K-theoretic Hall algebras, quantum groups and super quantum groups

Abstract

We realize the quantum loop groups and shifted quantum loop groups of arbitrary types, possibly non symmetric, via the K-theoretical critical convolution algebras introduced by . This yields a generalization of Nakajima's construction of symmetric quantum loop groups via quiver varieties to non symmetric types. We also construct some simple modules, in particular all Kirillov-Reshethikin modules, using critical K-theory.H lr e ir rk r `pl r ´1qd ir s.

K-theoretic Hall algebras of quivers with potential as Hopf algebras

“…For Q a quiver, let P(d) be the (derived) stack of representations of the preprojective algebra of Q of dimension d ∈ N I . The category d∈N I D b (P(d)) admits a Hall-type product [20]. The algebra obtained by taking K 0 of this category is called the preprojective K-theoretic Hall algebra (KHA) of Q.…”

“…The algebra obtained by taking K 0 of this category is called the preprojective K-theoretic Hall algebra (KHA) of Q. It was studied in [14], [20], [22] and it was seen to be related to positive parts of quantum affine algebras. Conjecturally, an equivariant version of the preprojective KHA is isomorphic to the positive part of the Okounkov-Smirnov quantum affine algebra [10].…”

“…For d ∈ N I , let X (d) be the stack of representations of Q of dimension d and let P(d) be the (derived) stack of representations of the preprojective algebra of Q of dimension d [20], [11, Subsection 7.6]. There is a Köszul equivalence [6], see also loc.…”

“…see [20] for more details. The categories D gr sg,T X (d) 0 and D sg,T X (d) 0 have the same K 0 by Propositions 2.1 and 3.9 for T as in (7), and thus…”

“…see [3], [20,Appendix C]. For X a quasi-smooth scheme with an action of G, let X = X/G and let X cl be its classical stack.…”