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 [Formula: see text] 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 [Formula: see text] for all dimension vectors [Formula: see text] and framing vectors [Formula: see text] 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.