On the universal enveloping algebra of a Lie algebroid
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- by I. Moerdijk and J. Mrčun PDF
- Proc. Amer. Math. Soc. 138 (2010), 3135-3145 Request permission
Abstract:
We review the extent to which the structure of the universal enveloping algebra of a Lie algebroid over a manifold $M$ resembles a Hopf algebra, and prove a Cartier-Milnor-Moore theorem for this type of structure.References
- Pierre Cartier, A primer of Hopf algebras, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 537–615. MR 2290769, DOI 10.1007/978-3-540-30308-4_{1}2
- Marius Crainic and Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. MR 1973056, DOI 10.4007/annals.2003.157.575
- Robert L. Grossman and Richard G. Larson, Differential algebra structures on families of trees, Adv. in Appl. Math. 35 (2005), no. 1, 97–119. MR 2141507, DOI 10.1016/j.aam.2005.01.001
- J.-C. Herz, Pseudo-algèbres de Lie. I, II. C. R. Acad. Sci. Paris 236 (1953) 1935–1937, 2289–2291.
- Mikhail Kapranov, Free Lie algebroids and the space of paths, Selecta Math. (N.S.) 13 (2007), no. 2, 277–319. MR 2361096, DOI 10.1007/s00029-007-0041-9
- V. K. Kharchenko, Automorphisms and derivations of associative rings, Mathematics and its Applications (Soviet Series), vol. 69, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian by L. Yuzina. MR 1174740, DOI 10.1007/978-94-011-3604-4
- Jean-Louis Loday, Generalized bialgebras and triples of operads, Astérisque 320 (2008), x+116 (English, with English and French summaries). MR 2504663
- Jiang-Hua Lu, Hopf algebroids and quantum groupoids, Internat. J. Math. 7 (1996), no. 1, 47–70. MR 1369905, DOI 10.1142/S0129167X96000050
- Kirill C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005. MR 2157566, DOI 10.1017/CBO9781107325883
- G. Maltsiniotis, Groupoïdes quantiques de base non commutative, Comm. Algebra 28 (2000), no. 7, 3441–3501 (French, with English summary). MR 1765327, DOI 10.1080/00927870008827035
- John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
- I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261, DOI 10.1017/CBO9780511615450
- Janez Mrčun, The Hopf algebroids of functions on étale groupoids and their principal Morita equivalence, J. Pure Appl. Algebra 160 (2001), no. 2-3, 249–262. MR 1836002, DOI 10.1016/S0022-4049(00)00071-2
- Janez Mrčun, On duality between étale groupoids and Hopf algebroids, J. Pure Appl. Algebra 210 (2007), no. 1, 267–282. MR 2311185, DOI 10.1016/j.jpaa.2006.09.006
- Warren D. Nichols, The Kostant structure theorems for $K/k$-Hopf algebras, J. Algebra 97 (1985), no. 2, 313–328. MR 812990, DOI 10.1016/0021-8693(85)90052-3
- W. Nichols and B. Weisfeiler, Differential formal groups of J. F. Ritt, Amer. J. Math. 104 (1982), no. 5, 943–1003. MR 675306, DOI 10.2307/2374080
- Victor Nistor, Alan Weinstein, and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152. MR 1687747, DOI 10.2140/pjm.1999.189.117
- Richard S. Palais, The cohomology of Lie rings, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 130–137. MR 0125867
- Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
- George S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195–222. MR 154906, DOI 10.1090/S0002-9947-1963-0154906-3
- Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
- Moss E. Sweedler, Groups of simple algebras, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 79–189. MR 364332
- Floris Takens, Derivations of vector fields, Compositio Math. 26 (1973), 151–158. MR 315723
- Mitsuhiro Takeuchi, Groups of algebras over $A\otimes \overline A$, J. Math. Soc. Japan 29 (1977), no. 3, 459–492. MR 506407, DOI 10.2969/jmsj/02930459
- David Winter, The structure of fields, Graduate Texts in Mathematics, No. 16, Springer-Verlag, New York-Heidelberg, 1974. MR 0389873
- Ping Xu, Quantum groupoids and deformation quantization, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 3, 289–294 (English, with English and French summaries). MR 1648433, DOI 10.1016/S0764-4442(97)82982-5
Additional Information
- I. Moerdijk
- Affiliation: Mathematical Institute, Utrecht University, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands
- Email: I.Moerdijk@uu.nl
- J. Mrčun
- Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
- Email: janez.mrcun@fmf.uni-lj.si
- Received by editor(s): September 28, 2009
- Received by editor(s) in revised form: December 17, 2009
- Published electronically: March 24, 2010
- Additional Notes: The second author was supported in part by the Slovenian Research Agency (ARRS) project J1-2247
- Communicated by: Gail R. Letzter
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 3135-3145
- MSC (2010): Primary 17B35, 16T10, 16T15
- DOI: https://doi.org/10.1090/S0002-9939-10-10347-5
- MathSciNet review: 2653938