Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
140 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Two-Parameter Quantum Groups and Ringel-Hall algebras of $A_{\infty}-$type (1106.1904v2)

Published 9 Jun 2011 in math.QA, math.RA, and math.RT

Abstract: 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 $U{+}_{r,s}(\mathfrak sl_{\infty})$.

Summary

We haven't generated a summary for this paper yet.