Taft Hopf algebra

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 n²-dimensional associative algebra generated over k by c and x with the relations cⁿ=1, xⁿ=0, xc=ζcx. 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⁻¹x: the order of the antipode is 2n (if n > 1).

References

• 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