We define a monoidal category and a closely related 2‐category using diagrammatic methods. We show that acts on the category of modules over Temperley–Lieb algebras, with its generating 1‐morphisms acting by induction and restriction. The Grothendieck groups of and a third category we define are closely related to the Weyl algebra. We formulate a sense in which acts asymptotically on .

In , Finkelberg and Tsymbaliuk introduced the notion of shifted quantum affine algebras and described their role in the study of quantized Coulomb branches associated to certain 3D N = 4 quiver gauge theories. We describe a new geometric construction of a deformation of one of these shifted quantum affine algebras as the Hall algebra of the category of representations of a certain quiver Q_Rud (modulo relations). This quiver first arose in the work of Rudakov in the study of the tame blocks of the category of restricted representations of the Lie algebra 𝔰𝔩_2(𝔽_q).

We define a monoidal category 𝐖 and a closely related 2-category 2𝐖𝐞𝐲𝐥 using diagrammatic methods. We show that 2𝐖𝐞𝐲𝐥 acts on the category 𝐓𝐋 :=⊕_n TL_n-mod of modules over Temperley-Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of 𝐖 and a third category we define 𝐖^∞ are closely related to the Weyl algebra. We formulate a sense in which K_0(𝐖^∞) acts asymptotically on K_0(𝐓𝐋).

We give a presentation of the Kauffman (BMW) skein algebra of the torus. This algebra is the “type $BCD$” analogue of the Homflypt skein algebra of torus, which was computed in earlier work of the 1st and 3rd authors [17]. This suggests the existence of a “type $BCD$” version of the Hall algebra of an elliptic curve [4]. In the appendix, we show this presentation is compatible with the Frohman–Gelca description of the Kauffman bracket (Temperley–Lieb) skein algebra of the torus [12].

We give a skein-theoretic realization of the $\mathfrak{gl}_n$ double affine Hecke algebra of Cherednik using braids and tangles in the punctured torus. We use this to provide evidence of a relationship we conjecture between the classical skein algebra of the punctured torus and the elliptic Hall algebra of Burban and Schiffmann.

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the $t=q$ specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.

We refine and prove the central conjecture of our first paper for annuli with at least two marked intervals on each boundary component by computing the derived Hall algebras of their Fukaya categories.

In (Compos. Math. 152(7): 1333–1384, 2016), Berest and Samuelson proposed a conjecture that the Kauffman bracket skein module of any knot in $$S^3$$ carries a natural action of a rank 1 double-affine Hecke algebra $$SH_{q,t_1, t_2}$$ depending on 3 parameters $$q, t_1, t_2$$ . As a consequence, for a knot K satisfying this conjecture, we defined a three-variable polynomial invariant $$J^K_n(q,t_1,t_2)$$ generalizing the classical coloured Jones polynomials $$J^K_n(q)$$ . In this paper, we give explicit formulas and provide a quantum group interpretation for the polynomials $$J^K_n(q,t_1,t_2)$$ . Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by Habiro (Invent. Math. 171(1): 1–81, 2008) : as in the classical case, they imply the integrality of $$J^K_n(q,t_1,t_2)$$ and, in fact, make sense for an arbitrary knot K independent of whether or not it satisfies the conjecture of Berest and Samuelson (Compos. Math. 152(7): 1333–1384, 2016). When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of $$J^K_n(q,t_1)$$ are expressed in terms of Macdonald orthogonal polynomials.

We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with m marked intervals and we give a conjectural description of the Hall algebras of all surfaces with enough marked intervals. Then we use a functoriality result to show that a graded version of the HOMFLY-PT skein relation holds among certain arcs in the Hall algebras of general surfaces.

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We give a topological realization of the (spherical) double affine Hecke algebra SH(q,)t of type sl(2), and we use this to construct a module over SHq,t for any knot K subset of S-3. As an application, we give a purely topological interpretation of Cherednik's two-variable polynomials P-n(r, s; q, t) of type sl(2) from [14] (where r, s is an element of Z are relatively prime). We then generalize the construction of these polynomials (for sl(2)) from torus knots to all iterated cables of the unknot and prove they specialize to the colored Jones polynomials of the knot. Finally, in the Appendix we compare this construction to a later construction of Cherednik and Danilenko.

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined in  Licata and Savage (Quantum Topol 4(2):125–185, 2013. arXiv:1009.3295). 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 (Duke Math J 161(7):1171–1231, 2012. arXiv:math/0505148), specialized at \(\sigma = {\bar{\sigma }}^{-1} = q\). 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 \(\mathrm{AH}_n^+\) 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.

It is known that the fundamental group homomorphism π1(T2)→π1(S3∖K) induced by the inclusion of the boundary torus into the complement of a knot K in S3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL2-character varieties of the corresponding fundamental groups. In our earlier work [3], we proposed a conjecture that the classical restriction map admits a canonical deformation into a two-parameter family of affine cubic surfaces in C3. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [1] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel–Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel–Hilden conjecture holds is closed under taking connect sums and knot coverings.

We give a generators and relations presentation of the HOMFLYPT skein algebra H of the torus T-2, and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that H is isomorphic to the sigma = (sigma) over bar (-1) specialization of the elliptic Hall algebra of Burban and Schiffmann.As an application, for an iterated cable K of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the lambda-colored HOMFLYPT polynomial of K. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko using the jlN double affine Hecke algebra. This proves one of the connection conjectures in their recent work.

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$ . We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$ ). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

Let S be a connected and locally 1-connected space, and let M subset of S. A decorated SL2(C)-local system is an SL2(C)-local system on S, together with a chosen element of the stalk at each component of M.We study the decorated SL2(C)-character algebra of (S, M): the algebra of poly-nomial invariants of decorated SL2(C)-local systems on (S, M). The character algebra is presented explicitly. The character algebra is shown to correspond to the C-algebra spanned by collections of oriented curves in S modulo local topological rules.As an intermediate step, we obtain an invariant-theory result of independent interest: a presentation of the algebra of SL2(C)-invariant functions on End (V)(m) circle plus V-n, where V is the tautological representation of SL2(C).

These are lecture notes of a minicourse given by the first author at the Summer School on Quantization at the University of Notre Dame in June 2011. The notes were written up and expanded by the second author who took the liberty of adding a few interesting results and proofs from the literature. In a broad sense, our goal is to give an introduction to representation theory of rational Cherednik algebras and some of its recent applications. More specifically, we focus on the two concepts featuring in the title (Dunkl operators and quasi-invariants) and explain the relation between them. The course was originally designed for graduate students and nonexperts in representation theory. In these notes, we tried to preserve an informal style, even at the expense of making imprecise claims and sacrificing rigor.

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We construct CAT(0) spaces on which various free-by-cyclic groups act. Let G be the free-by-cyclic group F(a,c1,…,cn,d)⋊φZ with φ determined by ϕ(a)=a, ϕ(cj)=cjaκ, and ϕ(d)=dw, where w is some word not containing d. Our main result is that if the exponent sum of cn in w is non-zero, then there is a CAT(0) metric space on which G acts properly discontinuously and cocompactly by isometries.