Source-reported preprojective CoHA actions and their type boundaries

For a finite quiver $Q$, its double adds an opposite arrow $h^{\ast }$ for each arrow $h$. The preprojective algebra is the path algebra of the double modulo the vertex moment-map relations ${\sum }_{h:t(h)=i}ϵ(h)B_h{B}_{h^{\ast }}=0$, with the sign fixed by an orientation of the original arrows. A framed Nakajima variety $\mathfrak{M}(\mathbf{v},\mathbf{w})$ is the geometric quotient of the stable locus of framed doubled-quiver representations satisfying the corresponding moment-map equation; stability ${\theta }^{+}$, framing dimensions $\mathbf{w}$, and the cohomology theory are part of the input.

Yang–Zhao report that for a finite quiver and an algebraic oriented cohomology theory $A$ with the formal-group-law and torus-action hypotheses in their §§1.1 and 3.1, the preprojective CoHA $P_A(Q)={⨁}_{\mathbf{v}}{A}_{G_{\mathbf{v}}\times {\mathbb{G}}_m^2}({\mu }_{\mathbf{v}}^{-1}(0))$ is associative and, for each fixed framing dimension vector $\mathbf{w}$, acts from the right on ${⨁}_{\mathbf{v}}{A}_{G_{\mathbf{w}}\times {\mathbb{G}}_m^2}(\mathfrak{M}(\mathbf{v},\mathbf{w}))$ with the fixed positive stability ${\theta }^{+}$ (their Theorems A and B). Here $G_{\mathbf{w}}={\prod }_{i\in Q_0}{\mathrm{GL}}_{w_i}$, and $A_H(-)$ denotes their $H$-equivariant $A$-homology. An element of dimension-vector degree $\mathbf{u}$ sends the $\mathbf{v}$ summand to the $\mathbf{v}+\mathbf{u}$ summand (Theorem 5.4). This action statement is for arbitrary finite $Q$ in their stated setup.

Their Yangian comparison is a narrower result and uses the specific arrow weights of Remark 3.2. For each numbered family of arrows $h_1,\dots ,h_a:i\to j$, the exponents are $m_{h_p}=a+2-2p$ and $m_{h_p^{\ast }}=-a+2p$, respectively, for $1\le p\le a$. After the diagonal specialization below, the corresponding weights are $m_{h_p}\mathrm{\hslash }\mathrm{/}2$ and $m_{h_p^{\ast }}\mathrm{\hslash }\mathrm{/}2$. This prescription is required throughout §§7–8; Assumption 3.1 alone is insufficient. With rational Chow theory, no edge loops, the Euler–Ringel sign twist ${\tilde{m}}_{{\mathbf{v}}_1,{\mathbf{v}}_2}=(-1{)}^{({\mathbf{v}}_2,C{\mathbf{v}}_1)+1}{m}_{{\mathbf{v}}_1,{\mathbf{v}}_2},C=I-A_D,$ the spherical extension, specialization $t_1=t_2=\mathrm{\hslash }\mathrm{/}2$, and quotient by the specified torsion, Yang–Zhao prove a surjection from the positive Borel of the quiver Yangian; it is an isomorphism for simply-laced ADE quivers, and the isomorphism intertwines the stated Nakajima-variety action (Theorems D and 8.3). Here ADE means the underlying graph is a finite Dynkin diagram of type $A,D,$ or $E$. The Yangian presentation is in their §7.1.

Nakajima’s ordinary Kac–Moody action uses a finite quiver with no edge loops. It acts on the direct sum of the top-degree homology groups ${⨁}_{\mathbf{v}}{H}_{d(\mathbf{v},\mathbf{w})}(\mathfrak{L}(\mathbf{v},\mathbf{w});\mathbb{C})$ of the Lagrangian cores, where $\mathfrak{L}(\mathbf{v},\mathbf{w})$ denotes the Lagrangian core and $d(\mathbf{v},\mathbf{w})={\dim }_{\mathbb{C}}\mathfrak{M}(\mathbf{v},\mathbf{w})$, and gives an integrable highest-weight representation with highest weight $\mathbf{w}$ (Theorem 17 of the review, referring to the original papers). The review omits the geometric proof of the Lagrangian theorem and refers the action relations to those original papers.

These are source reports with their stated conventions and hypotheses. This Remark supplies no local action theorem, Yangian comparison, or logical supplier; no general CoHA–Yangian identification follows from the untwisted associativity theorem.