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arXiv · 2601.01518

Shifted quantum groups via critical stable envelopes

Yalong Cao·Andrei Okounkov·Yehao Zhou·Zijun Zhou
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Abstract

Given a symmetric quiver with potential, we develop a geometric construction of shifted Yangians acting on the critical cohomologies of antidominantly framed quiver varieties with extended potentials, using the $R$-matrices constructed from critical stable envelopes. We relate such Reshetikhin type Yangians to Drinfeld type Yangians arising from critical cohomological Hall algebras. Several detailed examples, including the trivial, Jordan, and tripled Jordan quivers are explicitly computed. For symmetric quiver varieties with potentials, by using the smallness property of their affinization maps, we derive explicit formulas for quantum multiplication by divisors in terms of Casimir elements of the associated Lie (super)algebras, extending results from Nakajima quiver varieties to the critical setting. A similar formula in the antidominantly framed case is also obtained, which includes Hilbert schemes of points on $\mathbb C^3$ as examples.

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Yalong Cao, Andrei Okounkov, Yehao Zhou, Zijun Zhou. 2026-01-04. Shifted quantum groups via critical stable envelopes. https://arxiv.org/abs/2601.01518

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