Abstract
We consider a complex simple Lie algebra \({\mathfrak{g}}\), with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct \({\mathfrak{g}}\)-invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra \({{\rm U}(\mathfrak{g})}\). This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abellanas L., Martinez-Alonso L.: Quantization from the algebraic viewpoint. J. Math. Phys. 17(8), 1363–1365 (1976)
Agarwal G.S., Wolf E.: Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators. Phys. Rev. D2, 2161–2186 (1970)
Barr M.: Cohomology of Commutative Algebras. Dissertation, U. Penn. (1962)
Barr M.: Harrison homology, Hochschild homology and triples. J. Algebra 8, 314–323 (1968)
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111, 61–110, 111–151 (1978)
Bayen F., Fronsdal C.: Quantization on the sphere. J. Math. Phys. 22, 1345–1349 (1981)
Beilinson A., Ginsburg V., Schechtman V.: Koszul duality. J. Geom. Phys. 5, 317–350 (1988)
Berezin F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)
Bezrukavnikov, R.: Koszul property and Frobenius splitting of Schubert varieties. arXiv:alg-geom/9502021v1
Binegar B., Zierau R.: Unitarization of a singular representation of SO(p, q). Commun. Math. Phys. 138, 245–258 (1991)
Borho W., Brylinski J.: Differential operators on homogeneous spaces I: Irreducibility of the associated variety. Inv. Math. 69, 437–476 (1982)
Bourbaki: Groupes et algèbres de Lie. Masson, Paris (1981)
Braverman A., Joseph A.: The minimal realization from deformation Theory. J. Algebra 205, 13–16 (1998)
Brylinski, R.: Geometric quantization of real minimal nilpotent orbits, symplectic geometry. Diff. Geom. Appl. 9(1–2), 5–58 (1998). arXiv:math/9811033v1[math.SG]
Cahen M., Gutt S., Rawnsley J.: On tangential star products for the coadjoint Poisson structure. Commun. Math. Phys. 180, 99–108 (1996)
Cattaneo, A., Keller, B., Torossian, C., Bruguières, A.: Déformation, quantification, théorie de Lie. Panoramas et Synthèses, vol. 20. Société Mathématique de France, Paris (2005)
Collingwood D.H., McGovern W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold, New York (1993)
Dixmier J.: Algèbres Enveloppantes. Gauthier-Villars Editeur, Paris (1974)
Duflo M.: Sur la classification des idéaux primitifs dans l’algèbre enveloppante d’une algèbre de Lie semi-simple. Ann. Math. 105, 107–120 (1977)
De Wilde M., Lecomte P.B.A.: Existence of star-products and of formal deformations of the Poisson algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7, 487–496 (1983)
Fedosov B.V.: A simple geometrical construction of deformation quantization. J. Diff. Geom. 40, 213–238 (1994)
Fioresi R., Lledo M.A.: On the deformation quantization of coadjoint orbits of semisimple Lie groups. Pac. J. Math. 198(2), 411–436 (2001)
Fioresi R., Lledo M.A.: A comparison between star products on regular orbits of compact Lie groups. J. Phys. A 35, 5687–5700 (2002) arXiv:math/0106129v3 [math.QA]
Fioresi R., Levrero A., Lledo M.A.: Algebraic and differential star products on regular orbits of compact Lie groups. Pac. J. Math. 206, 321–337 (2002) arXiv:math/ 0011172v2 [math.QA]
Fioresi R., Lledo M.A., Varadarajan V.S.: On the deformation quantization of affine algebraic varieties. Int. J. Math. 16, 419–436 (2005) arXiv:math/0406196v1 [math.QA]
Flato M., Lichnerowicz A., Sternheimer D.: Déformations 1-différentiables d’algèbres de Lie attachées à une variété symplectique ou de contact. C. R. Acad. Sci. Paris Ser. A 279, 877–881 (1974)
Flato M., Lichnerowicz A., Sternheimer D.: Deformations of Poisson brackets, Dirac brackets and applications. J. Math. Phys. 17, 1754–1762 (1976)
Fleury P.J.: Splittings of Hochschild’s complex for commutative algebras. Proc. AMS 30, 323–405 (1971)
Fronsdal C.: Some ideas about quantization. Rep. Math. Phys. 15, 111–145 (1978)
Frønsdal, C.: Harrison cohomology and Abelian deformation quantization on algebraic varieties. Deformation quantization (Strasbourg, 2001). In: IRMA Lect. Math. Theor. Phys., vol. 1, pp. 149–161. de Gruyter, Berlin (2002). arXiv:hep-th/0109001v3
Fronsdal, C.: Abelian deformations. In: Proceedings of the IX’th International Conference on Symmetry Methods in Physics, Yerevan, July 2001
Fronsdal C., Galindo A.: The ideals of free differential algebras. J. Algebra 222, 708–746 (1999) arXiv:math/9806069v2 [math.QA]
Frønsdal C., Kontsevich M.: Quantization on curves. Lett. Math. Phys. 79, 109–129 (2007) arXiv:math-ph/0507021v2
Fulton W., Harris J.: Representation Theory. Springer, New York (1991)
Gan Wee Teck., Savin G.: Uniqueness of the Joseph ideal. Math. Res. Lett. 11, 589–597 (2004)
Garsia, A.M.: Combinatorics of the free Lie algebra and the symmetric group. In: Analysis, et cetera, Research Papers Published in Honor of Jürgen Moser’s 60’th Birthday, pp. 209–362. Academic Press, New York (1990)
Gerstenhaber M.: The cohomology structure of an associative ring. Ann. Math. (2) 78, 267–288 (1963)
Gerstenhaber M.: On the deformations of rings and algebras. Ann. Math. 79, 59–103 (1964)
Gerstenhaber M.: Developments from Barr’s thesis, presented at the celebration of the 60’th birthday of Michael Barr (30 June 1998). J. Pure Appl. Algebra 143, 205–220 (1999)
Gerstenhaber M., Schack S.D.: A Hodge-type decomposition for commutative algebra cohomology. J. Pure Appl. Algebra 48, 229–247 (1987)
Guieu, L., Roger, C.: Avec un appendice de Vlad Sergiescu. L’Algèbre et le Groupe de Virasoro: aspects géométriques et algébriques, généralisations, Publication du Centre de Recherches Mathématiques de Montréal, Monographies, notes de cours et Actes de conférences, PM28 (2007)
Gukov, S., Witten, E.: Branes and Quantization. arXiv:0809.0305v2[hep-th]
Gutt S.: An explicit star-product on the cotangent bundle of a Lie group. Lett. Math. Phys. 7, 249–258 (1983)
Halbout, G.: Oudom J.-M., Tang X., Deformations of Linear Poisson Orbifolds. arXiv:0807.0027v1[math.QA]
Halbout, G., Tang, X.: Noncommutative Poisson Structures on Orbifolds. arXiv: math/0606436v2[math.QA]
Harrison D.K.: Commutative algebras and cohomology. Trans. Am. Math. Soc. 104, 191–204 (1962)
Hochschild G., Kostant B., Rosenberg A.: Differential forms on regular affine algebras. Trans. Am. Math. Soc. 102, 383–408 (1962)
Humphreys J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1972)
Joseph A.: Minimal realizations and spectrum generating algebras. Commun. math. Phys. 36, 325–338 (1974)
Joseph A.: The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann. Sci. Ecol. Norm. Sup. 9, 1–30 (1976)
Joseph A.: Dixmier’s problem for Verma and principal series submodules. J. Lond. Math. Soc. 20, 193–204 (1979)
Joseph A.: On the associated variety of a primitive ideal. J. Algebra 93, 509–523 (1985)
Kirillov A.A.: Eléments de la Théorie des Représentations. Editions Mir, Moscou (1974)
Kirillov, A.A.: Lectures on the Orbit Method, Graduates Studies in Mathematics, vol. 64, American Mathematical Society, Providence, Rhode Island (2004)
Kontsevich M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35–72 (1999) arXiv:math/9904055v1 [math.QA]
Kontsevich M.: Deformation Quantization of algebraic varieties. In: Euro Conference Moshé Flato 2000, Part III (Dijon). Lett. Math. Phys. 56, 271–294 (2001). arXiv:math/0106006v1 [math.AG]
Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003) arXiv:q-alg/9709040v1
Kostant B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)
Kostant B.: Quantization and unitary representations. Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, vol. 170, pp. 87–208. Springer, Berlin (1970)
Lledo M.A.: Deformation quantization of non regular orbits of compact Lie groups. Lett. Math. Phys. 58, 57–67 (2001) arXiv:math/0105191v3 [math.QA]
Loday J.-L.: Opérations sur l’homologie cyclique des algèbres commutatives. Invent. math. 96, 205–230 (1989)
Moyal J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Phil. Soc. 45, 99–124 (1979)
Macfarlane J.A., Pfeiffer H.: Development of a unified tensor calculus for exceptional lie algebras. Int. J. Mod. Phys. A 19, 287–316 (2004) arXiv:math-ph/0212047v1
Marsden J.E., Ratiu T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, Heidelberg (1999)
Postnikov M.: Leçons de Géométrie, Groupes et algèbres de Lie. Editions Mir, Moscou (1982)
Souriau J.M.: Structures des Systèmes Dynamiques. Dunod, Paris (1970)
Tamarkin, D.E.: Another proof of M. Kontsevich formality theorem for \({\mathbb{R}^n}\). math.QA/ 9803025v4
Vey J.: Déformation du crochet de Poisson sur une variété symplectique. Comment. Math. Helv. 50, 421–454 (1975)
Weinstein, A.: Deformation quantization, Séminaire Bourbaki. Astérisque, vol.1993/ 1994, No. 227, Exp. No. 789, 5, 389–409 (1995)
Weyl H.: Theory of Groups and Quantum Mechanics. Dover, New York (1931)
Wigner E.P.: Quantum corrections for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Edited with Claude Roger and Frédéric Butin
Institut Camille Jordan, Université Lyon 1, CNRS UMR5208, 43 blvd du 11 novembre 1918, 69622 Villeurbanne-Cedex, France. e-mail: roger@math.univ-lyon1.fr; butin@math.univ-lyon1.fr
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Frønsdal, C. Deformation Quantization on the Closure of Minimal Coadjoint Orbits. Lett Math Phys 88, 271–320 (2009). https://doi.org/10.1007/s11005-009-0316-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-009-0316-5