Geometric construction of generators of CoHA of doubled quiver

Chen, Zongbin

doi:10.1016/j.crma.2014.09.025

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…Proof. As the argument is similar to [3], we will be brief. Poincaré duality for smooth Artin stacks gives a perfect pairing…”

Section: Remarkmentioning

confidence: 99%

“…The proof of injectivity in Theorem 2.4 relies on an interpretation of Ω Q in terms of the cohomology of (smooth) Nakajima quiver varieties [21]. Since smooth analogues of Nakajima varieties do not exist in the self-dual setting, it is not clear if the proof from [3] can be adapted to the present setting. In any case, it is natural to make the following conjecture.…”

Section: Remarkmentioning

confidence: 99%

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Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groups

Young¹

2016

Preprint

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We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution σ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are called orientifold Donaldson-Thomas invariants. We prove the integrality conjecture for orientifold Donaldson-Thomas invariants of σ-symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of these invariants and the freeness of the CoHM of a σ-symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type A 1 . We also verify the geometric conjecture in a number of examples. Finally, we describe the CoHM of finite type quivers by constructing explicit Poincaré-Birkhoff-Witt type bases of these representations.

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“…Proof. As the argument is similar to [3], we will be brief. Poincaré duality for smooth Artin stacks gives a perfect pairing…”

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groups

Young¹

2016

Preprint

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“…There are several geometric interpretations of the generators of the Cohomological Hall algebra. For instance, Chen identifies in [1] the primitive part of the Cohomological Hall algebra of a symmetric quiver with the invariant part under a Weyl group action of the cohomology of a variety introduced in [7]. Another interpretation can be given as follows.…”

Section: Generatorsmentioning

confidence: 99%

On the Cohomological Hall Algebra of the Kronecker Quiver

Franzen¹,

Reineke²

2019

Preprint

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“…The structure of H (and its modules) has been the subject of much study [KS11,Efi12,Rim13,Che14,Dav17,Fra16,Fra18,FR18,FR19]. In this paper, we restrict to the case of quivers Q which are acyclic, i.e., have no oriented cycles.…”

Section: Introductionmentioning

confidence: 99%

A family of structure isomorphisms for the Cohomological Hall Algebra of an acyclic quiver

Allman

2019

Preprint

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For any acyclic quiver, we establish a family of structure isomorphisms for its cohomological Hall algebra (CoHA). The family is parameterized by partitions of the quiver into Dynkin subquivers. For each such partition, we write the domain of our isomorphism as a tensor product of subalgebras in two ways. In the first, each tensor factor is isomorphic to the CoHA of the quiver with a single vertex and no arrows. In the second, the tensor factors are each isomorphic to the CoHAs of the corresponding Dynkin subquivers. When the quiver is already an orientation of a simply-laced Dynkin diagram, our results interpolate between isomorphisms proved by Rimányi. Such CoHA decompositions appear in prior work of Davison-Meinhardt and Franzen-Reineke, but our proof gives an explicit topological realization of these results. As a consequence of our method, we deduce that certain structure constants in the CoHA naturally arise as CoHA products of classes of Dynkin quiver polynomials.

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Geometric construction of generators of CoHA of doubled quiver

Cited by 5 publications

References 12 publications

Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groups

Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groups

On the Cohomological Hall Algebra of the Kronecker Quiver

A family of structure isomorphisms for the Cohomological Hall Algebra of an acyclic quiver

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