UNIVERSAL ENVELOPING ALGEBRAS OF PBW TYPE*

ARDIZZONI, ALESSANDRO

Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 9-26

Résumé

We continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (PBW) type, meaning that a suitable PBW-type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: as an application, we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.

Détail

DOI : 10.1017/S0017089511000310

Mots-clés : Primary 16W30, Secondary 16S30

Bibliographie

[1] 1.Ardizzoni, A., Separable functors and formal smoothness, J. K-theory 1 (3) (2008), 535–582. Google Scholar | DOI

[2] 2.Ardizzoni, A., A Milnor–Moore type theorem for primitively generated braided bialgebras, J. Algebra 327 (1) (2011), 337–365. Google Scholar | DOI

[3] 3.Ardizzoni, A., On primitively generated braided Bialgebras, Algebr. Represent. Theory, to appear (doi:10.1007/s10468-010-9257-z). Google Scholar | DOI

[4] 4.Ardizzoni, A., On the combinatorial rank of a graded braided bialgebra, J. Pure Appl. Algebra 215 (2011), 2043–2054. Google Scholar | DOI

[5] 5.Ardizzoni, A., Menini, C. and Ştefan, D., Braided bialgebras of Hecke-type, J. Algebra 321 (3) (2009), 847–865. Google Scholar | DOI

[6] 6.Ardizzoni, A., Menini, C. and Ştefan, D., PBW deformations of braided symmetric algebras and a Milnor–Moore type theorem for braided bialgebras, preprint (arXiv:math.QA/0604181v2). Google Scholar

[7] 7.Ardizzoni, A. and Stumbo, F., Quadratic Lie algebras, Commun. Algebra, to appear. Google Scholar

[8] 8.Baez, J. C., Hochschild homology in a braided tensor category, Trans. Amer. Math. Soc. 334 (2) (1994), 885–906. Google Scholar | DOI

[9] 9.Berger, R., The quantum Poincaré–Birkhoff–Witt theorem. Commun. Math. Phys. 143 (2) (1992), 215–234. Google Scholar | DOI

[10] 10.Braverman, S. and Gaitsgory, D., Poincaré–Birkhoff–Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (2) (1996), 315–328. Google Scholar | DOI

[11] 11.Gomez, X. and Majid, S., Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras, J. Algebra 261 (2) (2003), 334–388. Google Scholar | DOI

[12] 12.Gurevich, D. I., Generalized translation operators on Lie groups. (Russian. English, Armenian summary) Izv. Akad. Nauk Armyan. SSR Ser. Mat. 18 (4) (1983), 305–317. Google Scholar

[13] 13.Gurevich, D., Hecke symmetries and braided Lie algebras. in Spinors, twistors, Clifford algebras and quantum deformations (Sobótka Castle, 1992), 317–326, Fund. Theories Phys., Vol. 52 (Kluwer Academic Publishers, Dordrecht, 1993). Google Scholar

[14] 14.Helbig, M., A PBW basis criterion for pointed Hopf algebras, preprint (arXiv:1011.0140v1). Google Scholar

[15] 15.Humphreys, J. E., Introduction to Lie algebras and representation theory, in Graduate texts in mathematics, Vol. 9 (Springer-Verlag, New York–Berlin, 1972). Google Scholar

[16] 16.Jacobson, N., Lie algebras, in Interscience tracts in pure and applied mathematics, No. 10 (Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962). Google Scholar

[17] 17.Kharchenko, V. K., An algebra of skew primitive elements, Algebra Logic 37 (2) (1998), 101–126. Google Scholar | DOI

[18] 18.Kharchenko, V. K., A quantum analogue of the Poincaré–Birkhoff–Witt theorem. Algebra and Logic 38 (4) (1999), 259–276. Google Scholar | DOI

[19] 19.Kharchenko, V. K., Skew primitive elements in Hopf algebras and related identities, J. Algebra 238 (2) (2001), 534–559. Google Scholar | DOI

[20] 20.Kharchenko, V. K., Connected braided Hopf algebras, J. Algebra 307 (1) (2007), 24–48. Google Scholar | DOI

[21] 21.Lusztig, G., Canonical Bases Arising From Quantized Enveloping Algebras, J. Amer. Math. Soc. 3 (2) (1990), 447–498. Google Scholar | DOI

[22] 22.Majid, S., Quantum and braided-Lie algebras, J. Geom. Phys. 13 (4) (1994), 307–356. Google Scholar | DOI

[23] 23.Milnor, J. W. and Moore, J. C., On the structure of Hopf algebras, Ann. Math. 81 (2) (1965), 211–264. Google Scholar | DOI

[24] 24.Montgomery, S., Hopf Algebras and their actions on rings, CMBS Region. Conf. Ser. Math. 82 (1993). Google Scholar

[25] 25.Pareigis, B., On Lie algebras in the category of Yetter–Drinfeld modules, Appl. Categ. Struct. 6 (2) (1998), 151–175. Google Scholar | DOI

[26] 26.Ringel, C. M., PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51–88. Google Scholar

[27] 27.Rosso, M., An analogue of P.B.W. theorem and the universal RR-matrix for U sl(N+1), Commun. Math. Phys. 124 (2) (1989), 307–318. Google Scholar | DOI

[28] 28.Sweedler, M., Hopf algebras (Benjamin, New York, 1969). Google Scholar

[29] 29.Takeuchi, M., Survey of braided Hopf algebras, in New trends in Hopf algebra theory (La Falda, 1999), pp. 301–323; Contemp. Math., Vol. 267 (Amer. Math. Soc., Providence, RI, 2000). Google Scholar | DOI

[30] 30.Ufer, S., PBW bases for a class of braided Hopf algebras, J. Algebra 280 (1) (2004), 84–119. Google Scholar | DOI

[31] 31.Wambst, M., Complexes de Koszul quantiques, Ann. Inst. Fourier 43 (4) (1993), 1089–1156. Google Scholar | DOI

[32] 32.Woronowicz, S. L., Differential calculus on compact matrix pseudogroups (quantum groups), Commun. Math. Phys. 122 (1) (1989), 125–170. Google Scholar | DOI

[33] 33.Yamane, H., A Poincaré–Birkhoff–Witt theorem for the quantum group of type A . Proc. Japan Acad. Ser. A Math. Sci. 64 (10) (1988) 385–386. Google Scholar | DOI