Abstract
This paper aims at a geometric realization of the Yangian of non-simply laced type in terms of quiver with potentials. For every quiver with symmetrizer, there is an extended quiver with superpotential, whose Jacobian algebra is the generalized preprojective algebra of Geiß, Leclerc, and Schröer (Inventiones Mathematicae 209(1), 61–158, 2017). We study the cohomological Hall algebra of Kontsevich and Soibelman associated to this quiver with potential. In particular, we prove a dimensional reduction result, and provide a shuffle formula of this cohomological Hall algebra. In the case when the quiver with symmetrizer comes from a symmetrizable Cartan matrix, we prove that this shuffle algebra satisfies the relations of the Yangian associated to this Cartan matrix.
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The datasets generated during and/or analysed during the current study are available from the corresponding author (Yaping Yang) on reasonable request.
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Acknowledgements
This paper is motivated by a project in collaboration with Ivan Mirković on the quantization of the homogeneous coordinate ring of the zastava space [12], and is largely inspired by a talk of Paul Zinn-Justin in the Representation Theory Seminar at the University of Melbourne. The idea of doing dimensional reduction in the non-loop directions is suggested to us by Hiraku Nakajima. The authors thank the referee for a careful reading of the manuscript and suggestions to improve the paper.
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Open Access funding enabled and organized by CAUL and its Member Institutions. The two authors are partially supported by the Australian Research Council via the awards DE190101231 and DE190101222 respectively.
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Appendix A: Review of Dimensional Reduction
Appendix A: Review of Dimensional Reduction
We review a dimensional reduction procedure that describes the cohomological Hall algebra in the presence of a cut. Such a dimensional reduction on the level of cohomology groups is obtained by Davison [4, Appendix A], which is the main ingredient here.
In this section we study the behaviour of the algebra structure under the dimensional reduction, which is needed in the present paper. The argument here is similar to that in the proof of [22, Theorem 2.5]. However, the statement here is more general and more useful (see e.g., [16, Section 7.1]). We include a brief sketch of the argument in present generality, highlighting the difference to that in [22, Theorem 2.5].
Let Γ = (Γ0,Γ1) be a quiver, and W be the potential. In general, a cut C of (Γ,W) is a subset C ⊂Γ1 such that W is homogeneous of degree 1 with respect to the grading defined on arrows by
Consider the quotient of the path algebra \(\mathbb {C} ({{\Gamma }} \backslash C)\) by the relations {∂W/∂a∣a ∈ C}. The representation variety of this quotient algebra is denoted by
We view C as forming the edges of a new quiver, which we still denote by C for simplicity. Let MC, v be the representation variety of C with dimension vector v. Consider the trivial vector bundle π : MΓ,v = MΓ∖C, v ×MC, v →MΓ∖C, v carrying a scaling \(\mathbb {G}_{m}\) action of weight one on the fiber MC, v. Let \(\text {tr} W_{v}: \mathbf {M}_{{\Gamma }, v}= \mathbf {M}_{{\Gamma } \backslash C, v}\times \mathbf {M}_{C, v} \to \mathbb {A}^{1}\) be the function which is \(\mathbb {G}_{m}\)–equivariant. Define Z ⊂MΓ∖C, v to be the reduced scheme consisting of points z ∈MΓ∖C, v, such that π− 1(z) ⊂ (trWv)− 1(0). Then, we have Z = {x ∈MΓ∖C, v∣tr(Wv)(x, l) = 0,∀l ∈MC, v}. To summarize the notations, we have the diagram:
Lemma A.1
The subvariety Z of MΓ∖C, v is naturally identified with JΓ∖C, v.
Proof
The lemma follows from the same proof of [22, Lemma 3.1]. The difference in the current setting is the non-degenerate paring
given by the trace. □
Let \(\mathcal {D}\) be a torus \(\mathbb {G}_{m}^{r}\) for some \(r\in \mathbb {N}\). To each arrow in Γ1, we associate a \(\mathcal {D}\)-weight such that trWv is \(\mathcal {D}\)-invariant for any v. Then, ∂W/∂a is homogeneous for any a ∈ C. In particular, JΓ∖C, v is a \(\mathcal {D}\)-equivariant subvariety of MΓ∖C, v.
There is a canonical isomorphism of vector spaces [4, Theorem A.1]
Following [22], we now describe a multiplication mJ on the graded vector space
Let \(v_{1}, v_{2} \in {\mathbb {N}}_{0}^{{\Gamma }}\) be dimension vectors such that v = v1 + v2, let V1 ⊂ V be a |Γ0|-tuple of subspaces of V with dimension vector v1. Define \(\mathbf {M}_{{\Gamma }, v_{1}, v_{2}} := \{x\in \mathbf {M}_{{\Gamma }, v} \mid x(V_{1}) \subset V_{1}\}\). We write \(G:=G_{v} \times \mathcal {D}\) for short. Let \(P\subset G_{v} \times \mathcal {D}\) be the parabolic subgroup preserving the subspace V1 and \(L:=G_{v_{1}}\times G_{v_{2}}\times \mathcal {D}\) be the Levi subgroup of P. We have the following correspondence of L-varieties.
where pΓ is the natural projection and ηΓ is the embedding.
We have the following commutative diagram
where \(p_{1}=\text {id}_{\mathbf {M}_{{\Gamma } \backslash C, v_{1}}\times \mathbf {M}_{{\Gamma } \backslash C, v_{2}}} \times p_{C}\), and \(i_{1}=\text {id}_{\mathbf {M}_{{\Gamma } \backslash C, v_{1}}\times \mathbf {M}_{{\Gamma } \backslash C, v_{2}}} \times \eta _{C}\). Here pC,ηC are maps in the correspondence \(\mathbf {M}_{C, v_{1}}\times \mathbf {M}_{C, v_{2}}\xleftarrow {p_{C}} \mathbf {M}_{C, v_{1}, v_{2}} \xrightarrow {\eta _{C}} \mathbf {M}_{C, v}\). The vertical maps are natural inclusions, and \(\overline {p_{1}}, \overline {i_{1}}\) are the restrictions of p1,i1.
Identify \(\mathbf {M}_{C^{\text {op}}, v}\) with \({\mathbf {M}}_{C, v}^{*}\) via \( \mathbf {M}_{C^{\text {op}}, v}\cong {\mathbf {M}}_{C, v}^{*}, x\mapsto (y \mapsto \text {tr}(x y))\). For x ∈MΓ∖C, v, the cyclic derivative ∂W/∂a(x) is an element in \(\mathbf {M}_{C^{\text {op}}, v}\), for any a ∈ C. Thus, for any l ∈MC, v, we have the pairing (∂W/∂a(x),l) = tr(∂W/∂a(x) ⋅ l).
Recall that \(p_{C^{\text {op}}}: \mathbf {M}_{C^{\text {op}}, v_{1},v_{2}} \to \mathbf {M}_{C^{\text {op}}, v_{1}}\times \mathbf {M}_{C^{\text {op}}, v_{2}}\) is the natural projection. Introduce the following subvariety \(\mathbf {Y} \subset \mathbf {M}_{{\Gamma } \backslash C, v_{1}} \times \mathbf {M}_{{\Gamma } \backslash C, v_{2}} \times \mathbf {M}_{C^{\text {op}}, v_{1}, v_{2}}\).
There are two maps
Let \(\mathbf {J}_{{\Gamma } \backslash C, v_{1}, v_{2}} \subset \mathbf {M}_{{\Gamma } \backslash C, v_{1}, v_{2}}\) be the subvariety defined by the equation (∂W/∂a)a∈C(x), for all a ∈ C. We then have an embedding \(\overline {i_{2}}: \mathbf {J}_{{\Gamma } \backslash C, v_{1}, v_{2}}\subset \mathbf {J}_{{\Gamma } \backslash C, v}\). They fit into the following commutative diagram.
where the map \(\overline {\omega }\) is the restriction of ω. Note that by introducing the variety Y, the pullback of the two maps \(\omega : \mathbf {M}_{{\Gamma } \backslash C, v_{1}, v_{2}} \to \mathbf {Y}\) and \(\bar {\iota }: \mathbf {J}_{{\Gamma } \backslash C, v_{1}}\times \mathbf {J}_{{\Gamma } \backslash C, v_{2}}\to \mathbf {Y}\) is \(\mathbf {J}_{{\Gamma } \backslash C, v_{1}, v_{2}} \). In other words, the square in the diagram (4) formed by \(\omega , \bar {\iota }, \bar {\omega }\) is Cartesian.
The multiplication mJ is defined to be
The maps in the composition are the following.
-
(1)
The Künneth morphism \(H^{\text {BM}}_{G_{v_{1}} \times \mathcal {D}} (\mathbf {J}_{{\Gamma }\backslash C, v_{1}} \times \mathbf {M}_{C, v_{1}}) \otimes H^{\text {BM}}_{G_{v_{2}} \times \mathcal {D}}(\mathbf {J}_{{\Gamma }\backslash C, v_{2}} \times \mathbf {M}_{C, v_{2}}) \to {H}_{L}^{\text {BM}}(\mathbf {J}_{{\Gamma }\backslash C, v_{1}}\times \mathbf {J}_{{\Gamma }\backslash C, v_{2}} \times \mathbf {M}_{C, v_{1}}\times \mathbf {M}_{C, v_{2}})\). Here the tensor is over \(H_{\mathcal {D}}^{\text {BM}}(\text {pt})\).
-
(2)
\(\overline {i_{1}}_{*} \circ \overline {p_{1}}^{*}: {H}_{L}^{\text {BM}}(\mathbf {J}_{{\Gamma }\backslash C, v_{1}}\times \mathbf {J}_{{\Gamma }\backslash C, v_{2}} \times \mathbf {M}_{C, v_{1}}\times \mathbf {M}_{C, v_{2}})\to {H}_{L}^{\text {BM}}(\mathbf {J}_{{\Gamma }\backslash C, v_{1}} \times \mathbf {J}_{{\Gamma }\backslash C, v_{2}} \times \mathbf {M}_{C, v})\).
-
(3)
Denote by \((\omega \times \text {id}_{\mathbf {M}_{C,v}})_{\overline {\omega }\times \text {id}_{\mathbf {M}_{C,v}}}^{\sharp }\) the refined Gysin pullback of \(\omega \times \text {id}_{\mathbf {M}_{C,v}}\) along \(\overline {\omega }\times \text {id}_{\mathbf {M}_{C,v}}\). Let e(ι) be the L-equivariant Euler class of the normal bundle of ι. We have the following map
$$ \frac{1}{e(\iota)} \omega_{\overline{\omega}}^{\sharp}: {H}_{L}^{\text{BM}} (\mathbf{J}_{{\Gamma} \backslash C, v_{1}}\times \mathbf{J}_{{\Gamma}\backslash C, v_{2}} \times \mathbf{M}_{C, v}) \to {H}_{L}^{\text{BM}}(\mathbf{J}_{{\Gamma}\backslash C, v_{1}, v_{2}} \times \mathbf{M}_{C, v} )[\frac{1}{e(\iota)}]. $$ -
(4)
The pushforward \((\overline {i_{2}}\times \text {id}_{\mathbf {M}_{C,v}})_{*}: {H}_{L}^{\text {BM}} (\mathbf {J}_{{\Gamma } \backslash C, v_{1}, v_{2}} \times \mathbf {M}_{C, v}) \to {H}_{L}^{\text {BM}}(\mathbf {J}_{{\Gamma }\backslash C, v} \mathbf {M}_{C, v})\).
-
(5)
Pushforward along G ×P(JΓ∖C, v ×MC, v) →JΓ∖C, v ×MC, v, (g, m)↦gmg− 1, we get \({H}_{P}^{\text {BM}} (\mathbf {J}_{{\Gamma } \backslash C, v} \times \mathbf {M}_{C, v}) \cong {H}_{G}^{\text {BM}} (G\times _{P}(\mathbf {J}_{{\Gamma } \backslash C, v} \times \mathbf {M}_{C, v})) \to {H}_{G}^{\text {BM}} (\mathbf {J}_{{\Gamma } \backslash C, v} \times \mathbf {M}_{C, v})\).
This map mJ a priori is only defined after inverting e(ι). However, it follows from Theorem A.2 that it is well-defined before localization.
Theorem A.2
There is an isomorphism of algebras
where \({\mathscr{H}}_{\mathcal {D}}({\Gamma }, W)\) endowed with the Hall multiplication of Kontsevich-Soibelman and \({\bigoplus }_{v\in \mathbb {N}^{{\Gamma }_{0}}} H^{\text {BM}}_{G_{v} \times \mathcal {D}}(\mathbf {J}_{{\Gamma }\backslash C, v} \times \mathbf {M}_{C, v}, \mathbb {Q})\) has multiplication given by mJ.
As has been mentioned, a special case of this is [22, Theorem 2.5] and [17, Appendix, Corollary 4.5]. The proof of [22, § 2] goes through in the setting verbatim, with the following substitutions.
-
(1)
We need to distinguish between C and Cop in the present paper, where in [22], C consists of edge loops and hence C = Cop.
-
(2)
Consequently we need to identify \(\mathbf {M}_{C^{\text {op}}, v}\) with \(\mathbf {M}_{C, v}^{*}\) using the trace map.
-
(3)
The variety Y is changed to Eq. 3.
-
(4)
The [22, Lemma 3.1] is replaced by Lemma A.1.
-
(5)
The torus \(\mathcal {D}\)-action in the present generality is given in Section 2.2.
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Yang, Y., Zhao, G. The Cohomological Hall Algebras of a Preprojective Algebra with Symmetrizer. Algebr Represent Theor 26, 1067–1085 (2023). https://doi.org/10.1007/s10468-022-10125-6
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DOI: https://doi.org/10.1007/s10468-022-10125-6