Abstract
Masuoka proved (Proc Am Math Soc 137(6):1925–1932, 2009) that a finite-dimensional irreducible Hopf algebra H in positive characteristic is semisimple if and only if it is commutative semisimple if and only if the Hopf subalgebra generated by all primitives is semisimple. In this note, we give another proof of this result by using Hochschild cohomology of coalgebras.
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References
Andruskiewitsch, N.: About Finite Dimensional Hopf Algebras. Contemp. Math., vol. 294, pp. 1–57. Amer. Math. Soc., Providence (2002)
Blattner, R.J., Montgomery, S.: Crossed products and Galois extensions of Hopf algebras. Pac. J. Math. 137(1), 37–54 (1989)
Dăscălescu, S., Năstăsescu, C., Raianu, Ş.: Hopf Algebra: An Introduction, vol. 235. Marcel Dekker, New York (2001)
Demazure, M., Gabriel, P.: Groupes algébriques I. North Holland, Amsterdam (1970)
Hochschild, G.: Representations of restricted Lie algebras of characteristic \(p\). Proc. Am. Math. Soc. 5, 603–605 (1954)
Jacobson, N.: Lie Algebras. Dover Publications Inc., New York (1979)
Larson, R.G., Radford, D.E.: Finite dimensional cosemisimple Hopf algebras in characteristic \(0\) are semisimple. J. Algebra 117, 267–289 (1988)
Larson, R.G., Radford, D.E.: An associative orthogonal bilinear form for Hopf algebras. J. Algebra 91, 75–93 (1969)
Masuoka, A.: Harish–Chandra pairs for algebraic affine supergroup schemes over an arbitrary field. Transform. Gr. 17(4), 1085–1121 (2012)
Masuoka, A.: Semisimplicity criteria for irreducible Hopf algebras in positive characteristic. Proc. Am. Math. Soc. 137(6), 1925–1932 (2009)
Montgomery, S.: Hopf Algebras and Their Actions on Rings CBMS Regional Conference Series in Mathematics, vol. 82. Amer. Math. Soc., Providence (1993)
Nagata, M.: Complete reducibility of rational representations of a matric group. J. Math. Kyoto Univ. 1, 87–99 (1961/1962)
Nguyen, V., Wang, L., Wang, X.: Classification of connected Hopf algebras of dimension \(p^3\) I. J. Algebra 424, 473–505 (2015)
Nguyen, V.C., Wang, L., Wang, X.: Primitive deformations of quantum \(p\)-groups. Algebr. Represent. Theory (2018). https://doi.org/10.1007/s10468-018-9800-x
Nichols, W.D., Zoeller, M.B.: A Hopf algebra freeness theorem. Am. J. Math. 111(2), 381–385 (1989)
Ştefan, D., Oystaeyen, F.V.: Hochschild cohomology and the coradical filtration of pointed coalgebras: applications. J. Algebra 210(2), 535–556 (1998)
Sweedler, M.E.: Connected fully reducible affine group schemes in positive characteristic are abelian. J. Math. Kyoto Univ. 11, 51–70 (1971)
Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)
Wang, X.: Connected Hopf algebras of dimension \(p^2\). J. Algebra 391, 93–113 (2013)
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The author was partially supported by U. S. National Science Foundation.
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Wang, X. A note on Masuoka’s theorem for semisimple irreducible Hopf algebras. Arch. Math. 113, 11–20 (2019). https://doi.org/10.1007/s00013-019-01309-6
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DOI: https://doi.org/10.1007/s00013-019-01309-6