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Two-Parameter Quantum Groups and Ringel-Hall algebras of $A_{\infty}-$type

In this paper, we study the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ associated to the Lie algebra $\mathfrak sl_{\infty}$ of infinite rank. We shall prove that the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ 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 $H_{r,s}(A_{\infty})$ associated to the category of all finite dimensional representations of the infinite linear quiver $A_{\infty}$. In particular, we will establish an iterated skew polynomial presentation of $H_{r,s}(A_{\infty})$ and prove that $H_{r,s}(A_{\infty})$ is a direct limit of the directed system of the two-parameter Ringel-Hall algebras $H_{r,s}(A_{n})$ associated to the finite linear quiver $A_{n}$. As a result, we construct a PBW basis for $H_{r,s}(A_{\infty})$ and prove that all prime ideals of $H_{r,s}(A_{\infty})$ are completely prime. Furthermore, we will establish an algebra isomorphism from $U_{r,s}^{+}(\mathfrak sl_{\infty})$ to $H_{r,s}(A_{\infty})$, which enable us to obtain the corresponding results for $U_{r,s}^{+}(\mathfrak sl_{\infty})$. Finally, via the theory of generic extensions in the category of finite dimensional representations of $A_{\infty}$, we shall construct several monomial bases and a bar-invariant basis for $U^{+}_{r,s}(\mathfrak sl_{\infty})$.