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

Abstract

In this paper, we study the two-parameter quantum group $U_{r,s}(\mathfrak{s}{l}_{\infty })$ associated to the Lie algebra $\mathfrak{s}{l}_{\infty }$ of infinite rank. We shall prove that the two-parameter quantum group $U_{r,s}(\mathfrak{s}{l}_{\infty })$ 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 $U_{r,s}^{+}(\mathfrak{s}{l}_{\infty })$ to $H_{r,s}(A_{\infty })$, which enable us to obtain the corresponding results for $U_{r,s}^{+}(\mathfrak{s}{l}_{\infty })$. 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 $U_{r,s}^{+}(\mathfrak{s}{l}_{\infty })$.

Revised Version of the previous submission

References in corpus (1)

• Highest weight irreducible representations of the quantum algebra $U_h(A_{\infty })$