Braid actions on quantum toroidal superalgebras

Bezerra, Luan; Mukhin, E.

doi:10.48550/arxiv.1912.08729

Cited by 6 publications

(11 citation statements)

References 9 publications

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“…In [BM1], we studied representations of E s where K = 1 and C was not one. Thus, one can think that our generators of E s in this paper correspond to images of generators of [BM1] under the Miki automorphism, which exchanges K and C, see Theorem 5.9 in [BM2]. In particular, the vertical and horizontal subalgebras in this paper correspond to the horizontal and vertical subalgebras, respectively, in [BM1].…”

Section: 2mentioning

confidence: 79%

“…Definition of E s . The quantum toroidal superalgebra associated with gl m|n was introduced in [BM1] with standard parity, and in [BM2] for any choice of parity.…”

Section: 2mentioning

confidence: 99%

“…The subalgebra of E s generated by E i (z), F i (z), K ± i (z), i ∈ I, is isomorphic to U q sl s at level zero (that is, with central element c = 1, see for example Section 2.2 in [BM2] for the description of sl s ). We denote this subalgebra by U ver q sl s and call it the vertical quantum affine sl m|n .…”

Section: 2mentioning

confidence: 99%

“…The general quantum toroidal superalgebra of [BM2] has two central elements, K and C. In this paper, we set C = 1 since this happens in all representations we will be considering. In [BM1], we studied representations of E s where K = 1 and C was not one.…”

Section: 2mentioning

confidence: 99%

“…The superalgebras E s (with m = n) for arbitrary parity s were introduced in [BM2]. Fock representations of E s in the standard parity are given in [BM1].…”

Section: Introductionmentioning

confidence: 99%

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Representations of quantum toroidal superalgebras and plane $\mathbf{s}$-partitions

Bezerra,

Mukhin

2021

Preprint

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Section: 2mentioning

confidence: 79%

“…Definition of E s . The quantum toroidal superalgebra associated with gl m|n was introduced in [BM1] with standard parity, and in [BM2] for any choice of parity.…”

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…The superalgebras E s (with m = n) for arbitrary parity s were introduced in [BM2]. Fock representations of E s in the standard parity are given in [BM1].…”

Section: Introductionmentioning

confidence: 99%

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Representations of quantum toroidal superalgebras and plane $\mathbf{s}$-partitions

Bezerra,

Mukhin

2021

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Gauge/Bethe correspondence from quiver BPS algebras

Galakhov

Yamazaki

2022

J. High Energ. Phys.

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We study the Gauge/Bethe correspondence for two-dimensional $$ \mathcal{N} $$ N = (2, 2) supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we find that the Bethe ansatz equations for the crystal chain coincide with the vacuum equation of the quiver gauge theory, thus confirming the corresponding Gauge/Bethe correspondence. For more general chiral quivers, however, we find obstructions to the R-matrices satisfying the Yang-Baxter equations and the unitarity conditions, and hence to their corresponding Gauge/Bethe correspondence. We also discuss trigonometric (quantum toroidal) versions of the quiver BPS algebras, which correspond to three-dimensional $$ \mathcal{N} $$ N = 2 gauge theories and arrive at similar conclusions. Our findings demonstrate that there are important subtleties in the Gauge/Bethe correspondence, often overlooked in the literature.

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K-theoretic Hall algebras, quantum groups and super quantum groups

Varagnolo¹,

Vasserot²

2020

Preprint

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We prove that the K-theoretic Hall algebra of a preprojective algebra of affine type is isomorphic to the positive half of a quantum toroidal quantum group. An essential step consists to deform the K-theoretic Hall algebra so that the deformation is torsion free over some polynomial subalgebra. Next, we compare super toroidal quantum groups of type A with K-theoretic Hall algebras of quivers with potential, which are defined using the Grothendieck groups of categories of singularities of some Landau-Ginzburg models. K-THEORETIC HALL ALGEBRAS AND SUPER QUANTUM GROUPS 3. K-theoretic Hall algebras and quantum superalgebras 29 3.1. Quantum superalgebras 29 3.2. K-theoretic Hall algebras and quantum superalgebras 30 Appendix A. Proof of Lemma 2.5.1 35 Appendix B. Cohomological Hall algebra 36 B.1. The Yangian of g Q 36 B.2. The Yangian of p slpm|nq 37 References 39

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Braid actions on quantum toroidal superalgebras

Cited by 6 publications

References 9 publications

Representations of quantum toroidal superalgebras and plane $\mathbf{s}$-partitions

Representations of quantum toroidal superalgebras and plane $\mathbf{s}$-partitions

Gauge/Bethe correspondence from quiver BPS algebras

K-theoretic Hall algebras, quantum groups and super quantum groups

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