A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.q-alg/9507034

We derive a quantum deformation of the W N algebra and its quantum Miura transformation, whose singular vectors realize the Macdonald polynomials.q-alg/9508011

An extra term generally appears in the q-deformed su(2) algebra for the deformation parameter q = exp 2πiθ, if one combines the Biedenharn-Macfarlane construction of q-deformed su(2), which is a generalization of Schwinger's construction of conventional su(2), with the representation of the q-deformed oscillator algebra which is manifestly free of negative norm. This extra term introduced by the requirement of positive norm is analogous to the Schwinger term in current algebra.Implications of this extra term on the Bloch electron problem analyzed by Wiegmann and Zabrodin are briefly discussed.The notion of q-deformed algebra[1], which was originally introduced in connection with the inverse scattering problem and the Yang-Baxter equation [2], is going to be a standard machinery of theoretical physics. For example, the q-deformed su(2) for q = exp iπP/Q with mutually prime integers P and Q found a very interesting physical application to the Bloch electrons in two-dimensional lattice model [3][4][5]. Also, the qdeformed oscillator algebra, which was introduced by Biedenharn[6] and Macfarlane[7] to construct the q-deformed su(2) in the manner of Schwinger's construction of conventional * To be published in Modern Physics Letters A

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It is shown that an extended q-deformed su(2) algebra with an extra ("Schwinger ") term can describe Bloch electrons in a uniform magnetic field with an additional periodic potential. This is a generalization of the analysis of Bloch electrons by Wiegmann and Zabrodin. By using a representation theory of this q-deformed algebra, we obtain functional Bethe ansatz equations whose solutions should be functions of finite degree. It is also shown that the zero energy solution is expressed in terms of an Askey Wilson polynomial.The quantum deformation [1] of algebras was introduced in connection with inverse scattering problems and integrable models. It is also known that this notion has a deep relation to Yang-Baxter equations [2]. The U q (sl 2 ) algebra, one of the q-deformed algebras, was applied to an analysis of the two-dimensional Bloch electron problem [3] by Wiegmann and Zabrodin [4]. They found functional Bethe ansatz equations for this problem. From the representation theory of U q (sl 2 ) algebra on the functional space, they deduce that the solutions of the Bethe ansatz equations should be functions of finite degree. The zero energy solutions and the generalized functional Bethe ansatz equations have also been discussed in Refs. [5][6]. Biedenharn [7] and Macfarlane [8] constructed U q (sl 2 ) algebra, in the manner of Schwinger's construction of conventional su(2), by using two sets of qdeformed oscillator algebras. The q-deformed oscillator algebra H q (1), if suitably defined,