22 research outputs found
Shuffle algebras for quivers and wheel conditions
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 Negu\c{t}-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.Comment: The main theorem is substantially the same, but we added Section 4 on
an important variant of the construction which admits a Hopf algebra
structure, and Section 5 on non-generic parameters. Several applications are
discussed in Subsections 1.3 and 1.
Shuffle algebras for quivers as quantum groups
We define a quantum loop group associated to an arbitrary
quiver and maximal set of deformation parameters, with generators
indexed by and some explicit quadratic and cubic
relations. We prove that is isomorphic to the (generic, small)
shuffle algebra associated to the quiver and hence, by [Neg21a], to the
localized K-theoretic Hall algebra of . For the quiver with one vertex and
loops, this yields a presentation of the spherical Hall algebra of a
(generic) smooth projective curve of genus (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]).Comment: Added Section 5 (concerning special values of the parameters) and
Section 7 (on the Hall algebra of an arbitrary curve
Quantum loop groups for arbitrary quivers
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 include -theoretic
Hall algebras of quivers with 0 potential, quantum loop groups of Kac-Moody
type and quiver quantum toroidal algebras
Homomorphisms between different quantum toroidal and affine Yangian algebras
This paper concerns the relation between the quantum toroidal algebras and
the affine Yangians of , denoted by
and ,
respectively. Our motivation arises from the milestone work of Gautam and
Toledano Laredo, where a similar relation between the quantum loop algebra
and the Yangian has been established
by constructing an isomorphism of -algebras
(with standing for the
appropriate completions). These two completions model the behavior of the
algebras in the formal neighborhood of . The same construction can be
applied to the toroidal setting with for .
In the current paper, we are interested in the more general relation:
, where and
is an -th root of . Assuming is a
primitive -th root of unity, we construct a homomorphism
from the completion of the formal version of
to the completion
of the formal version of . We
propose two proofs of this result: (1) by constructing the compatible
isomorphism between the faithful representations of the algebras; (2) by
combining the direct verification of Gautam and Toledano Laredo for the
classical setting with the shuffle approach.Comment: v2: 30 pages, significant modifications from the previous version,
minor mistakes corrected. v3: Published version, 30 pages, minor corrections,
some details adde
Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes
Multi-loop scattering amplitudes in N=4 Yang-Mills theory possess cluster
algebra structure. In order to develop a computational framework which exploits
this connection, we show how to construct bases of Goncharov polylogarithm
functions, at any weight, whose symbol alphabet consists of cluster coordinates
on the cluster algebra. Using such a basis we present a new expression
for the 2-loop 6-particle NMHV amplitude which makes some of its cluster
structure manifest.Comment: 32 pages; v2: minor corrections and clarification
Generators of the preprojective CoHA of a quiver
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
Quantum loop groups for symmetric Cartan matrices
We introduce a quantum loop group associated to a general symmetric Cartan
matrix, by imposing just enough relations between the usual generators
in order for the natural
Hopf pairing between the positive and negative halves of the quantum loop group
to be perfect. As an application, we obtain a description of the localized
K-theoretic Hall algebra of any quiver without loops, endowed with a
particularly important action
Categorical and K-theoretic Donaldson-Thomas theory of (part II)
Quasi-BPS categories appear as summands in semiorthogonal decompositions of
DT categories for Hilbert schemes of points in the three dimensional affine
space and in the categorical Hall algebra of the two dimensional affine space.
In this paper, we prove several properties of quasi-BPS categories analogous to
BPS sheaves in cohomological DT theory.
We first prove a categorical analogue of Davison's support lemma, namely that
complexes in the quasi-BPS categories for coprime length and weight are
supported over the small diagonal in the symmetric product of the three
dimensional affine space. The categorical support lemma is used to determine
the torsion-free generator of the torus equivariant K-theory of the quasi-BPS
category of coprime length and weight.
We next construct a bialgebra structure on the torsion free equivariant
K-theory of quasi-BPS categories for a fixed ratio of length and weight. We
define the K-theoretic BPS space as the space of primitive elements with
respect to the coproduct. We show that all localized equivariant K-theoretic
BPS spaces are one dimensional, which is a K-theoretic analogue of the
computation of (numerical) BPS invariants of the three dimensional affine
space.Comment: 45 pages, to appear in Forum Math. Sigm