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We study a quantum (non-commutative) representation of the affine Weyl groupmainly of type $E_8^{(1)}$, where the representation is given by birationalactions on two variables $x$, $y$ with $q$-commutation relations. Using the tauvariables, we also construct quantum "fundamental" polynomials $F(x,y)$ whichcompletely control the Weyl group actions. The geometric properties of thepolynomials $F(x,y)$ for the commutative case is lifted distinctively in thequantum case to certain singularity structures as the $q$-difference operators.This property is further utilized as the characterization of the quantumpolynomials $F(x,y)$. As an application, the quantum curve associated withtopological strings proposed recently by the first named author is rederived bythe Weyl group symmetry. The cases of type $D_5^{(1)}$, $E_6^{(1)}$,$E_7^{(1)}$ are also discussed.

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves