Indecomposable coalgebras, simple comodules, and pointed Hopf algebras
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- by Susan Montgomery
- Proc. Amer. Math. Soc. 123 (1995), 2343-2351
- DOI: https://doi.org/10.1090/S0002-9939-1995-1257119-3
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Abstract:
We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C. This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element.References
- Eiichi Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. MR 594432
- William Chin and Ian M. Musson, Hopf algebra duality, injective modules and quantum groups, Comm. Algebra 22 (1994), no. 12, 4661–4692. MR 1285700, DOI 10.1080/00927879408825095
- Corrado De Concini and Victor G. Kac, Representations of quantum groups at roots of $1$, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506. MR 1103601
- J. Dieudonné, Introduction to the theory of formal groups, Pure and Applied Mathematics, vol. 20, Marcel Dekker, Inc., New York, 1973. MR 0332802 V. G. Drinfeld, Quantum groups, Proc. Internat. Congr. Math., Berkeley, Vol. 1, 1986, pp. 789-820.
- K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1020298
- Irving Kaplansky, Bialgebras, Lecture Notes in Mathematics, University of Chicago, Department of Mathematics, Chicago, Ill., 1975. MR 0435126
- Richard Gustavus Larson, Characters of Hopf algebras, J. Algebra 17 (1971), 352–368. MR 283054, DOI 10.1016/0021-8693(71)90018-4
- G. Lusztig, Modular representations and quantum groups, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp. 59–77. MR 982278, DOI 10.1090/conm/082/982278
- S. Montgomery, Some remarks on filtrations of Hopf algebras, Comm. Algebra 21 (1993), no. 3, 999–1007. MR 1204762, DOI 10.1080/00927879308824603
- 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
- S. Montgomery and S. Paul Smith, Skew derivations and $U_q(\textrm {sl}(2))$, Israel J. Math. 72 (1990), no. 1-2, 158–166. Hopf algebras. MR 1098987, DOI 10.1007/BF02764618
- Richard S. Pierce, Associative algebras, Studies in the History of Modern Science, vol. 9, Springer-Verlag, New York-Berlin, 1982. MR 674652, DOI 10.1007/978-1-4757-0163-0 D. E. Radford, Irreducible representations of ${U_q}(g)$ rising from $Mod_{{C^{1/2}}}^ \bullet$, Quantum Deformations of Algebras and Their Representations, Israel Math. Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1993.
- Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
- Moss Eisenberg Sweedler, Integrals for Hopf algebras, Ann. of Math. (2) 89 (1969), 323–335. MR 242840, DOI 10.2307/1970672
- Takefumi Shudo and Haruo Miyamoto, On the decomposition of coalgebras, Hiroshima Math. J. 8 (1978), no. 3, 499–504. MR 508357
- 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, Some topics on $\textrm {GL}_q(n)$, J. Algebra 147 (1992), no. 2, 379–410. MR 1161300, DOI 10.1016/0021-8693(92)90212-5
- Y. H. Xu and Y. Fong, On the decomposition of coalgebras, Words, languages and combinatorics (Kyoto, 1990) World Sci. Publ., River Edge, NJ, 1992, pp. 504–522. MR 1161046, DOI 10.1016/0378-4754(92)90087-w
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2343-2351
- MSC: Primary 16W30
- DOI: https://doi.org/10.1090/S0002-9939-1995-1257119-3
- MathSciNet review: 1257119