In algebra, a Taft Hopf algebra is a Hopf algebra introduced by Earl Taft (1971) that is neither commutative nor cocommutative and has an antipode of large even order.
Construction.
Suppose that "k" is a field with a primitive "n"'th root of unity ζ for some positive integer "n". The Taft algebra is the "n"2-dimensional associative algebra generated over "k" by "c" and "x" with the relations "c""n"=1, "x""n"=0, "xc"=ζ"cx". The coproduct takes "c" to "c"⊗"c" and "x" to "c"⊗"x" + "x"⊗1.
The counit takes "c" to 1 and "x" to 0. The antipode takes "c" to "c"−1 and "x" to –"c"−1"x": the order of the antipode is 2"n" (if "n" ; 1).