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Xin Tang's research paper on two-parameter quantum groups and Ringel-Hall algebras of infinite type

In this paper, we study the two-parameter quantum group Ur,s(sl∞) associated to the Lie algebra sl∞ of infinite rank. We shall prove that the two-parameter quantum group Ur,s(sl∞) admits both a Hopf algebra structure and a triangular decomposition. In particular, it can be realized as the Drinfeld double of it’s certain Hopf subalgebras. We will also study a two-parameter twisted Ringel-Hall algebra Hr,s(A∞) associated to the category of all finite dimensional representations of the infinite linear quiver A∞. In particular, we will establish an iterated skew polynomial presentation of Hr,s(A∞) and prove that Hr,s(A∞) is a direct limit of the directed system of the two-parameter Ringel-Hall algebras Hr,s(An) associated to the finite linear quiver An. As a result, we construct a PBW basis for Hr,s(A∞) and prove that all prime ideals of Hr,s(A∞) are completely prime. Furthermore, we will establish an algebra isomorphism from U r,s(sl∞) to Hr,s(A∞), which enable us to obtain the corresponding results for U r,s(sl∞). Finally, via the theory of generic extensions in the category of finite dimensional representations of A∞, we shall construct several monomial bases and a bar-invariant basis for U r,s(sl∞).