Quantization of coboundary Lie bialgebras

Enriquez, Benjamin; Halbout, Gilles

doi:10.4007/annals.2010.171.1267

Cited by 14 publications

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References 14 publications

(23 reference statements)

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“…Therefore it defines the same co-boundary structure on the category of U h (g)-modules as the quantization of the co-boundary element obtained by Enriquez and Halbout [7]. Theorem 6.11 is proved.…”

Section: Quantized Symmetric Algebras and Co-boundary Categoriesmentioning

confidence: 75%

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Equivariant quantizations of symmetric algebras

Zwicknagl

2009

Journal of Algebra

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Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra U h (g), resp. U q (g). We investigate, in particular, such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of g and V . We classify these structure in the important special case, when g is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce a more general notion, coPoisson module algebras and their quantizations, to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language.

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Section: Quantized Symmetric Algebras and Co-boundary Categoriesmentioning

confidence: 75%

“…Lemma 4.26. Let g = so (7) and V = V ω 3 . Then g V does not admit a semidirect Lie bialgebra structure corresponding to the standard Lie bialgebra structure on so (7).…”

Section: Lemma 422mentioning

confidence: 99%

Equivariant quantizations of symmetric algebras

Zwicknagl

2009

Journal of Algebra

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“…Etingof and Kazhdan gave a positive answer to this question in [8,9], where the Lie bialgebras that they considered including finite-and infinite-dimensional ones are the Lie algebras defined by generalized Cartan matrices. Enriquez and Halbout showed that any coboundary Lie bialgebra, in principle, can be quantized via a certain EtingofKazhdan quantization functor [7], and Geer extended Etingof and Kazhdan's work from Lie bialgebra to the setting of Lie superbialgebras [11]. After the work [8,9], it is natural to consider the quantizations of Cartan type Lie algebras which are defined by differential operators.…”

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confidence: 98%

Twists and quantizations of Cartan type S Lie algebras

Wang

2011

Journal of Pure and Applied Algebra

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a b s t r a c tWe construct explicit Drinfel'd twists for the generalized Cartan type S Lie algebras and obtain the corresponding quantizations. By modular reduction and base changes, we obtain certain quantizations of the restricted universal enveloping algebra u(S(n; 1)) in characteristic p. They are new Hopf algebras of truncated p-polynomial noncommutative and noncocommutative deformation of dimension p 1+(n−1)(p n −1) , which contain the well-known Radford algebra (Radford (1977) [23]) as a Hopf subalgebra. As a by-product, we also get some Jordanian quantizations for sl n .In [15], the authors studied quantizations both for the generalized Witt algebra W in characteristic 0 and for the Jacobson-Witt algebra W(n; 1) in characteristic p. In the present paper, we continue to treat the same questions both for the generalized Cartan type S Lie algebras in characteristic 0 (for the definition, see [4]) and for the restricted simple special algebras S(n; 1) in the modular case (for the definition, see [26,27]). Our techniques are somewhat different from those in [15] and the method of constructing the twist can be generalized to other Cartan type Lie algebras.We survey some previous related work. In [6], Drinfel'd raised the question of the existence of a universal quantization for Lie bialgebras. Etingof and Kazhdan gave a positive answer to this question in [8,9], where the Lie bialgebras that they considered including finite-and infinite-dimensional ones are the Lie algebras defined by generalized Cartan matrices. Enriquez and Halbout showed that any coboundary Lie bialgebra, in principle, can be quantized via a certain EtingofKazhdan quantization functor [7], and Geer extended Etingof and Kazhdan's work from Lie bialgebra to the setting of Lie superbialgebras [11]. After the work [8,9], it is natural to consider the quantizations of Cartan type Lie algebras which are defined by differential operators. In 2004, Grunspan [14] obtained the quantization of the (infinite-dimensional) Witt algebra W in characteristic 0 using the twist found by Giaquinto and Zhang [12], but his approach didn't work for the quantum version of its simple modular Witt algebra W(1; 1) in characteristic p. The authors in [15] obtained the quantizations of the generalized Witt algebra W in characteristic 0 and the Jacobson-Witt algebra W(n; 1) in characteristic p; they are new families of noncommutative and noncocommutative Hopf algebras of dimension p 1+np n in characteristic p, while in the rank 1 case, the work recovered Grunspan's work in characteristic 0 and gave the required quantum version in characteristic p.Although, in principle, the possibility of quantizing an arbitrary Lie bialgebra has been proved, an explicit formulation of Hopf operations remains nontrivial. In particular, only a few kinds of twists were known with an explicit expression; see [12,19,22,24] and the references therein. In this research, we start with an explicit Drinfel'd twist due to [12,14], which, we found recently, is essentially a variation of the Jorda...

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“…In [9] and [10] Etingof and Kazhdan gave a positive answer to this question for Lie bialgebras coming from finite-and infinite-dimensional Lie algebras defined by generalized Cartan matrices. Later, Enriquez-Halbout [8] showed that, in principle, any coboundary Lie bialgebra can be quantized via a certain EtingofKazhdan quantization functor, and Geer [12] extended the work of Etingof and Kazhdan from Lie bialgebras to the setting of Lie superbialgebras. In view of this, it is natural to consider the quantizations of Lie algebras of Cartan type which are defined by differential operators.…”

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confidence: 99%

“…Although, in principle, the possibility to quantize an arbitrary Lie bialgebra has been proved ( [9], [10], [11], [8], and [12]), it seems difficult to obtain explicit formulas for the Hopf algebra operations. In particular, only a few kinds of twists with explicit expressions for the Hopf algebra operations are known (see [24], [21], [13], [19], [1], and the references therein).…”

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confidence: 99%

Modular Quantizations of Lie Algebras of Cartan Type 𝐻 via Drinfel’d Twists

Tong¹,

Hu²,

Wang³

2015

Contemporary Mathematics

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Abstract. We construct explicit Drinfel'd twists for the Lie algebras of generalized Cartan type H in characteristic 0 and also obtain the corresponding quantizations and their integral forms. By using modular reduction and base changes, we derive certain quantizations of the restricted universal enveloping algebra u(H(2n; 1)) of the restricted Hamiltonian algebra H(2n; 1) in prime characteristic p. These quantizations are new non-pointed Hopf algebras of prime-power dimension p p 2n −1 and contain the well-known Radford algebras as Hopf subalgebras. As a by-product we also obtain some Jordanian quanti-This paper is a continuation of [16] and [17] in which modular quantizations of Lie algebras of Cartan types W and S were studied. In the present paper we consider the same questions, both for the Lie algebras of generalized Cartan type H in characteristic 0 (for the definition, see [22]) and for the restricted Hamiltonian algebras H(2n; 1) in the modular case (for the definition, see [25] and [26]).For the convenience of the reader we review some previous related work. In

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Quantization of coboundary Lie bialgebras

Cited by 14 publications

References 14 publications

Equivariant quantizations of symmetric algebras

Equivariant quantizations of symmetric algebras

Twists and quantizations of Cartan type S Lie algebras

Modular Quantizations of Lie Algebras of Cartan Type 𝐻 via Drinfel’d Twists

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