Deformed Kac-Moody and Virasoro algebras

Whenever the group Rn acts on an algebra A, there is a method to twist A to a new algebra Aθ which depends on an antisymmetric matrix θ (θμν = -θνμ = constant). The Groenewold-Moyal plane Aθ(Rd+1) is an example of such a twisted algebra. We give a general construction to realize this twist in terms of A itself and certain 'charge' operators Qμ. For Aθ(Rd+1), Qμ are translation generators. This construction is then applied to twist the oscillators realizing 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 an implication for statistics as well