×

On the universal enveloping algebra of a Lie algebroid. (English) Zbl 1241.17014

Let \(k\) be a field and \(R\) a unital commutative \(k\)-algebra. The main result of this note asserts that there is an equivalence between the full subcategory of \((k, R)\)-Lie algebras which are projective as left \(R\)-modules and that of cocomplete graded projective \(R/k\)-bialgebras. The result allows the authors to extend the Cartier-Milnor-Moore theorem from Lie algebras to Lie algebroids, and generalizes a similar result of Nichols proved in the weaker case when \(R/k\) is a field extension. Their proof is different from that of Nichols and is based on some properties of cocommutative non-counital coalgebras over a ring, which are proved in the appendix of the paper.

MSC:

17B35 Universal enveloping (super)algebras
16T10 Bialgebras
16T15 Coalgebras and comodules; corings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pierre Cartier, A primer of Hopf algebras, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 537 – 615. · Zbl 1184.16031 · doi:10.1007/978-3-540-30308-4_12
[2] Marius Crainic and Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575 – 620. · Zbl 1037.22003 · doi:10.4007/annals.2003.157.575
[3] Robert L. Grossman and Richard G. Larson, Differential algebra structures on families of trees, Adv. in Appl. Math. 35 (2005), no. 1, 97 – 119. · Zbl 1092.16023 · doi:10.1016/j.aam.2005.01.001
[4] J.-C. Herz, Pseudo-algèbres de Lie. I, II. C. R. Acad. Sci. Paris 236 (1953) 1935-1937, 2289-2291.
[5] Mikhail Kapranov, Free Lie algebroids and the space of paths, Selecta Math. (N.S.) 13 (2007), no. 2, 277 – 319. · Zbl 1149.14003 · doi:10.1007/s00029-007-0041-9
[6] 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. · Zbl 0746.16002
[7] Jean-Louis Loday, Generalized bialgebras and triples of operads, Astérisque 320 (2008), x+116 (English, with English and French summaries). · Zbl 1178.18001
[8] Jiang-Hua Lu, Hopf algebroids and quantum groupoids, Internat. J. Math. 7 (1996), no. 1, 47 – 70. · Zbl 0884.17010 · doi:10.1142/S0129167X96000050
[9] 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. · Zbl 1078.58011
[10] G. Maltsiniotis, Groupoïdes quantiques de base non commutative, Comm. Algebra 28 (2000), no. 7, 3441 – 3501 (French, with English summary). · Zbl 0956.18002 · doi:10.1080/00927870008827035
[11] John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211 – 264. · Zbl 0163.28202 · doi:10.2307/1970615
[12] I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. · Zbl 1029.58012
[13] 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. · Zbl 0986.16017 · doi:10.1016/S0022-4049(00)00071-2
[14] Janez Mrčun, On duality between étale groupoids and Hopf algebroids, J. Pure Appl. Algebra 210 (2007), no. 1, 267 – 282. · Zbl 1115.22003 · doi:10.1016/j.jpaa.2006.09.006
[15] Warren D. Nichols, The Kostant structure theorems for \?/\?-Hopf algebras, J. Algebra 97 (1985), no. 2, 313 – 328. · Zbl 0579.16005 · doi:10.1016/0021-8693(85)90052-3
[16] W. Nichols and B. Weisfeiler, Differential formal groups of J. F. Ritt, Amer. J. Math. 104 (1982), no. 5, 943 – 1003. · Zbl 0552.14010 · doi:10.2307/2374080
[17] Victor Nistor, Alan Weinstein, and Ping Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117 – 152. · Zbl 0940.58014 · doi:10.2140/pjm.1999.189.117
[18] Richard S. Palais, The cohomology of Lie rings, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 130 – 137.
[19] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205 – 295. · Zbl 0191.53702 · doi:10.2307/1970725
[20] George S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195 – 222. · Zbl 0113.26204
[21] Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. · Zbl 0194.32901
[22] Moss E. Sweedler, Groups of simple algebras, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 79 – 189. · Zbl 0314.16008
[23] Floris Takens, Derivations of vector fields, Compositio Math. 26 (1973), 151 – 158. · Zbl 0258.58005
[24] Mitsuhiro Takeuchi, Groups of algebras over \?\otimes \overline\?, J. Math. Soc. Japan 29 (1977), no. 3, 459 – 492. · Zbl 0349.16012 · doi:10.2969/jmsj/02930459
[25] David Winter, The structure of fields, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, No. 16. · Zbl 0292.12101
[26] 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). · Zbl 0911.17012 · doi:10.1016/S0764-4442(97)82982-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.