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*).

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

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