Deformed Kac-Moody and Virasoro algebras

Whenever the group R-n acts on an algebra A, there is a method to twist A to a new algebra A. which depends on an antisymmetric matrix. (theta(mu nu) = -theta(nu mu) = constant). The Groenewold - Moyal plane A(theta) ( 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 Q(mu). For A(theta) ( Rd+1), Q(mu) 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 KMalgebra. 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.