Quantum loop groups for symmetric Cartan matrices

Abstract We introduce a quantum loop group associated to a general symmetric Cartan matrix, by imposing just enough relations between the usual generators <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>e</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>f</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> <m:mo rspace="0.337em">,</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> </m:mrow> </m:msub> </m:math> \{e_{i,k},f_{i,k}\}_{i\in I,\,k\in\mathbb{Z}} in order for the natural Hopf pairing between the positive and negative halves of the quantum loop group to be perfect. As an application, we describe the localized 𝐾-theoretic Hall algebra of any quiver without loops, endowed with a particularly important <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> \mathbb{C}^{*} action.