Sugawara and Vertex Operator Constructions for Deformed Virasoro Algebras

From the defining exchange relations of the $$\mathcal{A}_{{q,p}} {\left( {\widehat{{gl}}_{N} } \right)}$$ elliptic quantum algebra, we construct subalgebras which can be characterized as q-deformed W_N algebras. The consistency conditions relating the parameters p, q, N and the central charge c are shown to be related to the singularity structure of the functional coefficients defining the exchange relations of specific vertex operators representations of $$\mathcal{A}_{{q,p}} {\left( {\widehat{{gl}}_{N} } \right)}$$ available when N = 2. Communicated by Petr Kulish