Quantizations of generalized-Witt algebra and of Jacobson–Witt algebra in the modular case

Abstract

We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song-Su [G. Song, Y. Su, Lie bialgebras of generalized-Witt type, arXiv: math.QA/0504168, Sci. China Ser. A 49 (4) (2006) 533-544]. Via a modulo p reduction and a modulo "p-restrictedness" reduction process, we get 2 n − 1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson-Witt algebra W(n; 1 ) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p 1+np n in characteristic p. Our results generalize a work of Grunspan [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145-161] in rank n = 1 case in characteristic 0. In the modular case, the argument for a refined version follows from the modular reduction approach (different from [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145-161]) with some techniques from the modular Lie algebra theory.

Three Études in QFT

“…Indeed, one immediately checks that r fulfills the YangBaxter equation. As direct consequence, we have Drinfel'd twists and quantized universal enveloping algebras [3] available for future study.…”

“…To this end, let us consider generalized Witt algebras [3]. We take Q as the underlying field (any field of characteristic zero would be possible) and let |R| be the number of amplitudes needing renormalization.…”

Quantization of the q-analog Virasoro-like algebras

“…Using the method twisting the coproduct by a Drinfel'd twist element but keeping the product unchanged, Grunspan [3] presented the quantization of a class of infinite dimensional Lie algebras containing Virasoro algebras studied in [4] (see also [5,6]). Using the same technique, Hu and Wang [7] quantized some Lie algebras presented in [8]. In a recent paper [9], the Lie bialgebra structures of q-analog Virasoro-like algebras L with the basis {L α , d 1 , d 2 | α ∈ Z 2 \{(0, 0)}} and brackets…”

“…In this paper, we will use the techniques developed in [3,7] to construct the quantization of this type of bialgebra. However, since in our case the Lie algebra is non-linear, some of our arguments may render rather technical.…”

Notes on 2-parameter quantum groups I

“…Quantum groups U q (g), depending on a single parameter q, are certain families of Hopf algebras that are deformations of universal enveloping algebras of symmetrizable Kac-Moody algebras. In the early 90s of the last century, much work had been done on their multiparameter generalizations, which can be obtained by twisting the algebra structure via a 2-cocycle on an indexed free abelian group (see [1]) or by twisting the coalgebra structure in the spirit of Drinfeld (see [2], [3]). Note that a 2-cocycle (or a Drinfeld twist) deformation is an important method to yield new (twisted) bialgebras from old ones.…”