--- title: 'Maulik–Okounkov Yangian: Geometric Quantum Group' url: https://www.emergentmind.com/topics/maulik-okounkov-yangian type: topic --- # Maulik–Okounkov Yangian: Geometric Quantum Group Searching arXiv for recent and foundational papers on the Maulik–Okounkov Yangian and Hall-algebra realizations. The **Maulik–Okounkov Yangian** is the Hopf algebra obtained from geometric \(R\)-matrices built from stable envelopes on holomorphic symplectic varieties, most prominently Nakajima quiver varieties. In the quiver setting, it is attached to the Dynkin type of a quiver \(Q\), is denoted \(Y^Q_{MO}\) or \(Y_{MO}(\mathfrak g_Q)\), and is generated by matrix elements of geometric \(R\)-matrices satisfying RTT relations. Its positive half is realized by Hall-algebra constructions, historically first conjecturally through cohomological Hall algebras of tripled quivers with potential and, for arbitrary quivers, by an isomorphism with the preprojective cohomological Hall algebra [1704.06039], [2312.15803]. ## 1. Geometric construction from stable envelopes The geometric input is a smooth quasi-projective complex variety \(X\) equipped with a holomorphic symplectic form \(\omega\), together with a torus \(A\subset T\) acting so that \(\omega\) is \(A\)-fixed, there is a proper \(T\)-equivariant map \(\pi:X\to X_0\) to an affine \(T\)-variety, and \(X\) is formal as a \(T\)-variety. Writing \(\Delta\subset \mathrm{Hom}(A,\mathbb C^\times)\) for the weights in the normal bundle to the fixed locus \(X^A\), the hyperplanes \(\alpha^\perp\) cut the real cocharacter space into chambers. For a chamber \(\mathfrak C\), Maulik and Okounkov define the stable-envelope map \[ \mathrm{Stab}_{\mathfrak C}:H_T^*(X^A)\longrightarrow H_T^*(X) \] characterized by support, normalization, and degree conditions [1704.06039]. For two chambers \(\mathfrak C,\mathfrak C'\), the corresponding geometric \(R\)-matrix is \[ R_{\mathfrak C',\mathfrak C}=(\mathrm{Stab}_{\mathfrak C'})^{-1}\circ \mathrm{Stab}_{\mathfrak C}. \] These operators satisfy the Yang–Baxter equation and unitarity. Botta’s formulation records that the collection of geometric \(R\)-matrices obtained by comparing stable envelopes for two chambers satisfies the Yang-Baxter equation with spectral parameter, and that by the usual RTT formalism this gives a Hopf algebra called the Maulik–Okounkov Yangian \(Y^Q_{MO}\) [2207.06280]. The same source records the classical expansion \[ R(a)=1+\frac{\hbar}{a}\,r+O(a^{-2}), \] whose leading term defines a Lie algebra \(\mathfrak g_Q\subset Y^Q_{MO}\) generated by the entries of the classical \(r\)-matrix [2207.06280]. In finite Dynkin type, when \(X\) is a Nakajima quiver variety, one can compare generators, coproducts, and Hecke-correspondence actions and obtain an algebra isomorphism \[ Y_Q\simeq Y(\mathfrak g), \] with \(\mathfrak g\) the simple Lie algebra of the same Dynkin type [1704.06039]. ## 2. RTT formalism, Drinfeld currents, and triangular structure Once a decomposition \(H_T^*(X^A)\simeq \bigoplus_i V_i\) is chosen, the matrix elements of the \(R\)-matrix \(R_{ij}(u):V_i\otimes V_j\to V_j\otimes V_i\), expanded in the spectral parameter \(u\), satisfy the RTT relation \[ R_{12}(u-v)\,T_1(u)\,T_2(v)=T_2(v)\,T_1(u)\,R_{12}(u-v), \] and the resulting Hopf algebra is the Yangian in Drinfeld new realization [1704.06039]. In this geometric setting, the generators are recovered as matrix elements of \(R(a)\) and its expansion at \(a=\infty\). The Drinfeld currents are written as \[ h_i(u)=1+\sum_{r\ge 0}h_i^{(r)}u^{-r-1},\qquad x_i^\pm(u)=\sum_{r\ge 0}x_i^{\pm,(r)}u^{-r-1}, \] with the standard relations \[ [\,h_i(u),h_j(v)\,]=0, \] \[ (u-v\pm\tfrac12 a_{ij})\,h_i(u)\,x_j^\pm(v) =(u-v\mp\tfrac12 a_{ij})\,x_j^\pm(v)\,h_i(u), \] \[ [\,x_i^+(u),x_j^-(v)\,] =\tfrac{\delta_{ij}}{u-v}\bigl(h_i(u)-h_i(v)\bigr), \] together with Serre-type identities [1704.06039]. In Hall-algebra realizations of the positive half, the positive generators are often packaged into formal series \[ e_i(z)=\sum_{n\ge 1} e_{i,n}z^{-n}, \] and satisfy Maulik–Okounkov–type quadratic relations \[ (z-w-\tfrac{\hbar}{2}a_{ij})\,e_i(z)e_j(w) =(z-w+\tfrac{\hbar}{2}a_{ij})\,e_j(w)e_i(z) \] for the symmetrized Cartan matrix \(a_{ij}\) of \(\mathfrak g_Q\) [2107.13642]. The internal structure is triangular. In the quiver-variety formulation one has \[ Y_R(Q)\cong Y_R^-(Q)\otimes Y_R^0(Q)\otimes Y_R^+(Q), \] where \(Y_R^+(Q)\) is spanned by positive-current generators, \(Y_R^-(Q)\) by negative currents, and \(Y_R^0(Q)\) by the Cartan generators [2312.15803]. Schiffmann–Vasserot likewise describe a triangular decomposition \(Y\simeq Y^-\otimes Y^0\otimes Y^+\) and identify the Cartan subalgebra \(Y^0\) through the Chern classes of the universal and framing bundles on Nakajima varieties [1705.07491]. ## 3. Quiver varieties, flag varieties, and geometric representations The primary representation spaces for the Maulik–Okounkov Yangian are the equivariant cohomologies of Nakajima quiver varieties. For each framing vector \(w\), one considers \[ F_w=\bigoplus_{v\in \mathbb N^I} H^*_{G_w\times T}(\mathfrak M(v,w)), \] and the Yangian acts on \(F_w\) through the stable-envelope \(R\)-matrix construction [1705.07491], [2312.15803]. Botta’s framed CoHA picture sharpens this by showing that the equivariant cohomology of the disjoint union of the Nakajima varieties \(\mathcal M_Q(v,w)\) for all dimension and framing vectors has a canonical structure of subalgebra of the framed CoHA, and that, restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map [2207.06280]. A fundamental family of examples is given by cotangent bundles of partial flag varieties. For \[ \mathcal X=\bigsqcup_{|\lambda|=n} T^*F_\lambda, \] Maulik–Okounkov introduced a \(Y_\hbar(\mathfrak{gl}_N)\)-module structure on \[ \bigoplus_{|\lambda|=n} H_T^*(T^*F_\lambda), \] and this module structure was identified with the Yangian action of Gorbounov–Rimányi–Tarasov–Varchenko. Stable-envelope maps \(\mathrm{Stab}_\sigma\) are expressed in terms of Yangian weight functions \(W_\sigma(I)\) divided by an explicit Euler class \(c_\lambda(\Theta)\), and the geometric \(R\)-matrix \(R_{\sigma,\sigma'}=\mathrm{Stab}_{\sigma'}^{-1}\circ \mathrm{Stab}_\sigma\) matches the algebraic \(R\)-matrix built from those weight functions [1212.6240]. The same framework connects the Yangian to quantum multiplication. In the partial-flag setting, quantum multiplication by divisors is identified with the action of dynamical Hamiltonians. More generally, Maulik–Okounkov show that the commutative Baxter subalgebra generated by transfer matrices \(\mathrm{Tr}_V(R(u))\) acting on \(H_T^*(X)\) coincides with quantum multiplication in the \(T\)-equivariant cohomology of \(X\) [1212.6240], [1704.06039]. This places the Maulik–Okounkov Yangian simultaneously in representation theory, symplectic geometry, and quantum cohomology. ## 4. Cohomological Hall algebras and the positive half A central theme is the realization of the positive half of the Maulik–Okounkov Yangian by Hall algebras. For a finite quiver \(Q=(I,E)\), one forms the tripled quiver \(\widetilde Q\) by adding opposite arrows and a loop \(w_i\) at each vertex, together with the cubic potential \[ \widetilde W=\sum_{e\in E}[\,\bar e,e\,]\,w_{t(e)}. \] The cohomological Hall algebra is \[ \mathrm{CoHA}(\widetilde Q,\widetilde W) =\bigoplus_{d\in A_\theta} H_T^*\bigl(X(d)^{ss},\varphi_{\mathrm{Tr}\widetilde W}\mathbb Q\bigr), \] with multiplication defined by the standard extension correspondence \[ m_{d,e}=p_{d,e*}\circ q_{d,e}^*\circ TS \] in vanishing-cycle cohomology [2107.13642]. Pădurariu’s K-theoretic paper presents the same construction as the Hall algebra on Borel–Moore homology with vanishing cycles for the tripled quiver [1911.05526]. Historically, this Hall algebra was conjectured to coincide with the positive half of the Maulik–Okounkov Yangian: \[ \mathrm{CoHA}(\widetilde Q,\widetilde W)\cong Y^+_{MO}(\mathfrak g_Q). \] Davison’s conjecture, refined and extended by later work, identifies the generators \(e_i(z)\) from the Hall algebra with the standard positive Drinfeld currents of \(Y_{MO}(\mathfrak g_Q)\) [2107.13642]. Schiffmann–Vasserot formulated the corresponding conjecture for the preprojective CoHA and constructed an embedding \[ \iota:Y_{1,K}\hookrightarrow Y_K^+ \] after extension of scalars, intertwining the two actions on quiver-variety cohomology [1705.07491]. The conjectural stage has a decisive modern refinement. Schiffmann–Vasserot construct an isomorphism \[ \Phi:\mathsf Y^R(Q)\xrightarrow{\sim} Y_R^+(Q) \] between the preprojective cohomological Hall algebra of an arbitrary quiver and the positive half of the corresponding Maulik–Okounkov Yangian, and they show that \(\Phi\) intertwines the Hall action and the Yangian action on the cohomology of Nakajima quiver varieties [2312.15803]. In this form, the Hall-algebraic positive half is no longer only conjectural. The Hall-algebra side also carries structural filtrations. Pădurariu records a perverse filtration \(P_{\le k}\) on the tripled-quiver CoHA such that \[ \mathrm{gr}^P \mathrm{CoHA}(\widetilde Q,\widetilde W)\simeq \mathrm{Sym}_{\mathbb Q}\bigl(P_{\le 1}(H^*),u\bigr), \] with \(u\) a degree-\(2\) formal generator, and \(P_{\le 1}\) closed under the commutator, forming the BPS Lie algebra [1911.05526]. This suggests a precise Lie-theoretic shadow of the positive half. ## 5. K-theoretic and categorical lifts The K-theoretic extension replaces vanishing-cycle cohomology by categories of singularities or, equivalently, equivariant matrix-factorization categories. For each dimension vector \(d\), \[ \mathrm{HAT}(d)=D_{sg}^T(X(d)_0) = D^b\mathrm{Coh}^T(\{\mathrm{Tr}\,W=0\})/\mathrm{Perf}, \] equivalently \(MF_T(X(d),\widetilde W)\), and these categories assemble into a monoidal dg-category \[ \mathrm{HAT}(\widetilde Q,\widetilde W)=\bigoplus_d \mathrm{HAT}(d). \] Its Grothendieck group \[ \mathrm{KHA}(\widetilde Q,\widetilde W)=\bigoplus_d K_0(\mathrm{HAT}(d)) \] is the K-theoretic Hall algebra [2107.13642]. The K-theoretic Hall product is again defined by pull–push along the extension correspondence, \[ m_{d,e}=[\,p_{d,e*}\circ q_{d,e}^*\,]: K_0(\mathrm{HAT}(d))\otimes K_0(\mathrm{HAT}(e)) \longrightarrow K_0(\mathrm{HAT}(d+e)), \] and it admits a shuffle form. For \(f\in K_0(X(a))\) and \(g\in K_0(X(b))\), \[ (f\star g)(x_1,\dots,x_{a+b}) =\sum_{w\in S_{a+b}/(S_a\times S_b)} w\!\Bigl[f(x_1,\dots,x_a)\,g(x_{a+1},\dots,x_{a+b})\, \prod_{i\le a\cdots>\mu(d^k)} \mathrm{KHA}_{d^1}(Q,W)\otimes\cdots\otimes \mathrm{KHA}_{d^k}(Q,W), \] equivalently a factorization by slopes [2107.13642]. The expected quantum-group target changes in K-theory. For the special tripled quivers \((\widetilde Q,\widetilde W)\), the KHA is expected to recover the positive part of the quantum affine algebra \(U_q(\widehat{\mathfrak g_Q})\) defined by Okounkov–Smirnov [2107.13642], [1911.05526]. Pădurariu also constructs a Chern-character map \[ ch:\mathrm{gr}_{\deg}\mathrm{KHA}(Q,\widetilde W)\to \mathrm{CoHA}(Q,\widetilde W) \] which intertwines products, coproducts, and braidings; after passing to associated graded, it is an injective bialgebra map [1911.05526]. In finite and affine ADE types, the conjectural K-theoretic identification with the positive part of \(U_q(\widehat{\mathfrak g_Q})\) is proved by Varagnolo–Vasserot [2107.13642]. ## 6. Special cases, extensions, and current status Several model cases organize the subject. In the one-loop or Jordan-quiver case, the Maulik–Okounkov Yangian becomes the affine Yangian of \(\widehat{\mathfrak{gl}}_1\), realized via the Heisenberg action on \(\bigoplus_n H^*(\mathrm{Hilb}^n)\) [1704.06039]. Tsymbaliuk gives a Drinfeld-current presentation with generators \[ e(u)=\sum_{r\ge 0} e_r u^{-r-1},\qquad f(u)=\sum_{r\ge 0} f_r u^{-r-1},\qquad \psi(u)=1+\sum_{r\ge 0}\psi_r u^{-r-1}, \] and closed-form relations \((Y0)-(Y6)\), including the additivized commutation \[ \psi(u)e(v)=\phi(u-v)e(v)\psi(u) \] and the cubic Serre-type identities [1404.5240]. Litvinov–Vilkoviskiy and Procházka relate the same affine Yangian to the Maulik–Okounkov \(R\)-matrix, \(RLL\) and current realizations, transfer matrices, Bethe ansatz, the Miura transformation, and \(\mathcal W_{1+\infty}\) structures [2007.00535], [1903.10372]. For cyclic quivers, the positive half has an integral Hall-algebra realization. Jindal proves that the \(\mathbb T\)-equivariant CoHA of the tripled cyclic quiver with sign twist is isomorphic to the positive half of an explicit integral form of Guay’s affine Yangian of \(\mathfrak{gl}(K+1)\), and equivalently, by recent results of Botta–Davison and Schiffmann–Vasserot, to the positive half of the Maulik–Okounkov Yangian for the cyclic quiver: \[ \mathcal A^{\mathbb T,\chi}_{\widetilde Q^K,\widetilde W^K} \cong \mathcal Y^{(K+1),+,\mathrm{CoHA}}_{-\hbar_2,\hbar_1+\hbar_2} \cong \mathbf Y^{\mathrm{MO},+}_{Q^K}. \] The same work embeds the classical limit into matrix differential operators on \(\mathbb C^*\) and exhibits a commutative polynomial subalgebra inside the additive shuffle algebra [2408.02618]. Stable-envelope methods also extend beyond ordinary Yangians. For \(\sigma\)-quiver varieties, Yiqiang Li’s construction and subsequent calculations produce coideal subalgebras of the Maulik–Okounkov Yangian, called twisted Yangians, together with \(K\)-matrices satisfying reflection equations [2510.13007]. This does not alter the definition of the Maulik–Okounkov Yangian itself, but it shows that the same stable-envelope formalism controls both the Yangian and its reflection-equation analogues. The status of the subject is therefore stratified. The geometric definition through stable envelopes and \(R\)-matrices is established, as is the resulting Hopf algebra structure [1704.06039], [2207.06280]. The identification of the positive half with the preprojective CoHA is established for arbitrary quivers [2312.15803]. The tripled-quiver-with-potential and categorical/K-theoretic formulations provide a parallel and highly structured Hall-algebraic framework, with the K-theoretic quantum-affine identification proved in finite and affine ADE and expected more generally [2107.13642], [1911.05526]. This suggests that the Maulik–Okounkov Yangian is best viewed not as a single presentation, but as a geometric quantum group simultaneously visible through stable envelopes, RTT matrices, Drinfeld currents, Hall correspondences, and symplectic representation theory. Source: https://www.emergentmind.com/topics/maulik-okounkov-yangian