Note on the Algebra of Screening Currents for the Quantum Deformed W -Algebra

With slight modifications in the zero modes contributions, the positive and negative screening currents for the quantum deformed W -algebra Wq,p(g) can be put together to form a single algebra which can be regarded as an elliptic deformation of the universal enveloping algebra of ĝ, where g is any classical simply-laced Lie algebra. Recently, various deformations of the classical and quantum Virasoro and W -algebras have received considerable interests. N.Reshetikhin and E.Frenkel [4] first introduced the Poisson algebras Wq(g), which are q-deformation of classical W -algebras. Later on, J.Shiraishi, H.Kubo, H.Awata and S.Odake [11] obtained a quantum version of the algebra Wq(sl2), which is a noncommutative algebra depending on two parameters p and q. B.Feigin and E.Frenkel [2] extended this result to general case, i.e. quantum deformedW -algebras Wq,p(g), where g is any classical semisimple Lie algebra. All these algebras were obtained together with their respective bosonic Fock space representations. Similar considerations with respect to the Yangian deformation were also carried out and have led to h̄ deformed Virasoro algebra [1] and quantum (ξ, h̄)-deformed W -algebras [6]. In their work [2], B.Feigin and E.Frenkel also obtained the screening currents for the algebra Wq,p(g) and found the elliptic relations between them. They noticed that the relations for the positive (resp. negative) screening currents form an elliptic deformation for the loop algebra n̂+ (resp. n̂−) of the nilpotent subalgebra n+ (resp. n−) of g. However, they did not consider whether these two nilpotent elliptic algebras can be put together to form a unified elliptic algebra. In this note we shall show that it is possible to combine the above nilpotent algebras into a single unified elliptic algebra if we introduce some new generating currents denoted by H i (z) and slightly modify the zero mode contributions in the bosonic representation of the screening currents (the modified “screening currents” are denoted by Ei(z) and Fi(z) respectively whilst the original ones by S i (z)). Unlike the unmodified screening currents, the modified currents Ei(z), Fi(z) and the newly introduced currents H ± i (z) do not commute with the W -algebra generating currents even up to total differences. Let us start from a brief description of the results of B.Feigin and E.Frenkel [2] which are necessary for our discussion. First we consider the simple case of g = slN . By definition, the algebra Wq,p(slN ) is generated by the Fourier coefficients of the currents T1(z), · · · , TN−1(z), which, in the free field realization, obey the quantum deformed Miura transformation Strictly speaking, the modified currents are no longer screening currents of the deformed W -algebra and hence we here use the quotation marks.