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Keywords:
two-parameter quantum group; quiver; complex; Hall algebra
two-parameter quantum group; quiver; complex; Hall algebra
Summary:
We construct the twisted Hall algebra ${\rm H}_{\rm tw}(\mathcal {C}(\mathcal {P}))$ involving $\mathbb {Z}_2$-graded projective complexes of quiver representations. Then we describe two-parameter quantum groups of symmetric Kac-Moody Lie algebras via the localization $\mathcal {DH}(\mathcal {A})$ of ${\rm H}_{\rm tw}(\mathcal {C}(\mathcal {P}))$ with respective to acyclic complexes. For a Dynkin quiver, we realized generic two-parameter quantum groups via corresponding Hall algebras defined by the Ringel-Hall polynomial.
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