Framed cohomological Hall algebras and cohomological stable envelopes.

There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver Q to the Yangian Y MO Q by Maulik-Okounkov, whose construction is based on the notion of stable envelopes of Nakajima varieties. In this article, we introduce the cohomological Hall algebra of the moduli stack of framed representations of a quiver Q (framed CoHA), and we show that the equivariant cohomology of the disjoint union of the Nakajima varieties M Q ( v , w ) for all dimension vectors v and framing vectors w has a canonical structure of subalgebra of the framed CoHA. Restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map. As a corollary, we deduce an explicit inductive formula to compute stable envelopes in terms of tautological classes.