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$q$-Virasoro Algebra and the Point-Splitting

It is shown that a particular $q$-deformation of the Virasoro algebra can be interpreted in terms of the $q$-local field $Φ(x)$ and the Schwinger-like point-splitted Virasoro currents, quadratic in $Φ(x)$. The $q$-deformed Virasoro algebra possesses an additional index $α$, which is directly related to point-splitting of the currents. The generators in the $q$-deformed case are found to exactly reproduce the results obtained by probing the fields $X(z)$ (string coordinate) and $Φ(z)$ (string momentum) with the non-splitted Virasoro generators and lead to a particular representation of the $SU_q (1,1)$ algebra characterized by the standard conformal dimension $J$ of the field. Some remarks concerning the $q$-vertex operator for the interacting $q$-string theory are made.