Quantum Representation Of Affine Weyl Groups And Associated Quantum Curves

We study a quantum (non-commutative) representation of the affine Weyl group mainly of type E-8((1)), where the representation is given by birational actions on two variables x, y with q-commutation relations. Using the tau variables, we also construct quantum "fundamental" polynomials F(x, y) which completely control the Weyl group actions. The geometric properties of the polynomials F(x, y) for the commutative case is lifted distinctively in the quantum case to certain singularity structures as the q-difference operators. This property is further utilized as the characterization of the quantum polynomials F(x, y). As an application, the quantum curve associated with topological strings proposed recently by the first named author is rederived by the Weyl group symmetry. The cases of type D-5((1)), E-6((1)), E-7((1)) are also discussed.