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.

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 n2-dimensional associative algebra generated over k by c and x with the relations cn=1, xn=0, xccx. The coproduct takes c to cc and x to cx + x⊗1. The counit takes c to 1 and x to 0. The antipode takes c to c−1 and x to –c−1x: the order of the antipode is 2n (if n > 1).

Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, rings and modules. Lie algebras and Hopf algebras, Mathematical Surveys and Monographs, 168, Providence, RI: American Mathematical Society, doi:10.1090/surv/168, ISBN 978-0-8218-5262-0, MR 2724822, Zbl 1211.16023
Taft, Earl J. (1971), "The order of the antipode of finite-dimensional Hopf algebra", Proc. Natl. Acad. Sci. U.S.A., 68: 2631–2633, doi:10.1073/pnas.68.11.2631, MR 0286868, PMC 389488, PMID 16591950, Zbl 0222.16012

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