Deformed Kac-Moody and Virasoro Algebras

Abstract

Whenever the group ${\mathbb{R}}^n$ acts on an algebra $\calA$, there is a method to twist $\mathcal{A}$ to a new algebra $\calA_θ$ which depends on an antisymmetric matrix $\theta$ (${\theta }^{\mu \nu }=-{\theta }^{\nu \mu }=\mathrm{constant}$). The Groenewold-Moyal plane $\calA_θ(\R^{d+1})$ is an example of such a twisted algebra. We give a general construction to realise this twist in terms of $\calA$ itself and certain ``charge'' operators $Q_{\mu }$. For $\calA_θ(\R^{d+1})$, $Q_{\mu }$ are translation generators. This construction is then applied to twist the oscillators realising the Kac-Moody (KM) algebra as well as the KM currents. They give different deformations of the KM algebra. From one of the deformations of the KM algebra, we construct, via the Sugawara construction, the Virasoro algebra. These deformations have implication for statistics as well.

arXiv commentFirst resubmission: Some important changes performed. Title and Abstract are changed. New references added. LateX, no figures, 18 pages. Second resubmission: small changes

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Cited by in corpus (6)

• Quantum Fields on the Groenewold-Moyal Plane
• Quantum Fields with Noncommutative Target Spaces
• Quantum Fields on the Groenewold-Moyal Plane: C, P, T and CPT
• Realization of the Noncommutative Seiberg-Witten Gauge Theory by Fields in Phase Space
• Deformed Kac-Moody Algebra and Noncommutative Fermi Theory in Two-Dimensions
• Review of Twisted Poincare Symmetry