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In this paper, the rational Ringel-Hall algebras for tame quivers areintroduced and are identified with the positive part of the quantum extendedKac-Moody algebras. By using the rational Ringel-Hall algebras, we show thatthe existence of Hall polynomials for tame quiver algebras. The PBW bases areconstructed and new classes of perverse sheaves are shown to have strong purityproperty. These allows us to construct the canonical bases of the positive partof the quantum extended Kac-Moody algebras.

Rational Ringel-Hall algebras, Hall polynomials of affine type and Canonical bases