arXiv subjects
Publications and source records attributed to Andrei Neguţ.
We formulate a connection between a topological and a geometric category. The former is the idempotent completion of the (horizontal) trace of the affine Hecke category, while the latter is the equivariant derived category of the (semi-nilpotent) commuting stack. This provides a more precise and improved version of our proposal in [13].
We survey the well-known Yangian of $\widehat{\mathfrak{gl}}_1$ /quantum toroidal $\mathfrak{gl}_1$ action on the cohomology / $K$-theory of moduli spaces of stable sheaves on surfaces, and give the generalization of this construction to elliptic cohomology.
We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We prove that $\mathbf{U}^+_Q$ is isomorphic to the (generic, small) shuffle algebra associated to the quiver $Q$ and hence, by [Neg21a], to the localized K-theoretic Hall algebra of $Q$. For the quiver with one vertex and $g$ loops, this yields a presentation of the spherical Hall algebra of a (generic) smooth projective curve of genus $g$ (invoking the results of [SV12]). We extend the above results to the case of non-generic parameters satisfying a certain natural metric condition. As an application, we obtain a description by generators and relations of the subalgebra generated by absolutely cuspidal eigenforms of the Hall algebra of an arbitrary smooth projective curve (invoking the results of [KSV17]).
We study PBW bases of the untwisted quantum loop group $U_q(L\mathfrak{g})$ (in the Drinfeld new presentation) using the combinatorics of loop words, by generalizing the treatment of [29,30,43] in the finite type case. As an application, we prove that Enriquez' homomorphism [11] from the positive half of the quantum loop group to the trigonometric degeneration of Feigin-Odesskii's elliptic algebra [15] associated to $\mathfrak{g}$ is an isomorphism.
As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation of the quantum affine group and its usual R-matrix. An upshot of this construction is a new topological coproduct on the quantum toroidal algebra which extends the Drinfeld-Jimbo coproduct on the horizontal quantum affine subalgebra.
We study the dual constructions of quantum loop groups and Feigin-Odesskii type shuffle algebras for an arbitrary quiver, for which the arrow parameters are arbitrary non-zero elements of any field. Examples of our setup include $K$-theoretic Hall algebras of quivers with 0 potential, quantum loop groups of Kac-Moody type and quiver quantum toroidal algebras.
In this short note, we refine a result of Schiffmann-Vasserot, by showing that the localized preprojective cohomological Hall algebra of any quiver is spherical, i.e. generated by elements of minimal dimension.
We consider the PBW basis of the type A quantum toroidal algebra developed by the author, and prove commutation relations between its generators akin to the ones studied by Burban-Schiffmann for n=1. This gives rise to a new presentation of the quantum toroidal algebra of type A.
We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of Varagnolo-Vasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized K-theoretic Hall algebra associated to the quiver by Schiffmann-Vasserot. With small modifications, our theorems also hold under certain specializations of the equivariant parameters, which will allow us in Neguţ-Sala-Schiffmann to give a generators-and-relations description of the Hall algebra of any curve over a finite field (which is a shuffle algebra due to Kapranov-Schiffmann-Vasserot). When the quiver has no edge loops or multiple edges, we show that the shuffle algebra, localized K-theoretic Hall algebra, and the positive half of the corresponding quantum loop group are all isomorphic; we also obtain the non-degeneracy of the Hopf pairing on the latter quantum loop group.
We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of $\mathfrak{gl}_n$, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e. a particular $\mathbb{Z}[q_1^{\pm 1}, q_2^{\pm 1}]$-submodule of the equivariant $K$-theory of a point) and the second as the $\mathbb{Z}[q_1^{\pm 1}, q_2^{\pm 1}]$-algebra generated by certain elements $\{\bar{H}_{n,d}\}_{(n,d) \in \mathbb{N} \times \mathbb{Z}}$.
We consider a natural generalization of the Carlsson-Okounkov Ext operator on the K-theory groups of the moduli spaces of stable sheaves on a smooth projective surface. We compute the commutation relations between the Ext operator and the action of the deformed W-algebra on K-theory, which was developed in [18]. The conclusion is that the Ext operator is closely related with a vertex operator, thus giving a mathematical incarnation of the Alday-Gaiotto-Tachikawa correspondence for a general algebraic surface.
We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.
We define functors on the derived category of the moduli space M of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the K-theory of the moduli space M, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.
We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an "affine" version of the construction of [14]. Explicitly, we show that the aforementioned trace is generated by the objects $E_{\textbf{d}} = \text{Tr}(Y_1^{d_1} \dots Y_n^{d_n} T_1 \dots T_{n-1})$ as $\textbf{d} = (d_1,\dots,d_n) \in \mathbb{Z}^n$, where $Y_i$ denote the Wakimoto objects of [9] and $T_i$ denote Rouquier complexes. We compute certain categorical commutators between the $E_{\textbf{d}}$'s and show that they match the categorical commutators between the sheaves $\mathcal{E}_{\textbf{d}}$ on the flag commuting stack, that were considered in [27]. At the level of $K$-theory, these commutators yield a certain integral form $\widetilde{\mathcal{A}}$ of the elliptic Hall algebra, which we can thus map to the $K$-theory of the trace of the affine Hecke category.
We consider the problem of the R-matrix of the quantum toroidal algebra $U_{q,t}(\ddot{\mathfrak{gl}}_1)$ in the Fock representation. Using the connection between the R-matrix $R(u)$ ($u$ being the spectral parameter) and the theory of Macdonald operators we obtain explicit formulas for $R(u)$ in the operator and matrix forms. These formulas are expressed in terms of the eigenvalues of a certain Macdonald operator which completely describe the functional dependence of $R(u)$ on the spectral parameter $u$. We then consider the geometric R-matrix (obtained from the theory of K-theoretic stable bases on moduli spaces of framed sheaves), which is expected to coincide with $R(u)$ and thus gives another approach to the study of the poles of the R-matrix as a function of u.
We consider the algebra of Hecke correspondences (elementary transformations at a single point) acting on the algebraic K-theory groups of the moduli spaces of stable sheaves on a smooth projective surface S. We derive quadratic relations between the Hecke correspondences, and compare the algebra they generate with the Ding-Iohara-Miki algebra (at a suitable specialization of parameters), as well as with a generalized shuffle algebra.
We define an integral form of the deformed W-algebra of type gl_r, and construct its action on the K-theory groups of moduli spaces of rank r stable sheaves on a smooth projective surface S, under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W-algebras by generators and relations is a new feature. Physically, this action encodes the AGT correspondence for 5d supersymmetric gauge theory on S x circle.
We construct an explicit, multiplicative Chow-Künneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga-Lunts-Verbitsky Lie algebra.