Frobenius extensions of subalgebras of Hopf algebras

Fischman, D.; Montgomery, S.; Schneider, H.-J.

doi:10.1090/S0002-9947-97-01814-X

Frobenius extensions of subalgebras of Hopf algebras
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by D. Fischman, S. Montgomery and H.-J. Schneider

Trans. Amer. Math. Soc. 349 (1997), 4857-4895

DOI: https://doi.org/10.1090/S0002-9947-97-01814-X

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Abstract:

We consider when extensions $S\subset R$ of subalgebras of a Hopf algebra are $\beta$-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra $H$, we show that when $S\subset R$ are Hopf algebras in the Yetter-Drinfeld category for $H$, the extension is $\beta$-Frobenius provided $R$ is finite over $S$ and the extension of biproducts $S\star H\subset R\star H$ is cleft.

More generally we give conditions for an extension to be $\beta$-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants.

We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.

References

Yuri A. Bahturin, Alexander A. Mikhalev, Viktor M. Petrogradsky, and Mikhail V. Zaicev, Infinite-dimensional Lie superalgebras, De Gruyter Expositions in Mathematics, vol. 7, Walter de Gruyter & Co., Berlin, 1992. MR 1192546, DOI 10.1515/9783110851205
Allen D. Bell and Rolf Farnsteiner, On the theory of Frobenius extensions and its application to Lie superalgebras, Trans. Amer. Math. Soc. 335 (1993), no. 1, 407–424. MR 1097163, DOI 10.1090/S0002-9947-1993-1097163-5
Jeffrey Bergen, Semisimplicity of restricted enveloping algebras of Lie superalgebras, Pacific J. Math. 162 (1994), no. 1, 1–11. MR 1247140, DOI 10.2140/pjm.1994.162.1
S. Caenepeel, G. Militaru, and S. Zhu, Doi-Hopf modules, Yetter-Drinfeld modules, and Frobenius-type properties, Trans. Amer. Math. Soc. 349 (1997), 4311–4342.
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Rolf Farnsteiner, On Frobenius extensions defined by Hopf algebras, J. Algebra 166 (1994), no. 1, 130–141. MR 1276820, DOI 10.1006/jabr.1994.1144
Davida Fischman and Susan Montgomery, A Schur double centralizer theorem for cotriangular Hopf algebras and generalized Lie algebras, J. Algebra 168 (1994), no. 2, 594–614. MR 1292782, DOI 10.1006/jabr.1994.1246
Friedrich Kasch, Projektive Frobenius-Erweiterungen, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1960/61 (1960/1961), 87–109 (German). MR 0132085
M. Koppinen, Coideal subalgebras in Hopf algebras: freeness, integrals, smash products, Comm. Algebra 21 (1993), no. 2, 427–444. MR 1199682, DOI 10.1080/00927879308824572
H. F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981), no. 5, 675–692. MR 625597, DOI 10.1512/iumj.1981.30.30052
Richard Gustavus Larson and Moss Eisenberg Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–94. MR 240169, DOI 10.2307/2373270
Shahn Majid, Doubles of quasitriangular Hopf algebras, Comm. Algebra 19 (1991), no. 11, 3061–3073. MR 1132774, DOI 10.1080/00927879108824306
Akira Masuoka and Yukio Doi, Generalization of cleft comodule algebras, Comm. Algebra 20 (1992), no. 12, 3703–3721. MR 1191974, DOI 10.1080/00927879208824536
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
Tadasi Nakayama and Tosiro Tsuzuku, On Frobenius extensions. I, Nagoya Math. J. 17 (1960), 89–110. MR 124375, DOI 10.1017/S0027763000002075
Barkley Rosser, On the first case of Fermat’s last theorem, Bull. Amer. Math. Soc. 45 (1939), 636–640. MR 25, DOI 10.1090/S0002-9904-1939-07058-4
Warren D. Nichols and M. Bettina Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 111 (1989), no. 2, 381–385. MR 987762, DOI 10.2307/2374514
Ulrich Oberst and Hans-Jürgen Schneider, Über Untergruppen endlicher algebraischer Gruppen, Manuscripta Math. 8 (1973), 217–241 (German, with English summary). MR 347838, DOI 10.1007/BF01297688
David E. Radford, The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (1976), no. 2, 333–355. MR 407069, DOI 10.2307/2373888
David E. Radford, The structure of Hopf algebras with a projection, J. Algebra 92 (1985), no. 2, 322–347. MR 778452, DOI 10.1016/0021-8693(85)90124-3
David E. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 157 (1993), no. 2, 285–315. MR 1220770, DOI 10.1006/jabr.1993.1102
P. Schauenburg, Hopf bi-Galois extensions, Comm. in Algebra 24 (1996), 3797–3825.
M. Scheunert, Generalized Lie algebras, J. Math. Phys. 20 (1979), no. 4, 712–720. MR 529734, DOI 10.1063/1.524113
Hans-Jürgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 167–195. Hopf algebras. MR 1098988, DOI 10.1007/BF02764619
Hans-Jürgen Schneider, Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992), no. 2, 289–312. MR 1194305, DOI 10.1016/0021-8693(92)90034-J
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
Earl J. Taft, The order of the antipode of finite-dimensional Hopf algebra, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 2631–2633. MR 286868, DOI 10.1073/pnas.68.11.2631
Mitsuhiro Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251–270. MR 321963, DOI 10.1007/BF01579722
Mitsuhiro Takeuchi, Relative Hopf modules—equivalences and freeness criteria, J. Algebra 60 (1979), no. 2, 452–471. MR 549940, DOI 10.1016/0021-8693(79)90093-0
William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117, DOI 10.1007/978-1-4612-6217-6
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
David N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 261–290. MR 1074714, DOI 10.1017/S0305004100069139