Quantization of Schrödinger-Virasoro Lie algebra

Su, Yucai; Yuan, Lamei

doi:10.1007/s11464-010-0072-y

Cited by 6 publications

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“…The Hopf algebraic structure problem stated below has led, in the past 30 years, to a multiplicity of new results in different fields of mathematics and physics. The theory of multiparameter quantum deformations of Lie algebras (Hu, 1999;Riahi et al, 2021), Lie bialgebras (Song and Su, 2006;Yue and Su, 2008), and quantization of Lie algebras (Chakrabarti and Jagannathan, 1991;Song et al, 2008;Su and Yuan, 2010) play an essential role in the quantum white noise literature.…”

Section: Introductionmentioning

confidence: 99%

Hopf Algebraic Structure of the (p,q)-Square Heizenberg White Noise Algebra

Riahi¹

2022

Journal of Mathematics and Statistics

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Section: Introductionmentioning

confidence: 99%

Hopf Algebraic Structure of the (p,q)-Square Heizenberg White Noise Algebra

Riahi¹

2022

Journal of Mathematics and Statistics

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q-deformation of W (2, 2) Lie algebra associated with quantum groups

Yuan

2012

Acta. Math. Sin.-English Ser.

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An explicit realization of the W (2, 2) Lie algebra is presented using the famous bosonic and fermionic oscillators in physics, which is then used to construct the q-deformation of this Lie algebra. Furthermore, the quantum group structures on the q-deformation of this Lie algebra are completely determined.Since 1990s, there have been intensive explorations of quantized universal enveloping algebras, namely, quantum groups, which were first introduced independently by Drinfeld [7, 8] and Jimbo [16, 17] around 1985 in order for them to construct solutions to the quantum Yang-Baxter equations. Since then quantum groups are found to have numerous applications in various areas ranging from statistical physics via symplectic geometry and knot theory to modular representations of reductive algebraic groups. For this reason, the interests in quantum groups, quantum deformations of Lie algebras as well as Lie bialgebras have been growing in the physical and mathematical literatures, especially those of Cartan type and Block type, which are closely related to the Virasoro algebra and the W -infinity algebra W 1+∞ (e.g., [20, 28, 29, 30, 31, 32, 33, 34, 35]). In particular, the q-deformed Virasoro algebra, q-deformed oscillator and q-deformed Heisenberg algebra have been investigated in a number of papers (see e.g. [1, 3, 4, 14, 26, 25]). Among these kinds of algebras, the q-deformation of the Virasoro algebra has been most intensively considered [1, 10, 13, 19, 23, 24], which can be viewed as a typical example of the physical application of the quantum group. In addition, two-parameter deformation of Lie algebras has also been considered by some authors (see e.g. [2, 5]), while the more general quantum Lie algebras have been investigated in [6, 9, 15, 27]. Roughly speaking, quantum Lie algebras in the context of these deformations are universal enveloping algebras deformed by one or more parameter(s) (q-deformation) and possess structures of Hopf algebras. However, the essential reason for the name "quantum" algebra is that it becomes the conventional Lie algebra in the q → 1 limit (classical limit).The W (2, 2) Lie algebra was introduced by Zhang-Dong in [36] for the study of classification of vertex operator algebras generated by vectors of weight 2. Later the Harish-Chandra modules of this Lie algebra were investigated in [21], while the classification of irreducible weight modules was discussed in [22]. The derivations, central extensions and automorphism groups of this Lie algebra were determined in [12]. Recently, the Verma modules over the W (2, 2) Lie algebra was investigated in [18]. A quantum group structure of the q-deformed W (2, 2) Lie algebra was also given in [11]. Nonetheless, there are still plenty rooms for our reconsideration on this matter, since our definition of the q-deformation W q (Proposition 2.2) of this Lie algebra with its origin from physics, is ratherAbstract. An explicit realization of the W (2, 2) Lie algebra is presented using the famous bosonic and fermionic oscillators in physics, ...

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Quantization of the Lie superalgebra of Witt type

Yue

Zhu

2021

Journal of Geometry and Physics

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Quantization of Schrödinger-Virasoro Lie algebra

Cited by 6 publications

References 20 publications

Hopf Algebraic Structure of the (p,q)-Square Heizenberg White Noise Algebra

Hopf Algebraic Structure of the (p,q)-Square Heizenberg White Noise Algebra

q-deformation of W (2, 2) Lie algebra associated with quantum groups

Quantization of the Lie superalgebra of Witt type

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