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 ${\mathcal{M}}_Q(\text{v},\text{w})$ for all dimension vectors $\text{v}$ and framing vectors $\text{w}$ 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.