x to câÂÂx + xâÂÂ1. The counit takes c to 1 and x to 0. The antipode takes c to c<sup>âÂÂ1</sup> and x to âÂÂc<sup>âÂÂ1</sup>x: the order of the antipode is 2n (if n > 1).