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Universal Enveloping Algebras of Lie–Rinehart Algebras as a Left Adjoint Functor

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Abstract

We prove how the universal enveloping algebra constructions for Lie–Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie–Rinehart algebra morphisms, of modules over Lie–Rinehart algebras and of the infinitesimal gauge algebra of a module.

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References

  1. Ardizzoni, A., El Kaoutit, L., Saracco, P.: Differentiation and integration between Hopf algebroids and Lie algebroids. Preprint (2019). (arXiv: 1905.10288)

  2. Bourbaki, N.: Éléments de mathématique. Groupes et algèbres de Lie. Chapitre 1. Reprint of the 1972 original. Berlin: Springer. 148 p. (2007)

  3. Dăscălescu, S., Năstăsescu, C., Raianu, Ş.: Hopf algebras. An introduction. Monographs and Textbooks in Pure and Applied Mathematics, 235. Marcel Dekker, Inc., New York, (2001)

  4. El Kaoutit, L., Gómez-Torrecillas, J.: On the finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory. Bull. Belg. Math. Soc. Simon Stevin 28 , no. 1, 53–121, (2021)

  5. El Kaoutit, L., Saracco, P.: The Hopf Algebroid Structure of Differentially Recursive Sequences. Quaestiones Mathematicae (2021). https://doi.org/10.2989/16073606.2021.1885520

    Article  Google Scholar 

  6. El Kaoutit, L., Saracco, P.: Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras. Commun. Contemp. Math. 21 , no. 6, 1850015, 53 pp, (2019)

  7. Grivaux, J.: Derived intersections and free dg-Lie algebroids. To appear in RIMS, special issue dedicated to the 70th birthday of Prof. Masaki Kashiwara (2020). (arXiv:2002.01285)

  8. Huebschmann, J.: Lie-Rinehart algebras, descent, and quantization. Galois theory, Hopf algebras, and semiabelian categories, 295–316, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI (2004)

  9. Huebschmann, J.: Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras. Ann. Inst. Fourier (Grenoble) 48(2), 425–440 (1998)

    Article  MathSciNet  Google Scholar 

  10. Huebschmann, J.: Poisson cohomology and quantization. J. Reine Angew. Math. 408, 57–113 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Jacobson, N.: Lie algebras. Interscience Tracts in Pure and Applied Mathematics, No. 10 Interscience Publishers (a division of John Wiley & Sons), New York-London (1962)

  12. Kapranov, M.: Free Lie algebroids and the space of paths. Selecta Math. (N.S.) 13 , no. 2, 277–319, (2007)

  13. Mac Lane, S.: Categories for the Working Mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, (1998)

  14. Mackenzie, K.C.H.: Lie algebroids and Lie pseudoalgebras. Bull. London Math. Soc. 27(2), 97–147 (1995)

    Article  MathSciNet  Google Scholar 

  15. Malliavin, M.-P.: Algèbre homologique et opérateurs différentiels. Ring theory (Granada, 1986), 173–186, Lecture Notes in Math., 1328, Springer, Berlin, (1988)

  16. Milnor, J.W., Moore, J.C.: On the Structure of Hopf Algebras. Ann. of Math. 81(2), 211–264 (1965)

  17. Moerdijk, I., Mrčun, J.: On the universal enveloping algebra of a Lie algebroid. Proc. Am. Math. Soc. 138(9), 3135–3145 (2010)

    Article  MathSciNet  Google Scholar 

  18. Rinehart, G.S.: Differential forms on general commutative algebras. Trans. Am. Math. Soc. 108, 195–222 (1963)

    Article  MathSciNet  Google Scholar 

  19. Saracco, P.: On anchored Lie algebras and the Connes-Moscovici bialgebroid construction. Preprint (2020). (arXiv: 2009.14656)

  20. Sweedler, M.E.: Groups of simple algebras. Inst. Hautes Études Sci. Publ. Math. 44, 79–189 (1974)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I am sincerely grateful to Damien Calaque and Julien Grivaux for pointing out the reference [7] and to the anonymous referee for pointing out [4].

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  1. Département de Mathématique, Université Libre de Bruxelles, Boulevard du Triomphe, Brussels, 1050, Belgium

    Paolo Saracco

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  1. Paolo Saracco
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Paolo Saracco is a Chargé de Recherches of the Fonds de la Recherche Scientifique - FNRS and a member of the “National Group for Algebraic and Geometric Structures and their Applications” (GNSAGA-INdAM)

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Saracco, P. Universal Enveloping Algebras of Lie–Rinehart Algebras as a Left Adjoint Functor. Mediterr. J. Math. 19, 92 (2022). https://doi.org/10.1007/s00009-022-01985-9

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  • DOI: https://doi.org/10.1007/s00009-022-01985-9

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