Framed cohomological Hall algebras and stable envelopes I
T. Botta. (2022)cite arxiv:2207.06280Comment: 34 pages.
Abstract
There are multiple conjectures relating the cohomological Hall algebras
(CoHAs) of certain substacks of the moduli stack of representations of a quiver
$Q$ to the Yangian $Y^Q_MO$ by Maulik-Okounkov, whose construction is based
on the notion of stable envelopes of Nakajima varieties. In this article, we
introduce the cohomological Hall algebra of the moduli stack of framed
representations of a quiver $Q$ (framed CoHA) and we show that the equivariant
cohomology of the disjoint union of the Nakajima varieties
$M_Q(v,w)$ for all dimension vectors $v$ and
framing vectors $w$ has a canonical structure of subalgebra of the
framed CoHA. Restricted to this subalgebra, the algebra multiplication is
identified with the stable envelope map. As a corollary, we deduce an explicit
inductive formula to compute stable envelopes in terms of tautological classes.
Description
Framed cohomological Hall algebras and stable envelopes I
%0 Generic
%1 botta2022framed
%A Botta, Tommaso Maria
%D 2022
%K Coha
%T Framed cohomological Hall algebras and stable envelopes I
%U http://arxiv.org/abs/2207.06280
%X There are multiple conjectures relating the cohomological Hall algebras
(CoHAs) of certain substacks of the moduli stack of representations of a quiver
$Q$ to the Yangian $Y^Q_MO$ by Maulik-Okounkov, whose construction is based
on the notion of stable envelopes of Nakajima varieties. In this article, we
introduce the cohomological Hall algebra of the moduli stack of framed
representations of a quiver $Q$ (framed CoHA) and we show that the equivariant
cohomology of the disjoint union of the Nakajima varieties
$M_Q(v,w)$ for all dimension vectors $v$ and
framing vectors $w$ has a canonical structure of subalgebra of the
framed CoHA. Restricted to this subalgebra, the algebra multiplication is
identified with the stable envelope map. As a corollary, we deduce an explicit
inductive formula to compute stable envelopes in terms of tautological classes.
@misc{botta2022framed,
abstract = {There are multiple conjectures relating the cohomological Hall algebras
(CoHAs) of certain substacks of the moduli stack of representations of a quiver
$Q$ to the Yangian $Y^{Q}_{MO}$ by Maulik-Okounkov, whose construction is based
on the notion of stable envelopes of Nakajima varieties. In this article, we
introduce the cohomological Hall algebra of the moduli stack of framed
representations of a quiver $Q$ (framed CoHA) and we show that the equivariant
cohomology of the disjoint union of the Nakajima varieties
$\mathcal{M}_Q(\text{v},\text{w})$ for all dimension vectors $\text{v}$ and
framing vectors $\text{w}$ has a canonical structure of subalgebra of the
framed CoHA. Restricted to this subalgebra, the algebra multiplication is
identified with the stable envelope map. As a corollary, we deduce an explicit
inductive formula to compute stable envelopes in terms of tautological classes.},
added-at = {2022-07-14T23:19:56.000+0200},
author = {Botta, Tommaso Maria},
biburl = {https://www.bibsonomy.org/bibtex/2711f9d9b9e0afb52abe1511a29717e39/dragosf},
description = {Framed cohomological Hall algebras and stable envelopes I},
interhash = {88bf967ed14b44f3d1185287e0f666e0},
intrahash = {711f9d9b9e0afb52abe1511a29717e39},
keywords = {Coha},
note = {cite arxiv:2207.06280Comment: 34 pages},
timestamp = {2022-07-14T23:19:56.000+0200},
title = {Framed cohomological Hall algebras and stable envelopes I},
url = {http://arxiv.org/abs/2207.06280},
year = 2022
}