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

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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

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* Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, rings and modules. Lie algebras and Hopf algebras, Mathematical Surveys and Monographs, vol. 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 (11): 2631–2633, Bibcode:1971PNAS...68.2631T, doi:10.1073/pnas.68.11.2631, MR 0286868, PMC 389488, PMID 16591950, Zbl 0222.160122631-2633&rft.date=1971&rft_id=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC389488#id-name=PMC&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0286868#id-name=MR&rft_id=info:bibcode/1971PNAS...68.2631T&rft_id=info:pmid/16591950&rft_id=info:doi/10.1073/pnas.68.11.2631&rft_id=https://zbmath.org/?format=complete&q;=an:0222.16012#id-name=Zbl&rft.aulast=Taft&rft.aufirst=Earl+J.&rfr_id=info:sid/en.wikipedia.org:Taft+Hopf+algebra" class="Z3988">