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Algebras and Representation Theory

, Volume 21, Issue 4, pp 703–716 | Cite as

Restricted Lie Algebras via Monadic Decomposition

  • Alessandro Ardizzoni
  • Isar Goyvaerts
  • Claudia Menini
Article
  • 31 Downloads

Abstract

We give a description of the category of restricted Lie algebras over a field \(\Bbbk \) of prime characteristic by means of monadic decomposition of the functor that computes the \(\Bbbk \)-vector space of primitive elements of a \(\Bbbk \)-bialgebra.

Keywords

Monads Restricted Lie algebras 

Mathematics Subject Classification (2010)

Primary 18C15 Secondary 16S30 

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Notes

Acknowledgements

This note was written while the first and the third authors were members of the ”National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA-INdAM). The first author was partially supported by the research grant “Progetti di Eccellenza 2011/2012” from the “Fondazione Cassa di Risparmio di Padova e Rovigo”. He thanks the members of the department of Mathematics of both Vrije Universiteit Brussel and Université Libre de Bruxelles for their warm hospitality and support during his stay in Brussels in August 2013, when the work on this paper was initiated. The second named author acknowledges the financial support of an INdAM Marie Curie Fellowship.

References

  1. 1.
    Adámek, J., Herrlich, H., Tholen, W.: Monadic decompositions. J. Pure Appl. Algebra 59, 111–123 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Appelgate, H., Tierney, M.: Categories with models. 1969 Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67). Springer, Berlin (1969)Google Scholar
  3. 3.
    Ardizzoni, A., Gómez-Torrecillas, J., Menini, C.: Monadic decompositions and classical lie theory. Appl. Categor. Struct. 23, 93–105 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ardizzoni, A., Menini, C.: Milnor-Moore categories and monadic decomposition. J. Algebra 448, 488–563 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Beck, J.M.: Triples, algebras and cohomology. Reprints Theory Appl. Categ. 2, 1–59 (2003)MathSciNetMATHGoogle Scholar
  6. 6.
    Berger, C.: Iterated wreath product of the simplex category and iterated loop spaces. Adv. Math. 213(1), 230–270 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Berger, C., Melliès, P.-A., Weber, M.: Monads with arities and their associated theories. J. Pure Appl. Algebra 216, 2029–2048 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Barr, M., Wells, C.: Toposes, triples and theories. Corrected reprint of the 1985 original. Repr. Theory Appl. Categ. 12, 288 (2005)MATHGoogle Scholar
  9. 9.
    Gray, J.W.: Formal Category Theory: Adjointness for 2-categories. Lecture Notes in Mathematics, vol. 391. Springer-Verlag, Berlin-New York (1974)CrossRefGoogle Scholar
  10. 10.
    Grünenfelder, L.: UBER DIE STRUKTUR VON HOPF-ALGEBREN. Dissertation ETH Zürich (1969)Google Scholar
  11. 11.
    Jacobson, N.: Abstract derivation and Lie algebras. Trans. Amer. Math. Soc. 42, 206–224 (1937)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jacobson, N.: Restricted Lie algebras of characteristic p. Trans. Amer. Math. Soc. 50, 15–25 (1941)MathSciNetMATHGoogle Scholar
  13. 13.
    Kharchenko, V.K.: Connected braided Hopf algebras. J. Algebra 307, 24–48 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Michaelis, W.: Lie coalgebras. Adv. Math. 38, 1–54 (1980)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Michaelis, W.: The Primitives of the Continuous Linear Dual of a Hopf Algebra as the Dual Lie Algebra of a Lie Coalgebra. Lie Algebra and Related Topics (Madison, WI, 1988), 125–176, Contemp Math., vol. 110. Amer. Math. Soc., Providence (1990)Google Scholar
  16. 16.
    Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. of Math. (2) 81, 211–264 (1965)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    MacDonald, J.L., Stone, A.: The tower and regular decomposition. Cahiers Topologie géom Différentielle 23, 197–213 (1982)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematics “G. Peano”University of TurinTorinoItaly
  2. 2.Department of Mathematics and Computer ScienceUniversity of FerraraFerraraItaly

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