The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra

Morton, H. R.; Samuelson, Peter

doi:10.1215/00127094-3718881

Cited by 40 publications

(59 citation statements)

References 35 publications

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“…On the other hand, by interpreting the affine Hecke algebra as the affine braid group modulo the HOMFLYPT skein relations, we construct a natural algebra map A → Sk q (T 2 ). We show in Theorem 3.25 that this map is injective, and by [MS14], this gives an algebra map A → E •,≥ . We define E + to be the image of this map, and we write ϕ + : E + → Tr(H) for the induced map.…”

Section: Introductionmentioning

confidence: 81%

“…Remark 3.26. This theorem implies combined with [MS14] shows that there are elements w a,b for a > 0 and b ∈ Z that generate Tr(AH) as an algebra and that satisfy the relations 3…”

Section: Theorem 316 ([Ms14]mentioning

confidence: 95%

“…Let Sk q (T 2 ) be the HOMFLYPT skein algebra of the torus, which is H 1 (T 2 ) = Z 2 -graded. We recall the following theorem from [MS14] (where their q 2 is our q).…”

Section: Traces Of Hecke Algebrasmentioning

confidence: 99%

“…We construct this isomorphism by relating a triangular piece of Tr(H) to the HOMFLYPT skein algebra Sk q (T 2 ) of the torus. This algebra comes from low dimensional topology and was related to the same specialization of E in [MS14]. Below we briefly describe these algebras, give a precise statement of the results, and provide a rough outline of the proof.…”

Section: Introductionmentioning

confidence: 99%

“…The algebra structure is given by stacking links on top of each other (in the [0, 1] direction). This algebra was described explicitly in [MS14], and it was shown to be isomorphic to the t = q specialization of E q,t (without the central extension). Under this isomorphism, the natural action of SL 2 (Z) on Sk q (T 2 ) agrees with the Burban-Schiffmann action on E q,t .…”

Section: Introductionmentioning

confidence: 99%

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The elliptic Hall algebra and the deformed Khovanov Heisenberg category

Cautis

Lauda

Licata

et al. 2018

Sel. Math. New Ser.

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We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined in [LS13]. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann [BS12], specialized at σ =σ −1 = q. A key step in the proof may be of independent interest: we show that the sum (over n) of the Hochschild homologies of the positive affine Hecke algebras AH + n is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the q-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.

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Section: Introductionmentioning

confidence: 81%

“…Remark 3.26. This theorem implies combined with [MS14] shows that there are elements w a,b for a > 0 and b ∈ Z that generate Tr(AH) as an algebra and that satisfy the relations 3…”

Section: Theorem 316 ([Ms14]mentioning

confidence: 95%

“…Let Sk q (T 2 ) be the HOMFLYPT skein algebra of the torus, which is H 1 (T 2 ) = Z 2 -graded. We recall the following theorem from [MS14] (where their q 2 is our q).…”

Section: Traces Of Hecke Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

Cautis

Lauda

Licata

et al. 2018

Sel. Math. New Ser.

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Basics of Skein Modules

Przytycki,

Bakshi,

Ibarra

et al. 2023

Universitext

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Perverse Sheaves with Nilpotent Singular Support on the Stack of Coherent Sheaves on an Elliptic Curve

Hennecart

2022

Transformation Groups

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We define a stratification of the moduli stack of coherent sheaves on an elliptic curve which allows us (1) to give an explicit description of the irreducible components of the global nilpotent cone of elliptic curves, (2) to establish an explicit bijection between the simple objects of the category of perverse sheaves defined by Schiffmann to categorify the elliptic Hall algebra (the so-called spherical Eisenstein sheaves) and the irreducible components of the global nilpotent cone and (3) to give an explicit description and parametrization of the perverse sheaves on the moduli stack of coherent sheaves on an elliptic curve having nilpotent singular support. Along the way, we find a combinatorial parametrization of the irreducible components of the semistable locus of the elliptic global nilpotent cone. The comparison with Bozec’s parametrization leads to an interesting combinatorial problem.

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The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra

Cited by 40 publications

References 35 publications

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

Basics of Skein Modules

Perverse Sheaves with Nilpotent Singular Support on the Stack of Coherent Sheaves on an Elliptic Curve

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