28 research outputs found
Tressages des groupes de Poisson formels à dual quasitriangulaire
No full textIn [V. G. Drinfeld, "Quantum groups", Proc. Intern. Congress of Math. (Berkeley, 1986) 1987, pp. 798-820], Drinfeld constructs a Quantum Formal Series Hopf Algebra (QFSHA) U'_h starting from a Quantum Universal Enveloping Algebra (QUEA) U_h . In this paper, we prove that if (U_h,R) is any quasitriangular QUEA, then U'_h with the restriction of Ad(R) to its tensor square is a braided QFSHA. As a consequence, we prove that if g is a quasitriangular Lie bialgebra over a field k of characteristic zero and g^* is its dual Lie bialgebra, then the algebra of functions F[[g^*]] on the formal group associated to g^* is a braided Hopf algebra. This result is a consequence of
the existence of a quasitriangular quantization (U_h,R) of U(g) and of the fact that U'_h is a quantization of F[[g^*]]
An ℏ-adic valuation property of universal R-matrices
No full textAbstractWe prove that if Uℏ(g) is a quasitriangular QUE algebra with universal R-matrix R, and Oℏ(G∗) is the quantized function algebra sitting inside Uℏ(g), then ℏlog(R) belongs to the tensor square Oℏ(G∗)⊗̄Oℏ(G∗). This gives another proof of the results of Gavarini and Halbout, saying that R normalizes Oℏ(G∗)⊗̄Oℏ(G∗) and therefore induces a braiding of the formal group G∗ (in the sense of Weinstein and Xu, or Reshetikhin)
Braiding structures on formal Poisson groups and classical solutions of the QYBE
No full textIf g is a quasitriangular Lie bialgebra, the formal Poisson group F[[g^*]] can be given a braiding structure. This was achieved by Weinstein and Xu using purely geometrical means, and independently by the authors by means of quantum groups. In this paper we compare these two approaches. First, we show that the braidings they produce share several similar properties (in particular, the construction is functorial); secondly, in the simplest case (G = SL_2) they do coincide. The question then rises of whether they are always the same this is positively answered in a separate paper
Construction par dualité des algèbres de Kac–Moody symétrisables
No full textRésuméWe know that there is a one to one correspondence between Kac–Moody algebras and generalized Cartan matrices. In Kac (“Infinite-Dimensional Lie algebras,” 3rd ed., Cambridge Univ. Press, Cambridge, UK, 1990), one can find a way to reconstruct such an algebra as a Lie algebra presented by generators and relations. The aim of the present work is to give another way to reconstruct those algebras when the Cartan matrix is symmetrisable. Our method will use a semi-classical version of techniques of quantum groups
Uniqueness of braidings of quasitriangular Lie bialgebras and lifts of classical r-matrices
No full textIt is known that any quantization of a quasitriangular Lie bialgebra g gives rise to a braiding on the dual Poisson-Lie formal group G*. We show that this braiding always coincides with the Weinstein-Xu braiding. We show that this braiding is the “time one automorphism” of a Hamiltonian vector field, corresponding to a certain formal function on G* × G*, the “lift of r”, which can be expressed in terms of r by universal formulas. The lift of r coincides with the classical limit of the rescaled logarithm of any R-matrix quantizing it
Quantization of coisotropic Lie subalgebras
No full textL’objet de cette thèse est l’étude de l’existence d’une quantification pour les sous-algèbres de Lie coisotropes des bigèbres de Lie. Une sous-algèbre de Lie coisotrope d’une bigèbre de Lie est une sous-algèbre de Lie qui est aussi un coidéal. Le problème de quantifications d’une sous-algèbre de Lie coisotrope fut posé par V. Drinfeld, lors de son étude de la quantification des espaces de Poisson homogènes G/C. Ces deux problèmes sont liés par le principe de dualité établi par N. Ciccoli et F. Gavarini. Dans cette thèse, nous cherchons à résoudre ce problème de quantification dans différents cadres. Premièrement, nous montrons qu’une quantification existe dans le cadre des bigèbres de Lie simple. Nous trouvons une quantification aux sous-algèbres de Lie coisotropes construites par M. Zambon. Puis nous établissons un lien entre ces quantifications et une classification des sous- algèbres coidéales à droite établie par I. Heckenberger et S. Kolb. Deuxièmement, nous trouvons une obstruction à la quantification universelle en utilisant une quantification d’ordre trois construite par V. Drinfeld. Nous montrons que cette obstruction disparait dans les exemples étudiés précédemment. Finalement, nous généralisons un résultat établi par P. Etingof et D. Kazhdan sur la quantification d’espaces de Poisson homogènes, liés aux sous-algèbres Lagrangiennes du double de Drinfeld.The aim of this thesis is the study of quantization of coisotropic Lie subalgebras of Lie bialgebras.A coisotropic Lie subalgebra of a Lie bialgebra is a Lie subalgebra which is also a Lie coideal. The problem of quantization of coisotropic Lie subalgebra was set forth by V. Drinfeld, in his study of quantization of Poisson homogeneous spaces G/C. These problems are closely related to the duality principle established by N. Ciccoli and F. Gavarini.In this thesis, we search for an answer to this quantization problem in different settings. Firstly, we show that a quantization exists for simple Lie bialgebras by constructing a quantization of examples provided by M. Zambon. We then establish a link between the quantization which we constructed and a classification of subalgebras right coideals established by I. Heckenberger and S. Kolb. Secondly, we find an obstruction to the quantization in the universal setting by using a third-order quantization constructed by V. Drinfeld. We show that this obstruction vanishes in the examples studied earlier. Finally, we generalize a result of P. Etingof and D. Kazhdan on the quantization of poisson homogeneous spaces, linked to Lagrangian Lie subalgebras of Drinfeld's double
Deformation quantization proceedings of the meeting of theoretical physicists and mathematicians, Strasbourg, May 31-June 2, 2001 = rencontre entre physiciens théoriciens et mathématiciens, Strasbourg, 31 mai-2 juin 2001
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