# Affine BPS algebras, W algebras, and the cohomological Hall algebra of
$\mathbb{A}^{2}$

Ben Davison  School of Mathematics
University of Edinburgh
Edinburgh EH9 3FD
UK<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce affinizations and deformations of the BPS Lie
algebra associated to a tripled quiver with potential, and use them to
precisely determine the $T$-equivariant cohomological Hall algebra
$\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}^{T}$ of compactly supported
coherent sheaves on $\mathbb{A}_{\mathbb{C}}^{2}$, acted on by a torus $T$. In
particular we show that this algebra is spherically generated for all $T$.

In honour of Ezra Getzler’s 60th birthday

##  1\. Introduction

###  1.1. Hall algebras

Let $\mathfrak{M}_{d}(\mathbb{C}[x,y])$ denote the moduli stack of coherent
sheaves of length $d$ on $\mathbb{A}^{2}\coloneqq\mathbb{A}^{2}_{\mathbb{C}}$.
We also consider the quotient $\mathfrak{M}^{T}_{d}(\mathbb{C}[x,y])$, by the
induced action of various tori $T$ acting linearly on the two coordinates of
$\mathbb{A}^{2}$. We consider the Borel–Moore homology of the stack

$\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}^{T}\coloneqq\bigoplus_{d\in\mathbb{Z}_{\geq
0}}\operatorname{\mathbf{H}}^{\operatorname{BM}}(\mathfrak{M}^{T}_{d}(\mathbb{C}[x,y]),\mathbb{Q}),$

for various $T$, which carries the Hall algebra product defined in [SV13].
This algebra plays a key intermediate role in the geometric representation
theory of $\operatorname{Hilb}_{d}(\mathbb{A}^{2})$ [Nak97, Gro96],
$W$-algebras [SV13, Mik07, AS13], Yangians and Hall algebras [RSYZ20b,
RSYZ20a, LY20, GY20]. Setting $T=\\{1\\}$, the resulting algebra is important
in the study of the cohomological Hall algebra of zero-dimensional coherent
sheaves for an arbitrary smooth surface [KV19], as well as partially
categorifying the degree zero motivic Donaldson–Thomas theory of
$\mathbb{A}^{3}$, as studied in [BBS13]. In this paper we fully describe this
algebra, for various $T$, via the introduction and study of deformed,
affinized, BPS Lie algebras $\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}^{T}$ of
tripled quivers with their canonical potentials, extending the original BPS
Lie algebra $\mathfrak{g}_{\tilde{Q},\tilde{W}}$ of [DM20].

###  1.2. Donaldson–Thomas theory

Donaldson–Thomas theory [Tho00], is a theory that assigns to a projective
Calabi–Yau threefold $X$, a Chern class
$\upalpha\in\operatorname{\mathbf{H}}(X,\mathbb{Z})$, and a stability
condition $\upzeta$, a number $\omega^{\upzeta}_{\upalpha}$ “counting”
$\upzeta$-semistable sheaves on $X$ of Chern class $\upalpha$. The inverted
commas are due to the fact that the moduli space
$\mathcal{M}_{\upalpha}^{\upzeta}(X)$ of such sheaves will generally have
strictly positive dimension, i.e. there are infinitely many such sheaves, and
so $\omega_{\upalpha}^{\upzeta}$ is defined by first constructing a virtual
fundamental class of the right expected dimension (i.e. zero) and then taking
its degree.

Subsequently it was realised by Kai Behrend [Beh09] that this virtual count
could be computed as the Euler characteristic of
$\mathcal{M}_{\upalpha}^{\upzeta}(X)$ weighted by a certain constructible
function $\upnu_{X}$. Furthermore, in the event that $X$ can be written111For
arbitrary moduli spaces of stable coherent sheaves on Calabi–Yau threefolds,
this global critical locus description is too much to ask for, but can still
be achieved locally: see [BBJ19, BBBBJ15]. as the critical locus of a function
$f$ on a smooth ambient variety $Y$, this weighted Euler characteristic is
equal to the Euler characteristic of the vanishing cycle cohomology
$\operatorname{\mathbf{H}}(Y,{}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q})$.
Firstly, these facts give us a way to define DT invariants for noncompact
moduli spaces, and secondly, they suggest a natural “categorification” of the
DT invariant: instead of considering the number
$\upchi(\operatorname{\mathbf{H}}(Y,{}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q}))=\sum_{i\in\mathbb{Z}}(-1)^{i}\dim(\operatorname{\mathbf{H}}^{i}(Y,{}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q}))$,
we may study the cohomology
$\operatorname{\mathbf{H}}(Y,{}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q})$
itself. In particular, being a vector space, and not a mere number, this
cohomology is expected to have (and indeed does have, in cases of interest in
this paper) a rich algebraic structure.

###  1.3. Noncommutative Donaldson–Thomas theory

Associated to a quiver with potential
$Q,W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]_{\operatorname{cyc}}$, we may
define the Jacobi algebra $\operatorname{Jac}(Q,W)$ as in §3.2 below. This is
a kind of noncommutative threefold, in the sense that the category of finite-
dimensional modules over $\operatorname{Jac}(Q,W)$ behaves in many ways like
the category of compactly supported coherent sheaves on a 3CY variety222I.e. a
smooth three-dimensional variety $X$ satisfying
$\mathscr{O}_{X}\cong\omega_{X}$.. We refer to [Gin06] for background on this
point of view. Furthermore, the moduli stack of finite-dimensional
$\operatorname{Jac}(Q,W)$-modules is realised as the critical locus of the
function $\operatorname{Tr}(W)$ on the smooth stack of $\mathbb{C}Q$-modules.
From the above discussion, it is thus natural to study the vanishing cycle
cohomology

$\operatorname{\mathcal{A}}_{Q,W}\coloneqq\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}(\mathfrak{M}_{\mathbf{d}}(Q),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}})$

a categorification of the noncommutative DT theory associated to the pair
$(Q,W)$, as initiated in [Sze08] in the case of the noncommutative conifold.
The cohomology $\operatorname{\mathcal{A}}_{Q,W}$ carries an action of
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\coloneqq\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*},\mathbb{Q})$
by tensoring with tautological classes. Furthermore, the object
$\operatorname{\mathcal{A}}_{Q,W}$ carries the cohomological Hall algebra
structure introduced by Kontsevich and Soibelman [KS11].

The connection with the algebra
$\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}^{T}$ comes from setting $Q^{(3)}$
to be the quiver with one vertex and three loops $a,a^{*},\omega$, and
considering the potential

$\tilde{W}=\omega[a,a^{*}].$

Then there is a natural isomorphism of Hall algebras (see §3.9)

$\operatorname{\mathcal{A}}_{Q^{(3)},\tilde{W}}\cong\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}.$

If $T$ scales the $x,y$ directions of $\mathbb{A}^{2}$ with weights
$w_{1},w_{2}$ respectively, we may consider the $T$-action on $Q^{(3)}$ that
scales the arrows $a,a^{*},\omega$ with weights $w_{1},w_{2},-w_{1}-w_{2}$,
and there is a similar isomorphism of deformed CoHAs
$\operatorname{\mathcal{A}}_{Q^{(3)},\tilde{W}}^{T}\cong\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}^{T}$.

Via a purity result of [Dav17b], the graded dimensions of
$\operatorname{\mathcal{A}}_{Q^{(3)},\tilde{W}}$ precisely encode the refined
DT invariants of the Jacobi algebra
$\operatorname{Jac}(Q^{(3)},\tilde{W})\cong\mathbb{C}[x,y,z]$ (equivalently,
the DT theory of coherent sheaves on $\mathbb{A}^{3}$) as defined in [KS08,
KS11], and calculated in [BBS13]. This means that the DT theory of
$\operatorname{Jac}(Q^{(3)},\tilde{W})$ enables us to calculate the
characteristic function encoding the graded dimensions of
$\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}$ (and from there, using purity
again, the characteristic function of
$\operatorname{\mathcal{A}}^{T}_{\mathbb{C}^{2}}$ — see Proposition 3.5). More
than this, the cohomological DT theory of the category of
$\operatorname{Jac}(Q^{(3)},\tilde{W})$-modules, and a thorough study of the
deformed affine BPS Lie algebra defined in this paper, enable us to precisely
determine $\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}^{T}$.

###  1.4. Results

The algebra $W_{1+\infty}$ is defined to be the universal central extension of
the Lie algebra of algebraic differential operators on the complex torus
$\mathbb{C}^{*}$. A basis for $W_{1+\infty}$ is provided by the set

$\\{z^{m}D^{n}\;\lvert\;m\in\mathbb{Z},n\in\mathbb{Z}_{\geq 0}\\}\cup\\{c\\}$

where we set $D=z(d/dz)$, and $c$ accounts for the central extension, which
will not feature in this paper. We define

(1) $W_{1+\infty}^{+}=\mathrm{Span}(z^{m}D^{n}\;\lvert\;m\in\mathbb{Z}_{\geq
1},n\in\mathbb{Z}_{\geq 0}),$

which is a Lie subalgebra of $W_{1+\infty}$. Explicitly, the Lie bracket with
respect to this basis is given by

(2) $[z^{m}D^{a},z^{n}D^{b}]=z^{m+n}((D+n)^{a}D^{b}-D^{b}(D+m)^{b}).$

The Lie algebra $W^{+}_{1+\infty}$ is $\mathbb{Z}$-graded, with $n$th graded
piece spanned by the operators $z^{n}D^{a}$ for $a\geq 0$. The Lie algebra
$W^{+}_{1+\infty}$ has a filtration given by setting333There is a doubling of
degree in our definition, so that the associated graded object lives only in
even degrees, in order to match a later, cohomological grading of Hall
algebras.

$F_{i}W^{+}_{1+\infty}=\mathrm{Span}(z^{m}D^{a}\;\lvert\;a\leq(i+2)/2,m\in\mathbb{Z}_{\geq
1}).$

We denote by $\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W_{1+\infty}^{+}$ the
associated graded Lie algebra. From (2), the induced Lie bracket on
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W_{1+\infty}^{+}$ is nontrivial;
it is given explicitly by

(3) $[z^{m}D^{a},z^{n}D^{b}]=(an-bm)z^{m+n}D^{a+b-1}.$

So the associated graded Lie algebra
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W_{1+\infty}^{+}$ is isomorphic to
(part of) the quantum torus associated to the lattice $\mathbb{Z}^{2}$ with
skew-symmetric form $\omega((m,a),(n,b))=an-bm$.

######  Theorem A (Theorem 5.4 and Corollary 5.5).

There is an isomorphism of Lie algebras

$\hat{\mathfrak{g}}_{Q^{(3)},\tilde{W}}\cong\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!\operatorname{W}_{1+\infty}^{+}$

between the affine BPS Lie algebra444As defined in §3.10. for the pair
$(Q^{(3)},\tilde{W})$ and the associated graded Lie algebra of
$\operatorname{W}_{1+\infty}^{+}$ with respect to the order filtration. As a
result, there is an isomorphism of algebras

$\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}\cong\operatorname{\mathbf{U}}(\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!\operatorname{W}_{1+\infty}^{+}).$

The algebras $\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2}}$ (for various
$T$) have appeared in a great deal of work in geometric representation theory
in the last decade. In particular, care is usually taken to restrict to the
spherical subalgebra of $\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2}}$,
i.e. the algebra generated by the subspace
$\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2},1}$, corresponding to coherent
sheaves on $\mathbb{A}^{2}$ of length one. One of the main consequences of
Theorem A is the following

######  Theorem B (Theorem 5.6).

Let $T$ act on the quiver $Q^{(3)}$, leaving the potential $\tilde{W}$
invariant. Then the CoHA $\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}^{T}$ is
spherically generated, i.e. it is generated by the piece
$\operatorname{\mathcal{A}}_{\mathbb{A}^{2},1}^{T}$ corresponding to coherent
sheaves of length one.

It may come as a temporary disappointment to the reader that the associated
graded algebra of the filtered algebra $W^{+}_{1+\infty}$ appears in Theorem
A, as opposed to $W^{+}_{1+\infty}$ itself. This is, however, somewhat
inevitable: the algebra $\operatorname{\mathcal{A}}_{Q^{(3)},\tilde{W}}$ is
bigraded, with one grading coming from the decomposition of the stack of
finite-dimensional $\mathbb{C}[x,y]$-modules according to their dimension, and
one grading coming from cohomological degree. The Lie algebra
$W^{+}_{1+\infty}$, by contrast, is graded according to the exponent of $z$,
but only filtered according to the exponent of $D$.

To improve upon this situation, we consider the action of $\mathbb{C}^{*}$ on
$\mathbb{A}^{2}$ that acts by scaling the first coordinate. This induces a
$\mathbb{C}^{*}$-action on $\mathfrak{M}_{d}(\mathbb{C}[x,y])$ for all $d$,
and we take the equivariant Borel–Moore homology of this stack with respect to
this action, and define the Hall algebra

$\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}^{\mathbb{C}^{*}}\coloneqq\bigoplus_{d\in\mathbb{N}}\operatorname{\mathbf{H}}^{\operatorname{BM}}(\mathfrak{M}^{\mathbb{C}^{*}}_{d}(\mathbb{C}[x,y]),\mathbb{Q})$

to be this equivariant Borel–Moore homology, equipped with the usual
multiplication via correspondences. The algebra
$\mathbb{Q}[t]\cong\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}\coloneqq\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*},\mathbb{Q})$
acts on $\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}^{\mathbb{C}^{*}}$, with
the action induced by the $\mathbb{C}^{*}$-action on $\mathbb{A}^{2}$, and the
Hall algebra multiplication is
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-linear. We denote by

(4) $\operatorname{\mathbf{R}}_{F}[W_{1+\infty}^{+}]\coloneqq\sum_{i\geq
0}F_{i}W_{1+\infty}^{+}t^{i}\mathbb{Q}[t]\subset
W_{1+\infty}^{+}\otimes_{\mathbb{Q}}\mathbb{Q}[t]$

the Rees Lie algebra. This is a $\mathbb{Q}[t]$-linear Lie algebra, and so may
be thought of as a family of Lie algebras over the affine line, deforming the
algebra $\operatorname{\mathbf{Gr}}_{F}\\!W_{1+\infty}^{+}$ to
$W_{1+\infty}^{+}$: for $\epsilon\neq 0$ the specialisation at $t=\epsilon$ is
isomorphic to $W^{+}_{1+\infty}$, and the specialisation at $t=0$ is
isomorphic to $\operatorname{\mathbf{Gr}}^{F}_{\bullet}W^{+}_{1+\infty}$.

Let $A$ be a commutative ring. For $\mathfrak{g}$ a $A$-linear Lie algebra, we
define the tensor algebra

$T_{A}(\mathfrak{g})\coloneqq\bigoplus_{m\geq
0}\underbrace{\mathfrak{g}\otimes_{A}\ldots\otimes_{A}\mathfrak{g}}_{m\;\;\mathrm{times}}$

and define

$\operatorname{\mathbf{U}}_{A}(\mathfrak{g})=T_{A}(\mathfrak{g})/\langle\upalpha\\!\cdot\\!\upbeta-\upbeta\\!\cdot\\!\upalpha-[\upalpha,\upbeta]_{\mathfrak{g}}\;\lvert\;\upalpha,\upbeta\in\mathfrak{g}\rangle.$

######  Theorem C (Corollary 5.9).

Let $\mathbb{C}^{*}$ act on $Q^{(3)}$ with weights $1,0,-1$ on the arrows
$a,a^{*},\omega$ respectively. There is an isomorphism of
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}=\mathbb{Q}[t]$-linear Lie algebras

$\hat{\mathfrak{g}}_{Q^{(3)},\tilde{W}}^{\mathbb{C}^{*}}\cong\operatorname{\mathbf{R}}_{F}[W^{+}_{1+\infty}]$

between the deformed affine BPS Lie algebra for $(Q^{(3)},\tilde{W})$,
introduced in §3.10, and the deformed $W^{+}_{1+\infty}$-algebra. As a result,
there is an isomorphism of $\mathbb{Q}[t]$-linear algebras:

$\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}}\cong\operatorname{\mathbf{U}}_{\mathbb{Q}[t]}(\operatorname{\mathbf{R}}_{F}[W^{+}_{1+\infty}]).$

We denote by $\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+}$
the strictly positive part of the affine Yangian; a presentation is recalled
in §3.13. Let $T=(\mathbb{C}^{*})^{2}$ act on $\mathbb{A}^{2}$, by scaling the
two coordinates independently. Our final theorem is a strengthening of a
result of Rapčák, Soibelman, Yau and Zhao [RSYZ20b] (in fact it is an
immediate consequence of their result, along with spherical generation):

######  Theorem D (Theorem 5.8).

There is an isomorphism of $\operatorname{\mathbf{H}}_{T}$-linear algebras

$\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2}}\cong\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+}.$

###  1.5. Notation and conventions

  * •

Every cohomological Hall algebra that we consider throughout the paper will
carry a
$\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*},\mathbb{Q})$-action
arising from the morphism from the stack of finite-dimensional modules over an
algebra to $\operatorname{B\\!}\mathbb{C}^{*}$ given by taking the
determinant. We denote this copy of $\operatorname{B\\!}\mathbb{C}^{*}$ by
$\operatorname{B\\!}\mathbb{C}^{*}_{u}$, to distinguish it from other
instances of $\operatorname{B\\!}\mathbb{C}^{*}$ arising from e.g. torus
actions on quivers. We write
$\operatorname{\mathbf{H}}_{\mathbb{C}_{u}^{*}}\coloneqq\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\cong\mathbb{Q}[u]$,
with $u$ in cohomological degree $2$.

  * •

More generally, if $T=(\mathbb{C}^{*})^{l}$ is a torus, we set
$\operatorname{\mathbf{H}}_{T}\coloneqq\operatorname{\mathbf{H}}(\operatorname{B\\!}T,\mathbb{Q})$.

  * •

If a torus $T$ acts on a stack $\mathfrak{X}$ with quotient $\mathfrak{Y}$, we
set $({}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
i},{}^{\mathfrak{p}^{\prime}}\\!\uptau^{\geq i})$ to be the shift of the usual
perverse t structure on the derived category of constructible complexes on
$\mathfrak{Y}$, so that $\mathcal{F}$ is in the heart if and only if
$(\mathfrak{X}\rightarrow\mathfrak{Y})^{*}\mathcal{F}$ is perverse (see §3.1).

###  1.6. Acknowledgements

Thanks go to Kevin Costello, Brian Williams, Sasha Minets and Shivang Jindal
for helpful conversations about the results of this paper, and encouragement
to write them down correctly. Special thanks go to Ezra Getzler, who
encouraged me in this endeavour along with many others. Thanks are also due to
KIAS, and Minhyong Kim, for organizing an extremely stimulating stay in Seoul
in Summer 2022, where this paper was finally written.

##  2\. Algebraic background

###  2.1. Heisenberg algebras

We denote by $\operatorname{\mathbf{Heis}}$ the Lie algebra over $\mathbb{Q}$
having basis $\\{p,q,c\\}$, subject to the relations

$\displaystyle[q,p]=$ $\displaystyle c$ $\displaystyle[c,p]=[c,q]=$
$\displaystyle 0.$

We denote by $\operatorname{\mathbf{Heis}}\operatorname{-mod}$ the category of
modules over $\operatorname{\mathbf{Heis}}$. We give
$\operatorname{\mathbf{Heis}}$ a (cohomological) $\mathbb{Z}$-grading, putting
$p,c,q$ in degrees $2,0,-2$ respectively. A graded
$\operatorname{\mathbf{Heis}}$-module is a
$\operatorname{\mathbf{Heis}}$-module $\rho$, and a direct sum decomposition

$\rho=\bigoplus_{n\in\mathbb{Z}}\rho_{n}$

of the underlying vector space, such that $p$ maps $\rho_{n}$ to $\rho_{n+2}$,
$q$ maps $\rho_{n}$ to $\rho_{n-2}$, and $c$ preserves the decomposition. We
denote by $\operatorname{\mathbf{Heis}}\operatorname{-mod}_{\mathbb{Z}}$ the
category of graded $\operatorname{\mathbf{Heis}}$-modules. For
$\mathbf{c}\in\mathbb{Z}$ we say that a
$\operatorname{\mathbf{Heis}}$-representation $\rho$ has central charge
$\mathbf{c}$ if $c$ acts on $\rho$ via scalar multiplication by $\mathbf{c}$.

For a quiver $Q$ with vertex set $Q_{0}$, we define the Lie algebra

$\operatorname{\mathbf{Heis}}_{Q}\coloneqq\bigoplus_{i\in
Q_{0}}\operatorname{\mathbf{Heis}}.$

For $i\in Q_{0}$ we denote by $p_{i},q_{i},c_{i}$ the basis elements in
$\operatorname{\mathbf{Heis}}_{Q}$ corresponding to the $i$th summand. For
$\mathbf{c}\in\mathbb{Z}^{Q_{0}}$ we say that a
$\operatorname{\mathbf{Heis}}_{Q}$-representation $\rho$ has central charge
$\mathbf{c}$ if the $i$th copy of $\operatorname{\mathbf{Heis}}$ acts with
central charge $\mathbf{c}_{i}$. There is a canonical inclusion
$\operatorname{\mathbf{Heis}}\hookrightarrow\operatorname{\mathbf{Heis}}_{Q}$
sending $p\mapsto\sum_{i\in Q_{0}}p_{i}$, $q\mapsto\sum_{i\in Q_{0}}q_{i}$ and
$c\mapsto\sum_{i\in Q_{0}}c_{i}$. We will often consider a
$\operatorname{\mathbf{Heis}}_{Q}$-module as a
$\operatorname{\mathbf{Heis}}$-module via this inclusion. Placing all
$p_{i},q_{i},c_{i}$ in degrees $2,0,-2$ respectively, we give
$\operatorname{\mathbf{Heis}}_{Q}$ a $\mathbb{Z}$-grading, and define the
category $\operatorname{\mathbf{Heis}}_{Q}\operatorname{-mod}_{\mathbb{Z}}$ of
graded modules as before.

The vector space $\mathbb{Q}_{0}=\mathbb{Q}$ forms a
$\operatorname{\mathbf{Heis}}_{Q}$-module of central charge $0$ if we let all
$p_{i},q_{i},c_{i}$ act via the zero matrix. The categories
$\operatorname{\mathbf{Heis}}_{Q}\operatorname{-mod}$ and
$\operatorname{\mathbf{Heis}}_{Q}\operatorname{-mod}_{\mathbb{Z}}$ are
symmetric tensor categories with monoidal unit $\mathbb{Q}_{0}$. If
$\mathscr{C}$ is a tensor category, a $\mathscr{C}$-algebra object is an
object $\mathcal{F}$ in $\mathscr{C}$, along with morphisms
$\mathbf{1}_{\mathscr{C}}\rightarrow\mathcal{F}$ from the monoidal unit, and a
multiplication morphism $\mathcal{F}\otimes\mathcal{F}\rightarrow\mathcal{F}$,
satisfying the standard properties. If $\mathscr{C}$ is a symmetric tensor
category, a $\mathscr{C}$-Lie algebra object is an object $\mathcal{F}$ in
$\mathscr{C}$ along with an antisymmetric morphism
$\mathcal{F}\otimes\mathcal{F}\rightarrow\mathcal{F}$ satisfying the Jacobi
identity. If $\rho$ is a Lie algebra object in
$\operatorname{\mathbf{Heis}}_{Q}$, the universal enveloping algebra
$\operatorname{\mathbf{U}}(\rho)$ is an algebra object in
$\operatorname{\mathbf{Heis}}_{Q}$, with underlying algebra the universal
enveloping algebra of the underlying Lie algebra of $\rho$.

We say that a $\operatorname{\mathbf{Heis}}_{Q}$-module $\rho$ is integrable
if it decomposes

(5) $\rho=\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\rho_{[\mathbf{d}]}$

with each $\rho_{[\mathbf{d}]}$ a $\operatorname{\mathbf{Heis}}_{Q}$-module of
central charge $\mathbf{d}$.

Let $\operatorname{\mathbf{Heis}}_{Q}$ act on the $\mathbb{N}^{Q_{0}}$-graded
algebra $\mathcal{A}$, where the $\mathbb{N}^{Q_{0}}$-grading is induced by a
decomposition of $\mathcal{A}$ into sub-modules of fixed central charge, as in
(5). Then $\mathcal{A}$ is an algebra object in
$\operatorname{\mathbf{Heis}}_{Q}$ if each $p_{i},q_{i}$ acts by derivations
on $\mathcal{A}$. Similarly, if $\mathfrak{n}$ is a
$\mathbb{N}^{Q_{0}}$-graded Lie algebra, and also an integrable
$\operatorname{\mathbf{Heis}}_{Q}$-module for which the
$\mathbb{N}^{Q_{0}}$-graded decomposition is provided by the decomposition
according to central charge, then $\mathfrak{n}$ is a Lie algebra object in
$\operatorname{\mathbf{Heis}}_{Q}$ if and only if for every $i\in Q_{0}$,
$p_{i}$ and $q_{i}$ act by Lie algebra derivations.

###  2.2. The Lie algebra $W_{1+\infty}^{+}$

Recall from (1) the definition of $W^{+}_{1+\infty}\subset W_{1+\infty}$. The
Lie bracket on $W^{+}_{1+\infty}$ is given by

(6) $[z^{m}f(D),z^{n}g(D)]=z^{m+n}(f(D+n)g(D)-f(D)g(D+m)).$

######  Proposition 2.1.

There is an action of $\operatorname{\mathbf{Heis}}$ on $W^{+}_{1+\infty}$ by
derivations, given by sending $p$ to $[D^{2},-]$ and sending $q$ to the
endomorphism

$\displaystyle\partial_{D}\colon$ $\displaystyle W^{+}_{1+\infty}\rightarrow
W^{+}_{1+\infty}$ $\displaystyle z^{m}f(D)\mapsto z^{m}f^{\prime}(D).$

The subspace $\mathrm{Span}(z^{m}D^{n}\;\lvert\;m\in\mathbb{Z}_{\geq 1})$ is
preserved by this action, on which it has central charge $2m$.

###### Proof.

The fact that $[D^{2},-]$ is a derivation is obvious, while the fact that
$\partial_{D}$ is a derivation follows from the formula (6). Now we calculate

$\displaystyle[\partial_{D},[D^{2},-]](z^{m}D^{n})$
$\displaystyle=\partial_{D}[D^{2},z^{m}D^{n}]-[D^{2},nz^{m}D^{n-1}]$
$\displaystyle=[2D,z^{m}D^{n}]+[D^{2},nz^{m}D^{n-1}]-[D^{2},nz^{m}D^{n-1}]$
$\displaystyle=2z^{m}((D+m)D^{n}-D^{1+n})$ $\displaystyle=2mz^{m}D^{n}$

as required. ∎

For even $m$ we denote by $V_{m}\subset W^{+}_{1+\infty}$ the
$\operatorname{\mathbf{Heis}}$-submodule with central charge $m$. By the above
proposition, it has a basis given by elements $z^{m}D^{n}$. If we let
$\mathbb{Q}[u]$ act on $V_{m}$ by letting $u$ act via the raising operator
$p=[D^{2},-]$, then all of the $V_{m}$ are isomorphic to $\mathbb{Q}[u]$,
considered as $\mathbb{Q}[u]$-modules.

We endow $V_{m}$ with a filtration by setting

$\displaystyle
F_{i}V_{m}=\begin{cases}\mathrm{span}(z^{m},z^{m}D,\ldots,z^{m}D^{i/2+1})&\textrm{if
}i\textrm{ is even}\\\ F_{i-1}V_{m}&\textrm{if }i\textrm{ is odd.}\end{cases}$

So if $i$ is odd, the $i$th piece of the associated graded object
$\operatorname{\mathbf{Gr}}^{F}_{i}\\!V_{m}$ is zero, and if
$i\in\mathbb{Z}_{\geq-1}$ then a basis of
$\operatorname{\mathbf{Gr}}^{F}_{2i}\\!V_{m}$ is given by
$\\{z^{m}D^{i+1}\\}$. We refer to the natural grading on
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{m}$ as the cohomological
grading. Summing over the filtrations above, we obtain the order filtration
$F$ on $W^{+}_{1+\infty}$ considered in the introduction. The Lie bracket on
$W^{+}_{1+\infty}$ preserves the order filtration.

Recall that we give $\operatorname{\mathbf{Heis}}$ a cohomological grading by
putting $p,c,q$ in degrees $2,0,-2$ respectively. We have $pF_{i}V_{m}\subset
F_{i+2}V_{m}$ and $qF_{i}V_{m}\subset F_{i-2}V_{m}$, so that
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{m}$ becomes a graded
$\operatorname{\mathbf{Heis}}$-module, where we place the monomial
$z^{m}D^{n}$ in cohomological degree $2n-2$.

######  Lemma 2.2.

The Lie algebras $\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$
and $\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]$ are spherically generated:
they are generated by their respective
$\operatorname{\mathbf{Heis}}$-submodules of central charge $2$.

###### Proof.

Let $\mathfrak{g}\subset\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]$ be the
Lie subalgebra generated by
$\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]_{[2]}=\mathrm{Span}(zD^{a}\;\lvert\;a\geq
0)$. The identity (2) yields

$\displaystyle[zD^{a},z^{m}]=$ $\displaystyle
z^{m+1}((D+mt^{2})^{a}-D^{a})t^{-2}$ (7) $\displaystyle=$ $\displaystyle
z^{m+1}(maD^{a-1}+\textrm{lower order terms in }D).$

Putting $a=1$ and arguing inductively we deduce that $z^{m}\in\mathfrak{g}$
for every $m$. Then fixing $m$ and arguing by induction on $a$, it also
follows from (7) that
$z^{m}D^{a}\in\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]$ for every $m\geq 1$
and $a\geq 0$, and this algebra is spherically generated. Spherical generation
of $\operatorname{\mathbf{Gr}}_{F}\\!W^{+}_{1+\infty}$ follows directly from
(3), or from the spherical generation of
$\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]$ and the isomorphism

$\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]\otimes_{\mathbb{Q}[t]}(\mathbb{Q}[t]/(t))\cong\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}.$

∎

We record a variant of the result, for later. It follows from (3).

######  Lemma 2.3.

The elements $\mathbf{ad}_{zD}^{m}(zD^{n})$ for $m,n\geq 0$ provide a basis
for $W^{+}_{1+\infty}$.

The Lie algebra $\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$
carries two gradings, one by central charge, and one given by the
cohomological grading, and
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$ is a Lie algebra
object in $\operatorname{\mathbf{Heis}}\operatorname{-mod}_{\mathbb{Z}}$. It
turns out that this object satisfies a simple universal property: the Lie
algebra $\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$ is the
universal Lie algebra object in
$\operatorname{\mathbf{Heis}}\operatorname{-mod}_{\mathbb{Z}}$ containing
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}$, and containing only
representations $\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{m}$ for various
$m$ as subrepresentations (i.e. no copies of
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}V_{m}[j]$ with $j\neq 0$)). More
precisely,

######  Lemma 2.4.

If $\mathcal{H}$ is a Lie algebra object in
$\operatorname{\mathbf{Heis}}_{Q}\operatorname{-mod}_{\mathbb{Z}}$ containing
only unshifted representations
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{m}$ as
$\operatorname{\mathbf{Heis}}_{Q}$-subrepresentations, and
$\upiota\colon\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}\rightarrow\mathcal{H}$
is an inclusion in
$\operatorname{\mathbf{Heis}}_{Q}\operatorname{-mod}_{\mathbb{Z}}$, then
$\upiota$ extends uniquely to a morphism of
$\operatorname{\mathbf{Heis}}_{Q}$-Lie algebra objects
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}\rightarrow\mathcal{H}$.

###### Proof.

The lemma is equivalent to the claim that
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$ is isomorphic to
$\mathcal{L}^{\prime}$, the free $\mathbb{Z}$-graded Lie algebra $\mathcal{L}$
generated by $\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}$, modulo the
ideal containing all the $\mathbb{Z}$-graded
$\operatorname{\mathbf{Heis}}_{Q}$-submodules of $\mathcal{L}$ that are not
isomorphic to unshifted copies of
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{m}$ for various $m$.

The algebra $\mathcal{L}^{\prime}$ carries a grading by tensor degree, and we
claim that
$\mathcal{L}^{\prime}_{m}\cong\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2m}$.
This obviously implies the lemma. We prove the claim by induction. The base
case $m=1$ is trivial, and so we assume the claim is proved for $i<m$, with
$m\geq 2$. By the inductive hypothesis, the $m$th piece of
$\mathcal{L}^{\prime}$ is either a quotient of

$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}\wedge\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}\cong\bigoplus_{i\geq
0}\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{4}[-4i]$

if $m=2$, or

$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}\otimes\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2m-2}\cong\bigoplus_{i\geq
0}\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2m}[2-2i]$

if $m\geq 3$. In either case, a single unshifted copy of
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{m}$ survives after we quotient
out by all of the shifted copies of
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{m}$, and the inductive claim
follows. ∎

Given an integrable cohomologically-graded
$\operatorname{\mathbf{Heis}}$-module $\rho$, we define the characteristic
function

$\upchi(\rho)\coloneqq\sum_{i,j\in\mathbb{Z}}\dim(\rho_{[i]}^{j})t^{i}q^{j}.$

######  Corollary 2.5.

Let $\mathcal{H}$ be an integrable $\mathbb{Z}$-graded
$\operatorname{\mathbf{Heis}}$-module generated by
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}\subset\mathcal{H}$ and
satisfying

$\displaystyle\upchi(\mathcal{H})=$
$\displaystyle\upchi(\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty})$
$\displaystyle=$ $\displaystyle\sum_{i\geq 1,j\geq-1}t^{2i}q^{2j}$
$\displaystyle=$ $\displaystyle t^{2}q^{-2}(1-t^{2})^{-1}(1-q^{2})^{-1}.$

Then
$\mathcal{H}\cong\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$.

###### Proof.

By the universal property of
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$ (Lemma 2.4)
there is a morphism of Lie algebra objects in
$\operatorname{\mathbf{Heis}}\operatorname{-mod}_{\mathbb{Z}}$

$f\colon\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}\rightarrow\mathcal{H}$

sending the copy of $\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}$ inside
$\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$ to the copy
inside $\mathcal{H}$. By supposition, $f$ is surjective. On the other hand,
the source and the target have the same characteristic function, so $f$ is an
isomorphism. ∎

##  3\. DT theory background

###  3.1. Vanishing cycle toolkit

If $\mathfrak{X}$ is a smooth stack, we denote by
$\mathbb{Q}_{\mathfrak{X},\operatorname{vir}}=\mathbb{Q}_{\mathfrak{X}}[\dim(\mathfrak{X})]$
the constant perverse sheaf on $\mathfrak{X}$. More generally, if an extra
torus $T$ acts on a smooth stack $\mathfrak{X}$ with quotient $\mathfrak{Y}$,
we define555The motivation for this slightly mixed convention is that we would
like $\mathbb{Q}_{\mathfrak{Y},\operatorname{vir}}$ to correspond to a
$T$-equivariant perverse sheaf on $\mathfrak{Y}$, i.e. we would like the
pullback
$(\mathfrak{X}\rightarrow\mathfrak{Y})^{*}\mathbb{Q}_{\mathfrak{Y},\operatorname{vir}}$
to be perverse.
$\mathbb{Q}_{\mathfrak{Y},\operatorname{vir}}=\mathbb{Q}_{\mathfrak{Y}}[\dim(\mathfrak{X})]$.
Under the same conditions, we modify the usual perverse truncation functors
for constructible complexes of sheaves on $\mathfrak{Y}$ by setting
${}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq i}={}^{\mathfrak{p}}\uptau^{\leq
i-\dim(T)}$.

Given a function $f$ on a smooth stack $X$, we define the vanishing cycle
functor ${}^{\mathfrak{p}}\\!\upphi_{f}$ as in [KS90]. It is an exact functor
with respect to the perverse t structure, and in particular
${}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q}_{X,\operatorname{vir}}$ is a
perverse sheaf on $X$, which we may refer to just as the sheaf of vanishing
cycles on $X$. We will often abuse notation by just writing
$\operatorname{\mathbf{H}}(X,{}^{\mathfrak{p}}\\!\upphi_{f})$ instead of
$\operatorname{\mathbf{H}}(X,{}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q}_{X,\operatorname{vir}})$.

Let $f,g$ be functions on smooth stacks $X,Y$ respectively. The
Thom–Sebastiani theorem states that there is a natural bifunctorial
equivalence of constructible complexes of sheaves on $X\times Y$

$\operatorname{\mathtt{TS}}\colon{}^{\mathfrak{p}}\\!\upphi_{f\boxplus
g}(\mathcal{F}\boxtimes\mathcal{G})\lvert_{f^{-1}(0)\times
g^{-1}(0)}\cong{}^{\mathfrak{p}}\\!\upphi_{f}\mathcal{F}\boxtimes{}^{\mathfrak{p}}\\!\upphi_{g}\mathcal{G}.$

Assuming that $\operatorname{supp}({}^{\mathfrak{p}}\\!\upphi_{f\boxplus
g}(\mathcal{F}\boxtimes\mathcal{G}))\subset f^{-1}(0)\times g^{-1}(0)$, and
applying the derived global sections functor yields the isomorphism

$\operatorname{\mathbf{H}}(X,{}^{\mathfrak{p}}\\!\upphi_{f}\mathcal{F})\otimes\operatorname{\mathbf{H}}(Y,{}^{\mathfrak{p}}\\!\upphi_{g}\mathcal{G})\cong\operatorname{\mathbf{H}}(X\times
Y,{}^{\mathfrak{p}}\\!\upphi_{f\boxplus g}(\mathcal{F}\boxtimes\mathcal{G})),$

the analogue in vanishing cycle cohomology of the Künneth isomorphism in
ordinary cohomology.

Given a morphism $q\colon X\rightarrow Y$ and a function $f$ on $Y$ there is a
natural transformation
$\uptheta\colon{}^{\mathfrak{p}}\\!\upphi_{f}q_{*}\rightarrow
q_{*}{}^{\mathfrak{p}}\\!\upphi_{fq}$, which is an isomorphism if $q$ is
projective. The natural transformation $\uptheta$ induces the natural
transformation $\upeta\colon{}^{\mathfrak{p}}\\!\upphi_{f}\rightarrow
q_{*}{}^{\mathfrak{p}}\\!\upphi_{f}q^{*}$ by adjunction. Applying the derived
global sections functor, $\upeta$ induces the pullback morphism

(8)
$q^{\star}\colon\operatorname{\mathbf{H}}(Y,{}^{\mathfrak{p}}\\!\upphi_{f}\mathcal{F})\rightarrow\operatorname{\mathbf{H}}(X,{}^{\mathfrak{p}}\\!\upphi_{fq}q^{*}\mathcal{F}).$

Given a proper morphism $p\colon X\rightarrow Y$ between smooth stacks and a
function $f$ on $Y$, we obtain the pushforward morphism

(9)
$p_{\star}\colon\operatorname{\mathbf{H}}(X,{}^{\mathfrak{p}}\\!\upphi_{pf}\mathbb{Q}_{X})[2\dim(X)]\rightarrow\operatorname{\mathbf{H}}(Y,{}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q})[2\dim(Y)]$

by applying ${}^{\mathfrak{p}}\\!\upphi_{f}$ to
$p_{*}\mathbb{Q}_{Y}[2\dim(Y)]\rightarrow\mathbb{Q}_{X}[2\dim(X)]$ (the
Verdier dual of $\mathbb{Q}_{Y}\rightarrow p_{*}\mathbb{Q}_{X}$), and
composing with $\uptheta$.

Let $\mathfrak{X}$ be a smooth stack, $f$ be a function on it, and
$\mathcal{F}\in\mathcal{D}^{b}_{c}(\mathfrak{X})$ be a constructible complex.
Let $\Delta\colon\mathfrak{X}\rightarrow\mathfrak{X}\times\mathfrak{X}$ be the
diagonal embedding. Then $f=(f\boxplus 0)\circ\Delta$ and there is a natural
isomorphism
$\mathcal{F}\cong\Delta^{*}(\mathcal{F}\boxtimes\mathbb{Q}_{\mathfrak{X}})$.
The pullback morphism $\Delta^{\star}$, composed with the Thom-Sebastiani
isomorphism, yields

$\operatorname{\mathbf{H}}(\mathfrak{X},{}^{\mathfrak{p}}\\!\upphi_{f}\mathcal{F})\otimes\operatorname{\mathbf{H}}(\mathfrak{X},\mathbb{Q})\rightarrow\operatorname{\mathbf{H}}(\mathfrak{X},{}^{\mathfrak{p}}\\!\upphi_{f}\mathcal{F}),$

and an action of the cohomology of $\mathfrak{X}$ on
$\operatorname{\mathbf{H}}(\mathfrak{X},{}^{\mathfrak{p}}\\!\upphi_{f}\mathcal{F})$.
If $f=0$ this is just the usual action of
$\operatorname{\mathbf{H}}(\mathfrak{X},\mathbb{Q})$ on
$\operatorname{\mathbf{H}}(\mathfrak{X},\mathcal{F})$, and if in addition
$\mathcal{F}=\mathbb{Q}_{\mathfrak{X}}$ this is the usual algebra structure on
$\operatorname{\mathbf{H}}(\mathfrak{X},\mathbb{Q})$. If
$q\colon\mathfrak{X}\rightarrow\mathfrak{Y}$ is a morphism of smooth stacks,
from the commutativity of

$\textstyle{\mathfrak{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{(\operatorname{id}\times
q)\circ\Delta}$$\textstyle{\mathfrak{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\mathfrak{X}\times\mathfrak{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q\times\operatorname{id}}$$\textstyle{\mathfrak{Y}\times\mathfrak{Y}}$

we deduce that (8) respects the
$\operatorname{\mathbf{H}}(\mathfrak{Y},\mathbb{Q})$-actions on the domain and
target. If $q$ is projective, one may show as in [Dav17a, Prop.2.14] that (9)
respects the $\operatorname{\mathbf{H}}(\mathfrak{Y},\mathbb{Q})$ actions on
the domain and target.

###  3.2. Quivers

Throughout the paper, $Q$ will denote a finite quiver, i.e. a pair of finite
sets $Q_{1}$ and $Q_{0}$ (the arrows and vertices, respectively) along with a
pair of morphisms $s,t\colon Q_{1}\rightarrow Q_{0}$. The free path algebra
$kQ$ of $Q$ over the field $k$ is the algebra having as basis the paths in
$Q$, including “lazy” paths $e_{i}$ of length zero at each of the vertices
$i\in Q_{0}$.

We define the Euler form

$\displaystyle\upchi_{Q}\colon$
$\displaystyle\mathbb{N}^{Q_{0}}\times\mathbb{N}^{Q_{0}}\rightarrow\mathbb{Z}$
$\displaystyle(\mathbf{d},\mathbf{e})\mapsto\sum_{i\in
Q_{0}}\mathbf{d}_{i}\mathbf{e}_{i}-\sum_{a\in
Q_{1}}\mathbf{d}_{s(a)}\mathbf{e}_{t(a)}.$

The double of $Q$ is obtained by setting $\overline{Q}_{0}=Q_{0}$ and
$\overline{Q}_{1}=Q_{1}\coprod Q_{1}^{*}$, where
$Q_{1}^{*}\coloneqq\\{a^{*}\;\lvert\;a\in Q_{1}\\}$ and the orientation of
$a^{*}$ is the opposite of the orientation of $a$. The preprojective algebra
is defined by

$\Pi_{Q}\coloneqq\mathbb{C}\overline{Q}/\langle\sum_{a\in
Q_{1}}[a,a^{*}]\rangle.$

The quiver $\tilde{Q}$ is defined by adding a loop $\omega_{i}$ to each vertex
$i\in\overline{Q}_{0}$.

######  Example 3.1.

If $Q=Q^{(1)}$ is the Jordan quiver with one vertex and one loop $a$, then
$\tilde{Q}=Q^{(3)}$ is the three loop quiver, with one vertex and loops
labelled $a,a^{*},\omega$.

A potential $W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]_{\mathrm{cyc}}$ is a
linear combination of cyclic words in $Q$. If $W=a_{1}\ldots a_{l}$ is a
single cyclic word then

$\partial W/\partial a\coloneqq\sum_{a_{i}=a}a_{i+1}\ldots a_{l}a_{1}\ldots
a_{i-1}.$

We extend this definition to general $W$ by linearity. The Jacobi algebra
associated to the pair $(Q,W)$ is defined by

$\operatorname{Jac}(Q,W)\coloneqq\mathbb{C}Q/\langle\partial W/\partial
a\;\lvert\;a\in Q_{1}\rangle.$

If $Q$ is an arbitrary quiver, the tripled quiver $\tilde{Q}$ carries the
canonical cubic potential

$\tilde{W}\coloneqq(\sum_{i\in Q_{0}}\omega_{i})(\sum_{a\in Q_{1}}[a,a^{*}]).$

Let $\Pi_{Q}[\omega]$ denote the polynomial algebra in one variable $\omega$
with coefficients in $\Pi_{Q}$. There is an isomorphism

(10) $\displaystyle j\colon$
$\displaystyle\Pi_{Q}[\omega]\rightarrow\operatorname{Jac}(\tilde{Q},\tilde{W})$

defined by $j(a)=a$, $j(a^{*})=a^{*}$, $j(\omega)=\sum_{i\in
Q_{0}}\omega_{i}$.

###  3.3. Moduli spaces

Given a quiver $Q^{\prime}$ and dimension vector
$\mathbf{d}\in\mathbb{N}^{Q_{0}}$ we define

$\mathbb{A}_{Q^{\prime},\mathbf{d}}\coloneqq\prod_{a\in
Q^{\prime}_{1}}\operatorname{Hom}(\mathbb{C}^{\mathbf{d}(s(a))},\mathbb{C}^{\mathbf{d}(t(a))})\quad\quad\operatorname{GL}_{\mathbf{d}}\coloneqq\prod_{i\in
Q^{\prime}_{0}}\operatorname{GL}_{\mathbf{d}(i)}.$

The group $\operatorname{GL}_{\mathbf{d}}$ acts on
$\mathbb{A}_{Q^{\prime},\mathbf{d}}$ by conjugation. The moduli stack
$\mathfrak{M}_{\mathbf{d}}(Q^{\prime})$, when evaluated on a scheme $X$,
yields the groupoid of coherent sheaves on $X$, flat over $X$, equipped with
an action of $\mathbb{C}Q^{\prime}$, such that at every geometric point $K$ of
$X$ the induced $KQ^{\prime}$-module is $\mathbf{d}$-dimensional. This stack
is naturally isomorphic to the stack quotient
$\mathbb{A}_{Q^{\prime},\mathbf{d}}/\operatorname{GL}_{\mathbf{d}}$.

Let $N=\mathbb{Z}^{r}$ be a lattice, and fix a function $\uptau\colon
Q^{\prime}_{1}\rightarrow N$. We fix the torus
$T=\operatorname{Hom}_{\mathrm{Grp}}(N,\mathbb{C}^{*})$. The torus $T$ acts on
$\mathbb{A}_{Q^{\prime},\mathbf{d}}$ by sending

$(\mathbf{t},(\rho_{a})_{a\in
Q^{\prime}_{1}})\mapsto(\mathbf{t}(\uptau(a))\cdot\rho_{a})_{a\in
Q^{\prime}_{1}},$

and this action commutes with the action of $\operatorname{GL}_{\mathbf{d}}$
by conjugation. We denote by

$\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})\cong\mathbb{A}_{Q^{\prime},\mathbf{d}}/(\operatorname{GL}_{\mathbf{d}}\times
T)$

the quotient of $\mathfrak{M}_{\mathbf{d}}(Q^{\prime})$ by the $T$-action.
Precisely, a homomorphism
$X\rightarrow\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})$ is provided by a
$T$-torsor $E\rightarrow X$, along with a $T$-equivariant family $\mathcal{F}$
of $\mathbf{d}$-dimensional $\mathbb{C}Q^{\prime}$ modules on $E$, along with
an action of $\mathbb{C}Q^{\prime}$ on the underlying coherent sheaf of
$\mathcal{F}$, such that for every $a\in Q_{1}$ the following diagram of
coherent sheaves on $G\times E$ commutes:

$\textstyle{l^{*}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{l^{*}\rho(a)}$$\textstyle{l^{*}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{p^{*}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod_{i\leq
r}z_{i}^{\uptau(a)_{i}}\cdot p^{*}\rho(a)}$$\textstyle{p^{*}\mathcal{F}.}$

Here, $l\colon T\times E\rightarrow E$ is the action, $p\colon T\times
E\rightarrow E$ is the projection, $\rho(a)$ is the action of $a$ on
$\mathcal{F}$, $z_{i}$ are coordinates on $T$, and the vertical isomorphisms
are defined by the $T$-equivariant structure on $\mathcal{F}$. Since we will
use it later, we also explicitly describe the natural morphism

(11)
$s\colon\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}(Q^{\prime})\times\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}(Q^{\prime})\rightarrow\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}(Q^{\prime})\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}(Q^{\prime})$

where
$\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}\in\mathbb{N}^{Q^{\prime}_{0}}$.
If $p^{\prime}\colon E^{\prime}\rightarrow X$ and $p^{\prime}\colon
E^{\prime\prime}\rightarrow X$ are a pair of $T$ torsors on a scheme $X$, we
start by giving the trivial vector bundle $\mathscr{O}_{E^{\prime\prime}}$ the
trivial $T$-equivariant structure. Then if $\mathcal{F}$ is a $T$-equivariant
coherent sheaf on $E^{\prime}$, we first consider $\uppi\hskip
0.50003pt^{*}\mathcal{F}$, where $\uppi\hskip 0.50003pt\colon
E^{\prime}\times_{X}E^{\prime\prime}\rightarrow E^{\prime}$ is the projection.
This is naturally a $T\times T$-equivariant sheaf (via the trivial
$T$-equivariant structure on $\mathscr{O}_{E^{\prime\prime}}$) that descends
to a $T$-equivariant sheaf on $E^{\prime}\times_{T}E^{\prime\prime}$ (along
with the action of $\rho(a)$ for each $a\in Q_{1}$), which we denote
$\mathcal{F}\times_{T}E^{\prime\prime}$. Then the morphism $s$ is given by

(12) $s((p^{\prime}\colon E^{\prime}\rightarrow
X,\mathcal{F}^{\prime}),(p^{\prime\prime}\colon E^{\prime\prime}\rightarrow
X,\mathcal{F}^{\prime\prime}))=(E^{\prime}\times_{T}E^{\prime\prime},\mathcal{F}^{\prime}\times_{T}E^{\prime\prime},\mathcal{F}^{\prime\prime}\times_{T}E^{\prime}).$

We denote by

$\mathcal{M}_{\mathbf{d}}(Q^{\prime})=\operatorname{Spec}(\Upgamma(\mathbb{A}_{Q^{\prime},\mathbf{d}})^{\operatorname{GL}_{\mathbf{d}}})$

the coarse moduli space. For $K\supset\mathbb{C}$ a field, $K$-points of
$\mathcal{M}_{\mathbf{d}}(Q^{\prime})$ correspond to isomorphism classes of
semisimple $\mathbf{d}$-dimensional $KQ^{\prime}$-modules. Since the
$T$-action commutes with the $\operatorname{GL}_{\mathbf{d}}$-action,
$\mathcal{M}_{\mathbf{d}}(Q^{\prime})$ carries a natural $T$-action. We define
the stack-theoretic quotient

$\mathcal{M}^{T}_{\mathbf{d}}(Q^{\prime})\coloneqq\mathcal{M}_{\mathbf{d}}(Q^{\prime})/T.$

In the case of trivial $T$, there is a natural morphism

$\mathtt{JH}\colon\mathfrak{M}_{\mathbf{d}}(Q^{\prime})\rightarrow\mathcal{M}_{\mathbf{d}}(Q^{\prime})$

which, at the level of points, takes modules to their semisimplifications, and
at the level of geometry is just the affinization morphism. This extends
naturally to a morphism

(13)
$\mathtt{JH}^{T}\colon\mathfrak{M}_{\mathbf{d}}^{T}(Q^{\prime})\rightarrow\mathcal{M}^{T}_{\mathbf{d}}(Q^{\prime}).$

Precisely, if the morphism $f\colon
X\rightarrow\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})$ is represented by the
$T$-torsor $E\rightarrow X$ and the morphism $\tilde{f}\colon
E\rightarrow\mathfrak{M}_{\mathbf{d}}(Q^{\prime})$, the morphism
$\mathtt{JH}^{T}\circ f$ is represented by the same torsor $E$, along with the
morphism $\mathtt{JH}\circ\tilde{f}$.

Given two dimension vectors
$\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}\in\mathbb{N}^{Q^{\prime}_{0}}$
with $\mathbf{d}^{\prime}+\mathbf{d}^{\prime\prime}=\mathbf{d}$ we define
$\mathbb{A}_{Q^{\prime},\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}\subset\mathbb{A}_{Q^{\prime},\mathbf{d}}$
and
$\operatorname{P}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}\subset\operatorname{GL}_{\mathbf{d}}$
to be the space of representations, and changes of basis, respectively,
preserving the flags
$0\subset\mathbb{C}^{\mathbf{d}^{\prime}(i)}\subset\mathbb{C}^{\mathbf{d}(i)}$
for each $i\in Q^{\prime}_{0}$. Denoting by
$\mathfrak{M}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}(Q^{\prime})$ the
stack of short exact sequences
$0\rightarrow\rho^{\prime}\rightarrow\rho\rightarrow\rho^{\prime\prime}\rightarrow
0$ of $\mathbb{C}Q^{\prime}$-modules, for which the dimension vectors of
$\rho^{\prime}$ and $\rho^{\prime\prime}$ are
$\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}$ respectively, there is a
natural isomorphism

$\mathfrak{M}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}(Q^{\prime})\cong\mathbb{A}_{Q^{\prime},\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}/\operatorname{GL}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}.$

We define
$\mathfrak{M}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}^{T}(Q^{\prime})\coloneqq\mathbb{A}_{Q^{\prime},\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}/(\operatorname{GL}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}\times
T)$.

###  3.4. Cohomological Hall algebras

Let $Q^{\prime}$ be a quiver. For the sake of exposition, we assume that
$Q^{\prime}$ is symmetric in this section, i.e. for every pair of vertices
$i,j\in Q^{\prime}_{0}$ there are as many arrows from $i$ to $j$ as from $j$
to $i$. Let
$W\in\mathbb{C}Q^{\prime}/[\mathbb{C}Q^{\prime},\mathbb{C}Q^{\prime}]_{\mathrm{cyc}}$
be a potential, which we assume to be $T$-invariant in the following sense:
for every cyclic word $a_{1}\ldots a_{c}$ appearing in $W$

(14) $\sum_{i=1}^{c}\uptau(a_{i})=0.$

The potential $W$ induces a function $\operatorname{Tr}(W)$ on
$\mathbb{A}_{Q^{\prime},\mathbf{d}}$ which is
$\operatorname{GL}_{\mathbf{d}}$-invariant by cyclic invariance of the trace
map, and $T$-invariant by (14). We define

(15)
$\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}\coloneqq\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q^{\prime}_{0}}}\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})})$

the underlying $\mathbb{N}^{Q^{\prime}_{0}}$-graded vector space of the
critical cohomological Hall algebra, defined by Kontsevich and Soibelman in
[KS11]. Consider the convolution diagram

$\textstyle{\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}(Q^{\prime})\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}(Q^{\prime})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathfrak{M}^{T}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}(Q^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt_{1}\times\uppi\hskip 0.35002pt_{3}}$$\scriptstyle{\uppi\hskip
0.35002pt_{2}}$$\textstyle{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}$

where $\uppi\hskip 0.50003pt_{1},\uppi\hskip 0.50003pt_{2},\uppi\hskip
0.50003pt_{3}$ take a flag $0\subset\mathcal{F}^{\prime}\subset\mathcal{F}$ of
flat families of $\mathbb{C}Q^{\prime}$-modules to the families
$\mathcal{F}^{\prime}$, $\mathcal{F}$, $\mathcal{F}/\mathcal{F}^{\prime}$
respectively. We define

$\displaystyle
m^{\prime}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}\colon$
$\displaystyle\operatorname{\mathbf{H}}\left(\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}(Q^{\prime})\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)\boxplus\operatorname{Tr}(W)}\right)\rightarrow\operatorname{\mathbf{H}}\left(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\right)$
$\displaystyle=\uppi\hskip 0.50003pt_{2,\star}\circ(\uppi\hskip
0.50003pt_{1}\times\uppi\hskip 0.50003pt_{3})^{\star}$

and
$m_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}=\operatorname{\mathtt{TS}}\circ
s^{\star}\circ m^{\prime}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}$
where $s$ is as in (11) and $\operatorname{\mathtt{TS}}$ is the
Thom–Sebastiani isomorphism. We then define the multiplication

$m\colon\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}\otimes\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}\rightarrow\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$

by summing $m_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}$ over all pairs
of dimension vectors.

Let $\mathcal{F}$ be a flat family of $\mathbf{d}$-dimensional
$\mathbb{C}Q^{\prime}$-modules on a scheme $X$. For each $i\in Q^{\prime}_{0}$
we obtain a vector bundle $\mathcal{F}_{i}=e_{i}\\!\cdot\\!\mathcal{F}$ of
rank $\mathbf{d}_{i}$. There is a determinant line bundle
$\operatorname{Det}(\mathcal{F}_{i})$ on $X$, which is defined to be the top
exterior power of the underlying bundle of $\mathcal{F}_{i}$. This defines a
morphism of stacks

$\operatorname{Det}_{(i)}\colon\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})\rightarrow\operatorname{B\\!}\mathbb{C}_{u}^{*}.$

Alternatively, we can construct this morphism as the composition

$\displaystyle\operatorname{Det}_{(i)}=$
$\displaystyle(\operatorname{Det}\colon\operatorname{B\\!}\operatorname{GL}_{\mathbf{d}_{i}}\rightarrow\operatorname{B\\!}\mathbb{C}^{*}_{u})\circ(\uppi\hskip
0.50003pt_{\operatorname{pt}/\operatorname{GL}_{\mathbf{d}_{i}}}\colon\operatorname{pt}/(\operatorname{GL}_{\mathbf{d}_{i}}\times
T)\rightarrow\operatorname{B\\!}\operatorname{GL}_{\mathbf{d}_{i}})$
$\displaystyle\circ(\mathbb{A}_{Q^{\prime},\mathbf{d}}/(\operatorname{GL}_{\mathbf{d}}\times
T)\rightarrow\operatorname{pt}/(\operatorname{GL}_{\mathbf{d}}\times T)).$

We denote by
$u_{(i)}\in\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),\mathbb{Q})$
the image of $u\in\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$ under
$\operatorname{Det}_{(i)}^{\star}$. Consider the morphism

$\displaystyle l_{i}=(\operatorname{Det}_{(i)},\operatorname{id})\colon$
$\displaystyle\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})\rightarrow\operatorname{B\\!}\mathbb{C}_{u}^{*}\times\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}).$

Since the function $\operatorname{Tr}(W)$ is pulled back from the function
$0\boxplus\operatorname{Tr}(W)$ on the target of $l$ we obtain the morphism in
vanishing cycle cohomology

$\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\otimes\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})\xrightarrow{l^{\star}}\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})$

which defines an action of the algebra
$\mathbb{Q}[u]=\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$ on
$\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$:

$p(u)\bullet_{i}\upalpha\coloneqq l_{i}^{\star}(p(u)\otimes\upalpha).$

Similarly, we define the morphism

$\displaystyle l=(\bigotimes_{i\in
Q_{0}}\operatorname{Det}_{(i)},\operatorname{id})\colon$
$\displaystyle\mathfrak{M}^{T}_{\mathbf{d}}(Q)\rightarrow\operatorname{B\\!}\mathbb{C}_{u}^{*}\times\mathfrak{M}^{T}_{\mathbf{d}}(Q).$

and an action of $\operatorname{\mathbf{H}}_{\mathbb{C}_{u}^{*}}$ on
$\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$ by
$p(u)\bullet\upalpha=l^{\star}(p(u)\otimes\upalpha)$.

###  3.5. PBW isomorphism

In this section we assume $T=\\{1\\}$ for simplicity. By [DM20] the direct
image
$\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}}$
splits: there is an isomorphism in
$\mathcal{D}^{+}(\operatorname{\mathbf{Perv}}(\mathcal{M}_{\mathbf{d}}(Q^{\prime})))$:

(16)
$\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}}\cong\bigoplus_{i\geq
1}{}^{\mathfrak{p}}\\!\operatorname{\mathcal{H}}^{i}(\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}})[-i].$

There is a natural morphism

(17)
$\mathfrak{P}_{i}\operatorname{\mathcal{A}}_{Q^{\prime},W,\mathbf{d}}\coloneqq\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\uptau^{\leq
i}\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}})\rightarrow\operatorname{\mathcal{A}}_{Q^{\prime},W,\mathbf{d}}$

induced by the natural transformation ${}^{\mathfrak{p}}\uptau^{\leq
i}\rightarrow\operatorname{id}$ and it follows from the splitting (16) that
(17) is injective, and the subspaces
$\mathfrak{P}_{i}\operatorname{\mathcal{A}}_{Q^{\prime},W}$ provide a
filtration of $\operatorname{\mathcal{A}}_{Q^{\prime},W,\mathbf{d}}$, called
the perverse filtration.

Now assume that $Q^{\prime}$ is symmetric. By [DM20, Sec.5.3] the
multiplication on $\operatorname{\mathcal{A}}_{Q^{\prime},W}$ respects the
perverse filtration. There is vanishing

(18) $\mathfrak{P}^{0}\\!\operatorname{\mathcal{A}}_{Q^{\prime},W}=0,$

and furthermore the subspace

$\mathfrak{g}_{Q^{\prime},W}\coloneqq\mathfrak{P}^{1}\\!\operatorname{\mathcal{A}}_{Q^{\prime},W}\subset\operatorname{\mathcal{A}}_{Q^{\prime},W}$

is preserved by the commutator Lie bracket $[-,-]$ by [DM20, Cor.6.11]. There
are isomorphisms

$\mathfrak{g}_{Q^{\prime},W,\mathbf{d}}=\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{d}}(Q^{\prime}),\mathcal{BPS}_{Q^{\prime},W,\mathbf{d}})[-1]$

where $\mathcal{BPS}_{Q^{\prime},W,\mathbf{d}}$ is the BPS sheaf defined in
[DM20, Thm.A]. We call $\mathfrak{g}_{Q^{\prime},W}$ the BPS Lie algebra
associated to $(Q^{\prime},W)$.

######  Theorem 3.2.

[DM20, Thm.C] The PBW morphism

(19)
$\Psi\colon\operatorname{\mathbf{Sym}}(\mathfrak{g}_{Q^{\prime},W}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}})\rightarrow\operatorname{\mathcal{A}}_{Q^{\prime},W}$

is an isomorphism of cohomologically graded
$\mathbb{N}^{Q^{\prime}_{0}}$-graded vector spaces.

The morphism $\Psi$ is constructed as follows. Firstly, we consider the
composition

$\Psi^{\prime}=\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\otimes\mathfrak{g}_{Q^{\prime},W}\hookrightarrow\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\otimes\operatorname{\mathcal{A}}_{Q^{\prime},W}\xrightarrow{l^{\star}}\operatorname{\mathcal{A}}_{Q^{\prime},W}.$

Since the target carries an algebra structure, this extends uniquely to a
morphism of algebras

$\Phi\colon\mathrm{T}\left(\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\otimes\mathfrak{g}_{Q^{\prime},W}\right)\rightarrow\operatorname{\mathcal{A}}_{Q^{\prime},W}$

from the free algebra generated by the domain of $\Psi^{\prime}$. Then $\Psi$
is defined to be the restriction of $\Phi$ to the subspace of (super666To be
entirely accurate: in general, we have to twist the Koszul sign rule in a
somewhat subtle way in order to make this theorem true, see [DM20, Sec.1.6].
For quivers of the form $\tilde{Q}$ this extra twist is unnecessary, since the
form $\uptau$ from Equation (15) of [DM20] vanishes.)symmetric tensors. Note
that the morphism $\Psi$ will typically not be a morphism of algebras; the
CoHA $\operatorname{\mathcal{A}}_{Q^{\prime},W}$ will in general have an
interesting algebra structure, and not be a free (super)commutative algebra.

###  3.6. Deformed BPS Lie algebras

Let $Q^{\prime}$ be a quiver, $T$ be a torus acting on the arrows of
$Q^{\prime}$, and let
$W\in\mathbb{C}Q^{\prime}/[\mathbb{C}Q^{\prime},\mathbb{C}Q^{\prime}]_{\operatorname{cyc}}$
be a $T$-invariant potential. Recall the map $\mathtt{JH}^{T}$ from (13). We
define777The perverse truncation functors are modified according to the
conventions introduced in §3.1. the $\operatorname{\mathbf{H}}_{T}$-module

$\mathfrak{g}^{T}_{Q^{\prime},W,\mathbf{d}}\coloneqq\operatorname{\mathbf{H}}(\mathcal{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}_{*}^{T}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}).$

Applying the derived global sections functor to the natural morphism of
complexes ${}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}_{*}^{T}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}\rightarrow\mathtt{JH}_{*}^{T}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}$
we obtain the morphism

$\upiota\colon\mathfrak{g}^{T}_{Q^{\prime},W}\rightarrow\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}.$

######  Theorem 3.3.

The morphism $\upiota$ is injective, so that there is a natural inclusion of
subspaces
$\mathfrak{g}^{T}_{Q^{\prime},W}\subset\operatorname{\mathcal{A}}_{Q^{\prime},W}^{T}$.
The subspace $\mathfrak{g}^{T}_{Q^{\prime},W}$ is closed under the commutator
Lie bracket on $\operatorname{\mathcal{A}}_{Q^{\prime},W}^{T}$. Assume,
furthermore, that $\mathfrak{g}_{Q^{\prime},W}^{T}$ is pure, as a mixed Hodge
structure. Then there is an isomorphism of vector spaces

(20)
$\mathfrak{g}^{T}_{Q^{\prime},W}\cong\mathfrak{g}_{Q^{\prime},W}\otimes\operatorname{\mathbf{H}}_{T},$

and the PBW morphism

$\operatorname{\mathbf{Sym}}_{\operatorname{\mathbf{H}}_{T}}(\mathfrak{g}_{Q^{\prime},W}^{T}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}})\rightarrow\operatorname{\mathcal{A}}_{Q^{\prime},W}^{T}$

constructed from the Hall algebra product on the target is an isomorphism.

###### Proof.

The proof is a minor modification of the analogous statements from [DM20], to
which we refer for more details. Firstly, we claim that $\mathtt{JH}^{T}$ is
approximated by projective morphisms in the sense of [DM20, Sec.4.1]. For this
the proof is as in [DM20], using the quotient of moduli spaces of framed
$\mathbb{C}Q^{\prime}$-modules by the $T$-action as auxiliary spaces. From
this it follows as in [DM20] that there is a splitting

$\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}}\cong\bigoplus_{i\in\mathbb{Z}}{}^{\mathfrak{p}}\\!\operatorname{\mathcal{H}}^{i}\\!\left(\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}}\right)[-i]$

and so the morphism

${}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}}\rightarrow\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}}$

has a left inverse, and the injectivity part of the theorem follows.

The monoid morphism
$\mathcal{M}(Q^{\prime})\times\mathcal{M}(Q^{\prime})\rightarrow\mathcal{M}(Q^{\prime})$
taking a pair of semisimple $\mathbb{C}Q^{\prime}$-modules to their direct sum
is $T$-equivariant, so that we can define the monoid structure

(21)
$\oplus\colon\mathcal{M}^{T}(Q^{\prime})\times_{\operatorname{B\\!}T}\mathcal{M}^{T}(Q^{\prime})\rightarrow\mathcal{M}^{T}(Q^{\prime})$

by passing to the stack-theoretic quotient by the $T$-action. Given a pair of
complexes of sheaves $\mathcal{F}$ and $\mathcal{G}$ on two stacks
$\mathfrak{X}$ and $\mathfrak{Y}$ over a stack $\mathfrak{Z}$ we define the
external product, a complex of sheaves on
$\mathfrak{X}\times_{\mathfrak{Z}}\mathfrak{Y}$ in the usual way:

$\mathcal{F}\boxtimes_{\mathfrak{Z}}\mathcal{G}\coloneqq\uppi\hskip
0.50003pt_{\mathfrak{X}}^{*}\mathcal{F}\otimes\uppi\hskip
0.50003pt_{\mathfrak{Y}}^{*}\mathcal{G}.$

This defines a symmetric tensor product on complexes of sheaves on
$\mathcal{M}^{T}(Q^{\prime})$: we define

$\mathcal{F}\boxtimes_{\oplus}\mathcal{G}=\oplus_{*}(\mathcal{F}\boxtimes_{\operatorname{B\\!}T}\mathcal{G}).$

Let $p\colon\mathcal{M}^{T}(Q^{\prime})\rightarrow\operatorname{B\\!}T$ be the
canonical map. This is a morphism of monoid objects over
$\operatorname{B\\!}T$, where $\operatorname{B\\!}T$ is considered as a monoid
over itself via the identity morphism. For arbitrary complexes of sheaves on
$\mathcal{M}^{T}(Q^{\prime})$ it is too much to hope that there is an
isomorphism

(22)
$\operatorname{\mathbf{H}}(\mathcal{M}^{T}(Q^{\prime}),\mathcal{F}\boxtimes_{\oplus}\mathcal{G})\cong\operatorname{\mathbf{H}}(\mathcal{M}^{T}(Q^{\prime}),\mathcal{F})\otimes_{\operatorname{\mathbf{H}}_{T}}\operatorname{\mathbf{H}}(\mathcal{M}^{T}(Q^{\prime}),\mathcal{G}).$

However, let us assume that the complexes $p_{*}\mathcal{F}$ and
$p_{*}\mathcal{G}$ lift to pure complexes of mixed Hodge modules. Then via the
usual spectral sequence argument (see [Dav17b, Sec.9]), there are isomorphisms

$\displaystyle\operatorname{\mathbf{H}}(\mathcal{M}^{T}(Q^{\prime}),\mathcal{F})\cong$
$\displaystyle\operatorname{\mathbf{H}}(\mathcal{M}(Q^{\prime}),\overline{\mathcal{F}})\otimes\operatorname{\mathbf{H}}_{T}$
$\displaystyle\operatorname{\mathbf{H}}(\mathcal{M}^{T}(Q^{\prime}),\mathcal{G})\cong$
$\displaystyle\operatorname{\mathbf{H}}(\mathcal{M}(Q^{\prime}),\overline{\mathcal{G}})\otimes\operatorname{\mathbf{H}}_{T}$
$\displaystyle\operatorname{\mathbf{H}}(\mathcal{M}^{T}(Q^{\prime}),\mathcal{F}\boxtimes_{\oplus}\mathcal{G})\cong$
$\displaystyle(\operatorname{\mathbf{H}}(\mathcal{M}(Q^{\prime}),\overline{\mathcal{F}})\otimes\operatorname{\mathbf{H}}(\mathcal{M}(Q^{\prime}),\overline{\mathcal{G}}))\otimes\operatorname{\mathbf{H}}_{T}$

where $\overline{\mathcal{F}}$ and $\overline{\mathcal{G}}$ denote the inverse
images of $\mathcal{F}$ and $\mathcal{G}$ along the canonical morphism
$\mathcal{M}(Q^{\prime})\rightarrow\mathcal{M}^{T}(Q^{\prime})$, and a natural
isomorphism (22).

Via the same morphisms as in [DM20, Sec.5.1], the complex
$\mathtt{JH}_{*}^{T}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}(Q^{\prime}),\operatorname{vir}}$
carries a relative Hall algebra structure, i.e. for
$\mathbf{d}=\mathbf{d}^{\prime}+\mathbf{d}^{\prime\prime}$ there is a morphism

$\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}(Q^{\prime})}\boxtimes_{\oplus}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}(Q^{\prime})}\rightarrow\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}$

satisfying the obvious associativity condition. We claim that

  * •

The perverse truncation ${}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
0}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}$
vanishes.

  * •

The morphism

${}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}[1]\boxtimes_{\oplus}{}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
0}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}[1]\rightarrow{}^{\mathfrak{p}^{\prime}}\\!\\!\operatorname{\mathcal{H}}^{2}\\!\left(\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}\right)$

given by restricting the commutator bracket on the relative CoHA
$\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}$
vanishes.

Taking derived global sections, these two claims prove that
$\mathfrak{g}^{T}_{Q^{\prime},W}$ is closed under the commutator Lie bracket.
The first claim concerns the vanishing of certain perverse sheaves on
$\mathcal{M}^{T}(Q^{\prime})$, and the second concerns the vanishing of
certain morphisms of perverse sheaves on $\mathcal{M}^{T}(Q^{\prime})$. As
such, both are local, and may be checked after pulling back along an open
cover $U\rightarrow\operatorname{B\\!}T$, at which point they both reduce to
the analogous claims on $\mathcal{M}(Q)$, which are proved as [DM20, Thm.A],
[DM20, Cor.6.11] respectively.

The isomorphism (20) follows from the usual spectral sequence argument (as in
[Dav17b]). Since $p$ is a morphism of monoids over $\operatorname{B\\!}T$, the
complex
$p_{*}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}(Q^{\prime})}$
inherits a Hall algebra structure, along with an action of
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$. We claim that the morphism

$\operatorname{\mathbf{Sym}}_{\boxtimes_{\operatorname{B\\!}T}}\left(p_{*}{}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}(Q^{\prime})}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\right)\rightarrow
p_{*}\mathtt{JH}^{T}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}(Q^{\prime})}$

is an isomorphism. Again, this can be proved by passing to an open cover of
$\operatorname{B\\!}T$, at which point it becomes the non-equivariant PBW
theorem (Theorem 3.2). The equivariant PBW theorem then follows. ∎

We refer to $\mathfrak{g}^{T}_{Q^{\prime},W}$ as the deformed BPS Lie algebra.
By the above theorem, as long as $\mathfrak{g}_{Q^{\prime},W}$ is pure, this
deformed Lie algebra provides a family of Lie algebras over
$\mathfrak{t}\cong\mathrm{\operatorname{Spec}}(\operatorname{\mathbf{H}}_{T})$
specialising to the original BPS Lie algebra $\mathfrak{g}_{Q^{\prime},W}$ at
the origin.

###  3.7. Kac polynomials and Kac–Moody Lie algebras

Given a quiver $Q$ without loops we denote by $\mathfrak{n}^{-}_{Q}$ the
negative half of the associated Kac–Moody Lie algebra. This is the free
$\mathbb{N}^{Q_{0}}$-graded Lie algebra with generators $F_{i}$ in
$\mathbb{N}^{Q_{0}}$-degree $1_{i}$, subject to the Serre relations

$[F_{i},-]^{a_{ij}+1}(F_{j})=0$

for $i\neq j$. Here $a_{ij}$ denotes the number of edges joining $i$ and $j$
in the graph obtained from $Q$ by forgetting orientations of edges. If the
underlying graph of $Q$ is a type ADE Dynkin diagram, $\mathfrak{n}^{-}_{Q}$
is (one half of) the associated simple Lie algebra.

Given a quiver $Q$ and a field $K$, a $KQ$-module $\rho$ is called
indecomposable if it cannot be written as
$\rho=\rho^{\prime}\oplus\rho^{\prime\prime}$ with
$\rho^{\prime},\rho^{\prime\prime}\neq 0$. It is called absolutely
indecomposable if $\rho\otimes_{K}\overline{K}$ is an indecomposable
$\overline{K}Q$-module. For $q=p^{n}$ a prime power, let
$\mathtt{a}_{Q,\mathbf{d}}(q)$ denote the number of isomorphism classes of
absolutely indecomposable $\mathbf{d}$-dimensional $\mathbb{F}_{q}Q$-modules.
Victor Kac showed [Kac83] that $\mathtt{a}_{Q,\mathbf{d}}(q)$ is a polynomial
in $q$, and moreover an element of $\mathbb{Z}[q]$.

If $Q$ is an orientation of a finite ADE graph, then

(23) $\mathtt{a}_{Q,\mathbf{d}}(q)=\dim(\mathfrak{n}^{-}_{Q,\mathbf{d}})$

In particular, it is constant. Conversely, if $Q$ is any other type of quiver,
$\mathtt{a}_{Q,\mathbf{d}}(q)$ is nonconstant for at least some values of
$\mathbf{d}$.

###  3.8. Preprojective CoHAs

We consider the class of special cases $(Q^{\prime},W)=(\tilde{Q},\tilde{W})$
for a finite quiver $Q$. Although we do not assume that $Q$ is symmetric,
$\tilde{Q}$ obviously is. We call these CoHAs preprojective CoHAs, for reasons
that will become evident in §3.9. By [Dav17b, Thm.4.7] there is an equality of
generating series

(24)
$\sum_{i\in\mathbb{Z}}\dim(\mathfrak{g}_{\tilde{Q},\tilde{W},\mathbf{d}}^{i})q^{i/2}=\mathtt{a}_{Q,\mathbf{d}}(q^{-1}).$

where $\mathtt{a}_{Q,\mathbf{d}}(q)$ is the Kac polynomial. So in particular,
$\mathfrak{g}_{\tilde{Q},\tilde{W}}$ is concentrated in even, negative
cohomological degrees.

######  Proposition 3.4.

[Dav20, Thm.6.6] Let $Q$ be a finite quiver, and denote by $Q^{\mathrm{re}}$
the quiver obtained by removing all vertices that support a loop, along with
any arrows incident to at least one such vertex. There is an isomorphism of
Lie algebras

(25)
$\mathfrak{g}_{\tilde{Q},\tilde{W}}^{0}\cong\mathfrak{n}^{-}_{Q^{\mathrm{re}}}.$

Choose a torus $T=(\mathbb{C}^{*})^{l}$, and a weighting function
$\uptau\colon\tilde{Q}_{1}\rightarrow\mathbb{Z}^{l}$ defining an action of $T$
on the category of $\mathbb{C}\tilde{Q}$-modules, for which $\tilde{W}$ is
invariant. We denote by $\mathfrak{m}_{T}$ the maximal homogeneous ideal in
$\operatorname{\mathbf{H}}_{T}$. The following theorem is a consequence of the
purity of the natural mixed Hodge structure on
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$, see [Dav17b] for details.

######  Proposition 3.5.

[Dav17b, Thm.9.6] Let
$l\colon\mathbb{Z}^{r}\rightarrow\mathbb{Z}^{r^{\prime}}$ be a split
surjection of lattices, with complement $\mathbb{Z}^{r^{\prime\prime}}$,
inducing the split inclusion of tori

$T^{\prime}=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb{Z}^{r^{\prime}},\mathbb{C}^{*})\hookrightarrow
T=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb{Z}^{r},\mathbb{C}^{*})$

with complement
$T^{\prime\prime}=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb{Z}^{r-r^{\prime}},\mathbb{C}^{*})$.
There is an isomorphism of $\mathbb{N}^{Q_{0}}$-graded, cohomologically graded
$\operatorname{\mathbf{H}}_{T}$-modules

(26)
$\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}\cong\operatorname{\mathcal{A}}^{T^{\prime}}_{\tilde{Q},\tilde{W}}\otimes\operatorname{\mathbf{H}}_{T^{\prime\prime}}$

and an isomorphism of algebras

$\operatorname{\mathcal{A}}^{T^{\prime}}_{\tilde{Q},\tilde{W}}\cong\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}\otimes_{\operatorname{\mathbf{H}}_{T}}\operatorname{\mathbf{H}}_{T^{\prime}}.$

For the special case $r^{\prime}=0$, the above proposition yields an
isomorphism of $\operatorname{\mathbf{H}}_{T}$-modules

(27)
$\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}\cong\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}\otimes\operatorname{\mathbf{H}}_{T}$

and isomorphism of algebras

$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}\cong\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}/\mathfrak{m}_{T}\cdot\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}.$

We will be particularly interested in the action of the torus
$T=(\mathbb{C}^{*})^{2}=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb{Z}^{2},\mathbb{C}^{*})$
acting via the weighting function

(28) $\displaystyle\tilde{\uptau}\colon$
$\displaystyle\tilde{Q}_{1}\rightarrow\mathbb{Z}^{2}$ $\displaystyle
a\mapsto(1,0)$ $\displaystyle a^{*}\mapsto(0,1)$
$\displaystyle\omega_{i}\mapsto(-1,-1).$

###  3.9. Dimensional reduction

We recall from [Dav17a, Appendix.A] the following result.

######  Theorem 3.6.

[Dav17a, Thm.A.1] Let $X=Y\times\mathbb{A}^{n}$ be a smooth variety, let $f$
be a function on $X$ of weight one for the scaling action on $\mathbb{A}^{n}$,
so that we can write

$f=\sum_{i=1}^{n}x_{i}f_{i}$

for $x_{1},\ldots,x_{n}$ a system of coordinates on $\mathbb{A}^{n}$ and
$f_{1},\ldots,f_{n}$ functions on $Y$. Set $Z=Z(f_{1},\ldots,f_{n})$, let
$i\colon Z\hookrightarrow Y$ be the inclusion, and let $\uppi\hskip
0.50003pt\colon X\rightarrow Y$ be the projection. Then there is a natural
isomorphism

$i_{*}i^{!}\mathcal{F}\rightarrow\uppi\hskip
0.50003pt_{*}{}^{\mathfrak{p}}\\!\upphi_{f}\uppi\hskip
0.50003pt^{*}\mathcal{F}[2\dim(\uppi\hskip 0.50003pt)].$

Applying the result to $\mathbb{Q}_{Y}$ and passing to global sections yields
the isomorphism

$\operatorname{\mathbf{H}}^{\operatorname{BM}}(Z,\mathbb{Q})\rightarrow\operatorname{\mathbf{H}}(X,{}^{\mathfrak{p}}\\!\upphi_{f}\mathbb{Q})[2\dim(\uppi\hskip
0.50003pt)].$

The theorem extends in the obvious way to affine fibrations over stacks
[Dav17a, Cor.A.9]. In particular, for the affine fibration

$\mathfrak{M}_{\mathbf{d}}(\tilde{Q})\rightarrow\mathfrak{M}_{\mathbf{d}}(\overline{Q})$

the function $\operatorname{Tr}(\tilde{W})$ has weight one with respect to the
action that scales the fibres (i.e. the space of choices of matrices for each
of the loops $\omega_{i}$). As such, we obtain an isomorphism

$\Psi_{\mathbf{d}}\colon\operatorname{\mathbf{H}}^{\operatorname{BM}}(\mathfrak{M}^{T}_{\mathbf{d}}(\Pi_{Q}),\mathbb{Q})[-2\upchi_{Q}(\mathbf{d},\mathbf{d})]\cong\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}).$

We slightly twist this isomorphism888See [RS17, Lem.4.1] for the origin of
this sign. by setting
$\Psi^{\prime}_{\mathbf{d}}=(-1)^{\binom{\lvert\mathbf{d}\lvert}{2}}\Psi_{\mathbf{d}}$.
We use these sign-twisted isomorphisms, and the algebra structure on
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$, to induce an algebra
structure on

$\operatorname{\mathcal{A}}^{T}_{\Pi_{Q}}\coloneqq\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}^{\operatorname{BM}}(\mathfrak{M}^{T}_{\mathbf{d}}(\Pi_{Q}),\mathbb{Q}_{\operatorname{vir}}).$

Recall that we have defined the complex $\mathbb{Q}_{\operatorname{vir}}$ on a
smooth stack to be the constant sheaf, shifted on each component of the stack
by the dimension of the stack. Since $\mathfrak{M}_{\mathbf{d}}^{T}(\Pi_{Q})$
is not smooth, we need a new convention to make sense of the above definition:
on $\mathfrak{M}_{\mathbf{d}}^{T}(\Pi_{Q})$ we define
$\mathbb{Q}_{\operatorname{vir}}=\mathbb{Q}[-2\upchi(\mathbf{d},\mathbf{d})]$.

The algebra structure on $\operatorname{\mathcal{A}}^{T}_{\Pi_{Q}}$ induced by
the algebra structure on
$\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}$ and the isomorphism
$\Psi^{\prime}$ is the same as the one defined by Schiffmann and Vasserot in
[SV13], see [RS17, Appendix], [YZ16] for proofs.

###  3.10. Affinizing BPS Lie algebras

For pairs $(\tilde{Q},\tilde{W})$ of a tripled quiver with potential there is
extra structure on the CoHA
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$, ultimately derived from a
factorization structure on the stack of representations of the Jacobi algebra.
We briefly describe this here; full details for more general 3-Calabi–Yau
completions of 2-Calabi–Yau categories will appear in joint work with Tasuki
Kinjo. In the special case in which $Q$ is the one-loop quiver, i.e. the case
we need for the main results of this paper, this structure is as presented in
[KV19].

Under the isomorphism $j$ from (10) the element $j(\omega)$ is central in
$\operatorname{Jac}(\tilde{Q},\tilde{W})$, so that every
$\operatorname{Jac}(\tilde{Q},\tilde{W})$-module $\rho$ admits a canonical
decomposition as a $\operatorname{Jac}(\tilde{Q},\tilde{W})$-module

$\rho\cong\bigoplus_{\lambda\in\mathbb{C}}\rho_{\lambda}$

where $j(\omega)$ acts on $\rho_{\lambda}$ with unique generalised eigenvalue
$\lambda$. From this one deduces (see [Dav17b, Lem.4.1]) that for
$U\subset\mathbb{C}$ an open ball, if one denotes by
$\mathfrak{M}_{\mathbf{d}}^{U}(\tilde{Q})$ the open substack satisfying the
condition that each $\omega_{i}$ acts with generalised eigenvalues inside $U$,
the restriction

$\operatorname{\mathbf{H}}(\mathfrak{M}_{\mathbf{d}}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})})\rightarrow\operatorname{\mathbf{H}}(\mathfrak{M}^{U}_{\mathbf{d}}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})})$

is an isomorphism. Let $U^{\prime}$ and $U^{\prime\prime}$ be disjoint open
balls in $\mathbb{C}$, then via the Thom–Sebastiani isomorphism one constructs
the composition of morphisms

$\displaystyle\Delta\colon\operatorname{\mathbf{H}}(\mathfrak{M}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})\mathbb{Q}_{\operatorname{vir}}})\rightarrow$
$\displaystyle\operatorname{\mathbf{H}}(\mathfrak{M}^{U^{\prime}\coprod
U^{\prime\prime}}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathbb{Q}_{\operatorname{vir}})$
$\displaystyle\cong\operatorname{\mathbf{H}}(\mathfrak{M}^{U^{\prime}}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathbb{Q}_{\operatorname{vir}})\otimes\operatorname{\mathbf{H}}(\mathfrak{M}^{U^{\prime\prime}}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathbb{Q}_{\operatorname{vir}})$
$\displaystyle\cong\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}\otimes\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$

defining a coproduct on $\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$. It
is cocommutative since the space of choices of $U^{\prime}$ and
$U^{\prime\prime}$ is connected, so that we can continuously swap them. It is
compatible with $m$, since $m$ lifts to an algebra structure on the object

$(\mathfrak{M}(\tilde{Q})\xrightarrow{\lambda}\operatorname{\mathbf{Sym}}(\mathbb{A}^{1}))_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathbb{Q}_{\operatorname{vir}}$

where $\lambda$ is the morphism recording the generalised eigenvalues of
$j(\omega)$. It follows by the Milnor–Moore theorem that
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$ is a universal enveloping
algebra. Furthermore, by the support lemma [Dav17b, Lem.4.1]
$\lambda_{*}\mathcal{BPS}_{\tilde{Q},\tilde{W}}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$
is supported on the subspace of $\mathcal{M}(\tilde{Q})$ for which all of the
generalised eigenvalues of $j(\omega)$ are identical, and so

$\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}\coloneqq\mathfrak{g}_{\tilde{Q},\tilde{W}}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\subset\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$

is primitive for the coproduct $\Delta$, i.e. for
$\upalpha\in\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}$,
$\Delta(\upalpha)=1\otimes\upalpha+\upalpha\otimes 1$. By Theorem 3.2 and the
PBW theorem for $\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$ the
subspace $\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}$ has the same graded
dimensions as the space of primitive elements for
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$. Since all elements of
$\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}$ are primitive, we have shown

######  Proposition 3.7.

The subspace
$\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}\subset\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$
is closed under the commutator Lie bracket, and
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}=\operatorname{\mathbf{U}}(\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}})$.

######  Definition 3.8.

We call $\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}$ the affine BPS Lie algebra
for $(\tilde{Q},\tilde{W})$.

######  Remark 3.9.

There are no known examples of quivers with potential for which
$\mathfrak{g}_{Q^{\prime},W}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$
can be shown not to be closed under the commutator Lie bracket. It would be
interesting to know what class of BPS Lie algebras can be affinized this way,
and in particular whether the “partially fermionized” cousins [Dav22] of the
BPS Lie algebras $\mathfrak{g}_{\tilde{Q},\tilde{W}}$ belong to this class.

Let $\mathbb{C}^{*}$ act on $\mathbb{C}\tilde{Q}$-modules via the weighting
function $\uptau\colon\tilde{Q}_{1}\rightarrow\mathbb{Z}$ which sends $a$ to
$1$ and $a^{*}$ to $-1$ for all $a\in Q_{1}$, and $\omega_{i}$ to $0$ for all
$i\in Q_{0}$. Then the morphism $\lambda$ is $\mathbb{C}^{*}$-invariant, so
that we can repeat the construction of the commutative coproduct above for the
CoHA

$\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}.$

Via the symmetry that permutes the arrows $a,a^{*},\omega$, the algebra
$\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{Q^{(3)},\tilde{W}}$ carries a
cocommutative coproduct, and is a universal enveloping algebra, as long as
$\mathbb{C}^{*}$ acts on one of $a,a^{*},\omega$ with weight zero. The
coproduct is $\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-linear by
construction, so that we obtain a
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-linear Lie algebra of primitive
elements $\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}$:

######  Definition 3.10.

We denote by
$\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}\subset\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}$
the Lie algebra of primitives, which we call the deformed affinized BPS Lie
algebra for $(\tilde{Q},\tilde{W})$.

The “deformed” in the definition is justified by the following

######  Proposition 3.11.

There is an isomorphism of $\mathbb{N}^{Q_{0}}$-graded, cohomologically graded
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-modules.

(29)
$\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}\cong\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$

as well as an isomorphism of Lie algebras

$\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}\cong\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}/\mathfrak{m}_{\mathbb{C}^{*}}\cdot\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}.$

###### Proof.

Before we start, we remind the reader that
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}\neq\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$:
the first algebra has to do with the extra $\mathbb{C}^{*}$-action on
$\mathbb{C}\tilde{Q}$, the second has to do with the action studied at the end
of §3.4.

Since the coproduct is $\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-linear,
and $\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}$ is
free as a $\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-module by Proposition
3.5, it follows that the space of primitives for the coproduct (i.e.
$\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}$) is free as a
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-module, and $\upalpha$ is
primitive if and only if $t\cdot\upalpha$ is primitive, for $t$ a generator of
$\mathfrak{m}_{\mathbb{C}^{*}}$. The graded dimensions of this module are
determined by the existence of the isomorphisms (30) and (26), yielding (29).

Again by $\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-linearity of the
coproduct, the composition

$\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}/\mathfrak{m}_{\mathbb{C}^{*}}\\!\cdot\\!\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}\hookrightarrow\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}/\mathfrak{m}_{\mathbb{C}^{*}}\\!\cdot\\!\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}\xrightarrow{\cong}\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$

realises
$\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}/\mathfrak{m}_{\mathbb{C}^{*}}\cdot\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}$
as a subspace of primitive elements in
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$. But the graded dimensions
of
$\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}/\mathfrak{m}_{\mathbb{C}^{*}}\cdot\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}$
match those of $\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}$, by the first
statement, and the second statement follows. ∎

The following is a basic application of the Milnor–Moore theorem:

######  Proposition 3.12.

There is an isomorphism of
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$-algebra objects:

(30)
$\operatorname{\mathbf{U}}_{\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}}\\!(\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}})\rightarrow\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\tilde{Q},\tilde{W}}.$

###  3.11. Affinized finite type BPS algebras

Let $Q$ be an orientation of a finite type Dynkin diagram. We calculate the
(undeformed) affine BPS Lie algebra
$\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}$. We start with an easy lemma

######  Lemma 3.13.

Let $\mathcal{H}$ be an algebra object in
$\operatorname{\mathbf{Heis}}_{Q}\operatorname{-mod}$, such that there is an
isomorphism of underlying $\operatorname{\mathbf{Heis}}$-modules

$\mathcal{H}\cong\bigoplus_{m\in\mathbb{N}}\operatorname{\mathbf{Gr}}_{\bullet}^{F}(V_{m})[-2]^{\oplus
a_{m}}$

for integers $a_{m}$. Then up to isomorphism, $\mathcal{H}$ is determined by
the sub-algebra $\mathcal{H}^{0}$. Similarly, if $\mathfrak{g}$ is a Lie
algebra object in
$\operatorname{\mathbf{Heis}}\operatorname{-mod}_{\mathbb{Z}}$ such that there
is an isomorphism

$\mathfrak{g}\cong\bigoplus_{m\in\mathbb{N}}\operatorname{\mathbf{Gr}}_{\bullet}^{F}(V_{m})[-2]^{\oplus
b_{m}}$

for integers $b_{m}$, then $\mathfrak{g}$ is determined up to isomorphism by
the Lie sub-algebra $\mathfrak{g}^{0}$.

###### Proof.

Under the conditions of the Lemma, $\mathcal{H}$ is concentrated in
cohomological degrees indexed by $2\mathbb{Z}_{\geq 0}$, and the lowering
operator $q\colon\mathcal{H}^{i}\rightarrow\mathcal{H}^{i-2}$ is injective for
$i>0$. As such, if $\upalpha,\upbeta\in\mathcal{H}$ are not both of
cohomological degree zero, $\upalpha\star\upbeta$ is determined by
$q(\upalpha\star\upbeta)=q(\upalpha)\star\upbeta+\upalpha\star q(\upbeta)$,
and the result follows by induction. ∎

Given a $\mathbb{N}^{Q_{0}}$-graded Lie algebra $\mathfrak{g}$ we form the Lie
algebra $\mathfrak{g}\otimes_{\mathbb{Q}}\mathbb{Q}[D]$ by
$\mathbb{Q}[D]$-linear extension. We make this into a
$\operatorname{\mathbf{Heis}}_{Q}$-module by setting $q_{i}(g\otimes
D^{m})=mg\otimes D^{m-1}$ and $p_{i}(g\otimes D^{m})=\mathbf{d}_{i}g\otimes
D^{m+1}$, for $g$ of homogeneous $\mathbb{N}^{Q_{0}}$-degree $\mathbf{d}$. In
particular, for $Q$ a finite type Dynkin diagram with associated Lie algebra
$\mathfrak{g}$, the Lie algebra object
$\mathfrak{g}\otimes_{\mathbb{Q}}\mathbb{Q}[D]$ is a Lie algebra object in
$\operatorname{\mathbf{Heis}}\operatorname{-mod}_{\mathbb{Z}}$ satisfying the
assumptions of Lemma 3.13.

######  Proposition 3.14.

Let $Q$ be a quiver obtained by orienting a finite type Dynkin diagram. There
is an isomorphism of Lie algebras

$\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}\cong\mathfrak{n}_{Q}^{-}\otimes_{\mathbb{Q}}\mathbb{Q}[D]$

where $\mathfrak{n}_{Q}^{-}$ is half of the simple Lie algebra associated to
the underlying Dynkin diagram of $Q$.

###### Proof.

By Proposition 3.4 there is an isomorphism
$\hat{\mathfrak{g}}_{\tilde{Q},\tilde{W}}^{0}\cong\mathfrak{n}_{Q}^{-}$, and
then the result follows from Lemma 3.13. ∎

We leave it to the enthusiastic reader to calculate the deformed affine BPS
Lie algebra $\mathfrak{g}_{\tilde{Q},\tilde{W}}^{\mathbb{C}^{*}}$.

###  3.12. Nakajima quiver varieties

Fix a finite quiver $Q$, and a framing vector
$\mathbf{f}\in\mathbb{N}^{Q_{0}}$. We define the quiver $Q_{\mathbf{f}}$ to be
the quiver obtained by adding a single vertex $\infty$ to the set $Q_{0}$, and
for each $i\in Q_{0}$, adding $\mathbf{f}_{i}$ arrows from $\infty$ to $i$,
which we label $a_{i,m}$ for $m=1,\ldots,\mathbf{f}_{i}$. We define
$\widetilde{Q_{\mathbf{f}}}$ to be the quiver obtained by tripling
$Q_{\mathbf{f}}$, and set $Q^{+}_{\mathbf{f}}$ to be the quiver obtained by
removing the loop $\omega_{\infty}$ from $\widetilde{Q_{\mathbf{f}}}$. We
define

$W^{+}=\left(\sum_{a\in Q_{1}}[a,a^{*}]+\sum_{i\in
Q_{0}}\sum_{m=1}^{\mathbf{f}_{i}}a_{i,m}a^{*}_{i,m}\right)\left(\sum_{i\in
Q_{0}}\omega_{i}\right).$

Given a dimension vector $\mathbf{d}\in\mathbb{N}^{Q_{0}}$ we extend
$\mathbf{d}$ to a dimension vector $\mathbf{d}^{+}$ for $Q^{+}_{\mathbf{f}}$
by setting $\mathbf{d}^{+}_{\infty}=1$. We define
$\mathbb{A}_{Q^{+}_{\mathbf{f}},\mathbf{d}^{+}}^{\upzeta\operatorname{-ss}}\subset\mathbb{A}_{Q^{+}_{\mathbf{f}},\mathbf{d}^{+}}$
to be the subspace of representations $\mathcal{F}$ such that the one-
dimensional vector space $e_{\infty}\cdot\mathcal{F}$ generates $\mathcal{F}$
under the action of $\mathbb{C}Q^{+}_{\mathbf{f}}$. We call such modules
$\upzeta$-stable. Then we define the scheme

$\mathcal{M}_{\mathbf{f},\mathbf{d}}(Q)=\mathbb{A}_{Q^{+}_{\mathbf{f}},\mathbf{d}^{+}}^{\upzeta\operatorname{-ss}}/\operatorname{GL}_{\mathbf{d}}.$

This is the moduli scheme parameterising pairs of a
$\mathbf{d}^{+}$-dimensional $\upzeta$-stable
$\mathbb{C}Q^{+}_{\mathbf{f}}$-module $\mathcal{F}$, along with a
trivialisation $e_{\infty}\\!\cdot\\!\mathcal{F}\cong\mathbb{C}$.

We define
$\mathbb{A}_{\overline{Q_{\mathbf{f}}},\mathbf{d}^{+}}^{\upzeta\operatorname{-ss}}\subset\mathbb{A}_{\overline{Q_{\mathbf{f}}},\mathbf{d}^{+}}$
to be the subspace of representations $\mathcal{F}$ of the doubled framed
quiver such that the one-dimensional space $e_{\infty}\\!\cdot\\!\mathcal{F}$
generates $\mathcal{F}$ under the action of
$\mathbb{C}\overline{Q_{\mathbf{f}}}$. We consider the function

$\displaystyle\upmu_{\mathbf{d}}\colon$
$\displaystyle\mathbb{A}_{\overline{Q_{\mathbf{f}}},\mathbf{d}^{+}}^{\upzeta\operatorname{-ss}}\rightarrow\mathfrak{gl}_{\mathbf{d}}$
$\displaystyle((A_{a},A_{a}^{*})_{a\in Q_{1}},(I_{i,m},J_{i,m})_{i\in
Q_{0},m\leq\mathbf{f}_{i}})\mapsto\sum_{a\in
Q_{1}}[A_{a},A_{a}^{*}]+\sum_{i\in Q_{0},m\leq\mathbf{f}_{i}}I_{i,m}J_{i,m}.$

The group $\operatorname{GL}_{\mathbf{d}}$ acts freely on
$\mathbb{A}_{\overline{Q_{\mathbf{f}}},\mathbf{d}^{+}}^{\upzeta\operatorname{-ss}}$,
and thus on $\upmu^{-1}_{\mathbf{d}}(0)$, and we define the Nakajima quiver
variety
$\mathcal{N}_{\mathbf{f},\mathbf{d}}(Q)=\upmu_{\mathbf{d}}^{-1}(0)/\operatorname{GL}_{\mathbf{d}}$
[Nak94]. Via dimensional reduction, one may show that there is a natural
isomorphism (see [Dav20, Prop.6.3]).

(31)
$\operatorname{\mathbf{H}}(\mathcal{N}_{\mathbf{f},\mathbf{d}}(Q),\mathbb{Q}_{\operatorname{vir}})\cong\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{f},\mathbf{d}}(Q),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W^{+})}).$

Let $\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}\in\mathbb{N}^{Q_{0}}$ be a
pair of dimension vectors satisfying
$\mathbf{d}^{\prime}+\mathbf{d}^{\prime\prime}=\mathbf{d}$. We define
$\mathcal{M}_{\mathbf{f},\mathbf{d},\mathbf{d}^{\prime\prime}}(Q)$ to be the
moduli scheme of pairs of a short exact sequence

$0\rightarrow\mathcal{F}^{\prime}\rightarrow\mathcal{F}\rightarrow\mathcal{F}^{\prime\prime}\rightarrow
0$

along with a trivialisation $e_{\infty}\\!\cdot\\!\mathcal{F}\cong\mathbb{C}$,
where $\mathcal{F}$ is a $\mathbf{d}^{+}$-dimensional $\upzeta$-stable
$\mathbb{C}Q^{+}_{\mathbf{f}}$-module, and $\mathcal{F}^{\prime}$ is a
$\mathbf{d}$-dimensional $\mathbb{C}\tilde{Q}$-module, considered as a
$\mathbb{C}Q^{+}_{\mathbf{f}}$-module with dimension vector zero at the vertex
$\infty$. The module $\mathcal{F}^{\prime\prime}$ appearing in such a short
exact sequence is a $\mathbf{d}^{\prime\prime+}$-dimensional $\upzeta$-stable
$\mathbb{C}Q^{+}_{\mathbf{f}}$-module. We consider the standard correspondence
diagram

$\textstyle{\mathfrak{M}_{\mathbf{d}^{\prime}}(\tilde{Q})\times\mathcal{M}_{\mathbf{f},\mathbf{d}^{\prime\prime}}(Q)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}_{\mathbf{f},\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt_{1}\times\uppi\hskip 0.35002pt_{3}}$$\scriptstyle{\uppi\hskip
0.35002pt_{2}}$$\textstyle{\mathcal{M}_{\mathbf{f},\mathbf{d}}(Q).}$

Composing the Thom–Sebastiani isomorphism with the pushforward $\uppi\hskip
0.50003pt_{2,\star}$ and pullback $(\uppi\hskip 0.50003pt_{1}\times\uppi\hskip
0.50003pt_{3})^{\star}$ in vanishing cycle cohomology we define the operation

$\star\colon\operatorname{\mathbf{H}}(\mathfrak{M}_{\mathbf{d}^{\prime}}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})})\otimes\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{f},\mathbf{d}^{\prime\prime}}(Q),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W^{+})})\rightarrow\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{f},\mathbf{d}}(Q),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W^{+})})$

######  Proposition 3.15.

Let $Q$ be a finite quiver.

  * •

Via the isomorphism (31), the operation $\star$ defines an action of the CoHA
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$ on the vector space

$M_{\mathbf{f}}(Q)=\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}(\mathcal{N}_{\mathbf{f},\mathbf{d}}(Q),\mathbb{Q}_{\operatorname{vir}})$

  * •

The operation is $\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$-linear:
there are equalities
$u\bullet(\upalpha\star\upbeta)=(u\bullet\upalpha)\star\upbeta+\upalpha\star(u\bullet\upbeta)$.

###### Proof.

For the first part, see e.g. [YZ18]. The second part is proved as in
Proposition 4.1 below. ∎

For $Q$ the one loop quiver, and framing vector $\mathbf{f}=1$, we have
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W},1}\cong\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}\cong\mathbb{Q}[x]$.
One shows as in [Dav20] that for $1_{1}$ the copy of
$1\in\mathbb{Q}[x]\cong\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W},1}$ we
have

(32) $1\star=\mathfrak{p}_{1}$

where $\mathfrak{p}_{1}$ is the Nakajima raising operator [Nak97] on
$\operatorname{Hilb}_{n}(\mathbb{A}^{2})\cong\mathcal{M}_{1,n}(Q)$. From the
second part of Proposition 3.15 it follows that

(33) $(u\bullet\upalpha)\star-=[u\bullet,\upalpha\star](-)$

as operators on $\operatorname{Hilb}_{n}(\mathbb{A}^{2})$. We recall the
following theorem of Manfred Lehn:

######  Theorem 3.16 ([Leh99] Thm.3.10).

Write $\mathfrak{p}_{n}$ for Nakajima’s $n$th raising operator on
$\operatorname{Hilb}(\mathbb{A}^{2})$. Write
$\mathfrak{p}^{\prime}_{n}=[u\cdot,\mathfrak{p}_{n}]$. Then

$[\mathfrak{p}^{\prime}_{1},\mathfrak{p}_{n}]=-n\mathfrak{p}_{n+1}.$

We refer to [Nak97, Leh99] for the definition of the operators
$\mathfrak{p}_{n}$. Alternatively, via (33), (32) and Theorem 3.16, these
operators may be defined in terms of the action of
$1\in\operatorname{\mathcal{A}}_{Q^{(3)},\tilde{W},1}$ and $u\bullet$ on
$\operatorname{Hilb}(\mathbb{A}^{2})$.

###  3.13. Affine Yangian and shuffle algebra embedding

Let $T$ be a torus, acting on the quiver $Q^{\prime}$. The superpotential
$W=0$ is obviously $T$-invariant, and the vanishing cycle sheaf
${}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})}$
on $\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})$ is isomorphic to the constant
(shifted) perverse sheaf
$\mathbb{Q}_{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),\operatorname{vir}}$
and so there is an easy description of the vanishing cycle cohomology (leaving
out the overall cohomological shifts for now):

$\displaystyle\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})\cong$
$\displaystyle\operatorname{\mathbf{H}}(\operatorname{pt}/(\operatorname{GL}_{\mathbf{d}}\times
T),\mathbb{Q})$ $\displaystyle\cong$ $\displaystyle\mathbb{Q}[x_{1,1},\ldots
x_{1,\mathbf{d}_{1}},x_{2,1},\ldots,x_{r,\mathbf{d}_{r}}]^{\mathfrak{S}_{\mathbf{d}}}\otimes\mathbb{Q}[t_{1},\ldots
t_{l}]$

where $Q^{\prime}_{0}=\\{1,\ldots,r\\}$ and
$\mathfrak{S}_{\mathbf{d}}=\prod_{i\in
Q^{\prime}_{0}}\mathfrak{S}_{\mathbf{d}_{i}}$ is the product of symmetric
groups generated by permutations of the variables preserving their first
subscript.

######  Proposition 3.17.

Let the torus
$T=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb{Z},(\mathbb{C}^{*})^{2})$ act on
$\tilde{Q}$ via the function $\tilde{\uptau}$ from (28). The natural morphism
$\Phi\colon\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}\rightarrow\operatorname{\mathcal{A}}^{T}_{\tilde{Q}}$
of algebras is embedding of algebras.

The definition of $\upphi$ and proof that $\Phi$ preserves the algebra product
is given in [SV13]. The fact that it is an embedding of algebras is proved in
[Dav17b, Thm.10.2], slightly generalising and fixing a gap the proof in
[SV20].

For what follows, we introduce (the strictly positive part) of the affine
Yangian $\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})$, for
which our reference is [Tsy17]. Let $t_{1},t_{2},t_{3}$ be formal parameters
satisfying $t_{1}+t_{2}+t_{3}=0$. Then
$\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+}$ is the
algebra over $\mathbb{Q}[t_{1},t_{2},t_{3}]/(t_{1}+t_{2}+t_{3})$ generated by
symbols $e_{i}$ with $i\in\mathbb{Z}_{\geq 0}$ subject to the two relations

(34)
$\displaystyle[e_{i+3},e_{j}]-3[e_{i+2},e_{j+1}]+3[e_{i+1},e_{j+2}]-[e_{i},e_{j+3}]$
$\displaystyle+\upsigma_{2}([e_{i+1},e_{j}]-[e_{i},e_{j+1}])=-\upsigma_{3}(e_{i}e_{j}+e_{j}e_{i})$
(35)
$\displaystyle\mathrm{Sym}_{\mathfrak{S}_{3}}[e_{i_{1}},[e_{i_{2}},e_{i_{3}+1}]]=0$

where $\upsigma_{2}=t_{1}t_{2}+t_{1}t_{3}+t_{2}t_{3}$ and
$\upsigma_{3}=t_{1}t_{2}t_{3}$. The shuffle algebra embedding $\Phi$ is used
in [RSYZ20b] to prove that the relations (34) and (35) hold in
$\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}^{T}$:

######  Theorem 3.18.

[RSYZ20b, Thm.7.1.1] There is an algebra morphism

$\upiota\colon\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+}\rightarrow\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}^{T}$

sending $e_{i}$ to $\upalpha_{1}^{(i)}$. Furthermore, this morphism is
injective, so that
$\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+}$ may be
identified with the spherical subalgebra
$\mathcal{S}\\!\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2}}\subset\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2}}$
under $\upiota$, i.e. the subalgebra generated by
$\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2},1}$.

##  4\. Heisenberg actions on $\operatorname{\mathcal{A}}_{Q^{\prime},W}$

It turns out that $\operatorname{\mathbf{Heis}}_{Q^{\prime}}$-actions on
cohomological Hall algebras exist at a very high level of generality. We
explain this first, before zooming in on the CoHAs that this paper is most
concerned with, i.e. $\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}^{T}$.

###  4.1. Raising operators

######  Proposition 4.1.

Let $Q^{\prime}$ be an arbitrary quiver, with a $T$-action for some torus
$T=\operatorname{Hom}(N,\mathbb{C}^{*})$, and with $T$-invariant potential
$W\in\mathbb{C}Q^{\prime}/[\mathbb{C}Q^{\prime},\mathbb{C}Q^{\prime}]_{\mathrm{cyc}}$.
For each $i\in Q_{0}$ the morphism $\upalpha\mapsto
u\bullet_{i}\upalpha\coloneqq u_{(i)}\\!\cdot\\!\upalpha$ defines a derivation
of $\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$.

###### Proof.

Fix $\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}\in\mathbb{N}^{Q_{0}}$ and
set $\mathbf{d}=\mathbf{d}^{\prime}+\mathbf{d}^{\prime\prime}$. Consider the
following commutative diagram

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}(Q^{\prime})\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}(Q^{\prime})}$$\scriptstyle{\operatorname{Det}_{(i)}\times\operatorname{Det}_{(i)}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathfrak{M}^{T}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}(Q^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt_{1}\times\uppi\hskip 0.35002pt_{3}}$$\scriptstyle{\uppi\hskip
0.35002pt_{2}}$$\textstyle{\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Det}_{(i)}}$$\textstyle{\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\operatorname{B\\!}\mathbb{C}^{*}_{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes}$$\textstyle{\operatorname{B\\!}\mathbb{C}^{*}_{u}.}$

The morphisms $(\uppi\hskip 0.50003pt_{1}\times\uppi\hskip
0.50003pt_{3})^{\star}$ and $\uppi\hskip 0.50003pt_{2,\star}$ respect the
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$-actions induced by the
morphism to $\operatorname{B\\!}\mathbb{C}^{*}_{u}$ in the above diagram. On
the other hand, under the morphism

$\mathbb{Q}[u]=\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\xrightarrow{\otimes^{*}}\mathbb{Q}[u_{1},u_{2}]=\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})$

we have

(36) $\otimes^{*}\\!(u)=u_{1}\otimes 1+1\otimes u_{2}.$

So

$\displaystyle u\bullet_{i}(m(\upalpha\otimes\upbeta))=$ $\displaystyle
m(((u_{1}\otimes 1)+(1\otimes u_{2}))\bullet_{i}(\upalpha\otimes\upbeta))$
$\displaystyle=$ $\displaystyle m(u\bullet_{i}(\upalpha\otimes\upbeta)).$

∎

###  4.2. Lowering operators

The morphism $\otimes$ in the proof of Proposition 4.1 makes
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$ into a cocommutative
coalgebra. Let $\mathcal{L}$ be a line bundle on a scheme $X$, and let
$\mathcal{F}$ be a flat family of $\mathbf{d}$-dimensional
$\mathbb{C}Q^{\prime}$-modules. Then $\mathcal{L}\otimes\mathcal{F}$ is a flat
family of $\mathbf{d}$-dimensional $\mathbb{C}Q^{\prime}$-modules, where the
$\mathbb{C}Q^{\prime}$-action is induced by the given action on $\mathcal{F}$
and the natural isomorphism
$\mathcal{E}nd_{X}(\mathcal{L}\otimes\mathcal{F},\mathcal{L}\otimes\mathcal{F})\cong\mathcal{E}nd_{X}(\mathcal{F},\mathcal{F})$.
Recall that an $X$-point of $\mathfrak{M}_{\mathbf{d}}^{T}(Q^{\prime})$ is
defined by a $T$-torsor $p\colon E\rightarrow X$ along with a flat family
$\mathcal{F}$ of $\mathbf{d}$-dimensional $\mathbb{C}Q^{\prime}$-modules on
$E$ compatible with the $T$-action. We define a morphism

$\displaystyle A\colon$
$\displaystyle\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})\rightarrow\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})$
$\displaystyle(\mathcal{L},(p\colon E\rightarrow
X,\mathcal{F}))\mapsto(E,p^{*}\mathcal{L}\otimes\mathcal{F})$

which makes $\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})$ into a module over the
algebra object $\operatorname{B\\!}\mathbb{C}^{*}_{u}$ in stacks. We obtain an
induced morphism in vanishing cycle cohomology

$A^{\star}\colon\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})\rightarrow\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\otimes\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}).$

making $\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W,\mathbf{d}}$ into a
comodule for the coalgebra $\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$.
Via our fixed identification
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}=\mathbb{Q}[u]$, for
$\upalpha\in\operatorname{\mathcal{A}}_{Q^{\prime},W}$ we define coefficients
$\upalpha_{n}$ via the expansion

(37) $A^{\star}(\upalpha)=\sum_{n\geq 0}u^{n}\otimes\upalpha_{n}.$

######  Proposition 4.2.

The morphism

$\displaystyle\partial_{u}\colon$
$\displaystyle\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}\rightarrow\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$
$\displaystyle\upalpha\mapsto\upalpha_{1}$

defines a derivation of $\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$, where
$\upalpha_{1}$ is as defined in (37).

###### Proof.

We abbreviate
$\mathfrak{M}^{T}_{\mathbf{d}}=\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})$ etc.
to remove clutter. Consider the commutative diagram

$\textstyle{\operatorname{B\\!}\mathbb{C}^{*}\times\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{B\\!}\mathbb{C}^{*}\times\mathfrak{M}^{T}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{\operatorname{B\\!}\mathbb{C}^{*}\times\mathfrak{M}^{T}_{\mathbf{d}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathfrak{M}^{T}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}^{T}_{\mathbf{d}}}$

with morphisms of points over a scheme $X$ defined by

$\displaystyle a\colon$
$\displaystyle(\mathcal{L},(E\xrightarrow{p}X,\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime}))\mapsto(E\rightarrow
X,p^{*}\mathcal{L}\otimes\mathcal{F}^{\prime},p^{*}\mathcal{L}\otimes\mathcal{F}^{\prime\prime})$
$\displaystyle b\colon$
$\displaystyle(\mathcal{L},(E\xrightarrow{p}X,\mathcal{F}^{\prime}\rightarrow\mathcal{F}\rightarrow\mathcal{F}^{\prime\prime}))\mapsto(E\rightarrow
X,(p^{*}\mathcal{L}\otimes\mathcal{F}^{\prime}\rightarrow
p^{*}\mathcal{L}\otimes\mathcal{F}\rightarrow
p^{*}\mathcal{L}\otimes\mathcal{F}^{\prime\prime})).$

This induces the commutative diagram

$\textstyle{\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\otimes\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}},{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\otimes
m^{\prime}}$$\textstyle{\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\otimes\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}},{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}},{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\star}}$$\scriptstyle{m^{\prime}}$$\textstyle{\operatorname{\mathbf{H}}(\mathfrak{M}^{T}_{\mathbf{d}},{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A^{\star}}$

from which we deduce that $(m(\upalpha\otimes\upbeta))_{1}$ is equal to $m$
applied to the $u$-coefficient of $a^{\star}(\upalpha\otimes\upbeta)$. Here we
have also used that the pullback $s^{\star}$ commutes with the
$\operatorname{\mathbf{H}}_{\mathbb{C}_{u}^{*}}$-action, which follows from
the formula (12). On the other hand $a$ factorises as

$\operatorname{B\\!}\mathbb{C}_{u}^{*}\times\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}\xrightarrow{\Delta\times\operatorname{id}\times\operatorname{id}}\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}\times_{\operatorname{B\\!}T}\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}\xrightarrow{A\times
A}\mathfrak{M}^{T}_{\mathbf{d}^{\prime}}\times\mathfrak{M}^{T}_{\mathbf{d}^{\prime\prime}}$

and so since $\Delta^{\star}(p(u_{1},u_{2}))=p(u,u)$ the $u$ coefficient of
$a^{*}(\upalpha\otimes\upbeta)$ is equal to
$\upalpha_{1}\otimes\upbeta+\upalpha\otimes\upbeta_{1}$. ∎

######  Proposition 4.3.

For each $\mathbf{d}\in\mathbb{N}^{Q^{\prime}_{0}}$ there is an action of
$\operatorname{\mathbf{Heis}}_{Q^{\prime}}$ on
$\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W,\mathbf{d}}$ defined by letting
$p_{i}$ act by $u\bullet_{i}$, and letting $q_{i}$ act by $\partial_{u}$. This
action has central charge $\mathbf{d}$.

###### Proof.

Consider the composition defined on $X$-points by

$\displaystyle\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})\xrightarrow{A}\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})\xrightarrow{l_{i}}\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\mathfrak{M}^{T}_{\mathbf{d}}(Q^{\prime})$
$\displaystyle(\mathcal{L},(E\xrightarrow{p}X,\mathcal{F}))\mapsto(\mathcal{L}^{\otimes\mathbf{d}_{i}}\otimes\operatorname{Det}(\mathcal{F}_{i}),p^{*}\mathcal{L}\otimes\mathcal{F}).$

The induced endomorphism $F=(l_{i}\circ A)^{\star}$ of
$\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\otimes\operatorname{\mathbf{H}}(\mathfrak{M}_{\mathbf{d}}(Q^{\prime}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)})$
satisfies

$\displaystyle F(u\otimes\upalpha)=(\mathbf{d}_{i}(u\otimes 1)+(1\otimes
u_{(i)}))\bullet A^{\star}(\upalpha)$

from which we deduce that

$\displaystyle\partial_{u}(u\bullet_{i}\upalpha)=$
$\displaystyle(u_{(i)}\cdot\upalpha)_{1}$ $\displaystyle=$
$\displaystyle\mathbf{d}_{i}\upalpha_{0}+u_{(i)}\cdot\upalpha_{1}$
$\displaystyle=$
$\displaystyle\mathbf{d}_{i}\upalpha+u\bullet_{i}(\partial_{u}\upalpha)$

as required. ∎

Combining Propositions 4.1, 4.2 and 4.3 yields

######  Theorem 4.4.

The algebra $\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$ is an algebra
object in the category
$\operatorname{\mathbf{Heis}}_{Q^{\prime}}\operatorname{-mod}_{\mathbb{Z}}$,
for which the underlying $\operatorname{\mathbf{Heis}}_{Q^{\prime}}$-module is
integrable. The decomposition

$\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}\cong\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q^{\prime}_{0}}}(\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W})_{[2\mathbf{d}]}$

according to central charge of the
$\operatorname{\mathbf{Heis}}_{Q^{\prime}}$-action is given by the standard
decomposition (15) according to dimension vector.

######  Proposition 4.5.

Let $Q^{\prime}$ be an arbitrary quiver with potential
$W\in\mathbb{C}Q^{\prime}/[\mathbb{C}Q^{\prime},\mathbb{C}Q^{\prime}]_{\mathrm{cyc}}$.
The inclusion
$\mathfrak{g}^{T}_{Q^{\prime},W}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}\hookrightarrow\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$
is an inclusion of $\operatorname{\mathbf{Heis}}$-modules.

###### Proof.

Since the image of
$\mathfrak{g}^{T}_{Q^{\prime},W}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}_{u}}$
is just the closure of the image of
$\mathfrak{g}^{T}_{Q^{\prime},W}\subset\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$
under the raising operator $u\bullet$, the proposition follows from the claim
that the lowering operator $\partial_{u}$ annihilates
$\mathfrak{g}^{T}_{Q^{\prime},W}$. Consider the following diagram

$\textstyle{\operatorname{B\\!}\mathbb{C}^{*}_{u}\times\mathfrak{M}_{\mathbf{d}}^{T}(Q^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p\coloneqq\mathtt{JH}^{T}\circ\uppi\hskip
0.35002pt_{\mathfrak{M}_{\mathbf{d}}^{T}(Q^{\prime})}}$$\scriptstyle{A}$$\textstyle{\mathfrak{M}_{\mathbf{d}}(Q^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathtt{JH}^{T}}$$\textstyle{\mathcal{M}^{T}_{\mathbf{d}}(Q^{\prime}).}$

We claim that the diagram commutes: this is easy to see at the level of
geometric points, from which the claim follows. Thus there is a natural
isomorphism

$\mathtt{JH}^{T}_{*}A_{*}(\mathbb{Q}\boxtimes{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}})\cong\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\otimes\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}}$

and so passing to the first perverse truncation and using the perverse
vanishing result (18)

${}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}^{T}_{*}A_{*}(\mathbb{Q}\boxtimes{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}})\cong{}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}^{T}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}}.$

It follows that the morphism

${}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}^{T}_{*}\left({}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}}\rightarrow
A_{*}(\mathbb{Q}_{\operatorname{B\\!}\mathbb{C}^{*}_{u}}\boxtimes{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}})\right)$

factors through the morphism

$\operatorname{\mathbf{H}}^{0}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\otimes{}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}_{*}^{T}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}}\rightarrow\operatorname{\mathbf{H}}(\operatorname{B\\!}\mathbb{C}^{*}_{u},\mathbb{Q})\otimes{}^{\mathfrak{p}^{\prime}}\\!\uptau^{\leq
1}\mathtt{JH}_{*}^{T}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\operatorname{vir}}$

and $A^{\star}(\upalpha)=1\otimes\upalpha$ for
$\upalpha\in\mathfrak{P}_{1}\\!\operatorname{\mathcal{A}}^{T}_{Q^{\prime},W}$.
As such, $\upalpha_{1}=0$ as required. ∎

##  5\. Proofs of main results

Let $Q=Q^{(1)}$ be the Jordan quiver, with one vertex and one loop, so
$\tilde{Q}=Q^{(3)}$ is the quiver with three loops. Then for every $d\geq 1$
there are $q$ absolutely indecomposable $d$-dimensional
$\mathbb{F}_{q}Q$-modules, i.e. $\mathtt{a}_{Q,d}(q)=q$, and so

(38) $\mathfrak{g}_{\tilde{Q},\tilde{W},d}\cong\mathbb{Q}[2]$

by (24). We define

$\mathfrak{g}_{\mathbb{A}^{2}}^{T}\coloneqq\mathfrak{g}_{Q^{(3)},\tilde{W}}^{T}\quad\hat{\mathfrak{g}}_{\mathbb{A}^{2}}^{T}\coloneqq\hat{\mathfrak{g}}_{Q^{(3)},\tilde{W}}^{T}$

the deformed and the deformed affinized BPS Lie algebras for the pair
$(Q^{(3)},\tilde{W})$, respectively.

######  Proposition 5.1.

Let the torus
$T=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb{Z},(\mathbb{C}^{*})^{r})$ act on
the quiver $Q^{(3)}$, with $\tilde{W}$ $T$-invariant. Then the Lie bracket on
$\mathfrak{g}^{T}_{\mathbb{A}^{2}}$ vanishes.

###### Proof.

By (38) and Theorem 3.3 there is an isomorphism of cohomologically graded
vector spaces

$\mathfrak{g}^{T}_{\mathbb{A}^{2},d}\cong\operatorname{\mathbf{H}}_{T}[2]$

for every $d\geq 1$. Let
$\upalpha^{(0)}_{d^{\prime}}\in\mathfrak{g}^{T}_{\mathbb{A}^{2},d^{\prime}}$
and
$\upalpha^{(0)}_{d^{\prime\prime}}\in\mathfrak{g}^{T}_{\mathbb{A}^{2},d^{\prime\prime}}$
be the unique (up to scalar) nonzero elements of cohomological degree $-2$.
Then

$[\upalpha_{d^{\prime}}^{(0)},\upalpha_{d^{\prime\prime}}^{(0)}]\in\mathfrak{g}^{T}_{\mathbb{A}^{2},d^{\prime}+d^{\prime\prime}}$

has cohomological degree $-4$, and so is zero. For arbitrary
$p(\mathbf{t}),q(\mathbf{t})\in\operatorname{\mathbf{H}}_{T}$ it follows that

$[p(\mathbf{t})\\!\cdot\\!\upalpha_{d^{\prime}}^{(0)},q(\mathbf{t})\\!\cdot\\!\upalpha_{d^{\prime\prime}}^{(0)}]=p(\mathbf{t})q(\mathbf{t})\\!\cdot\\![\upalpha_{d^{\prime}}^{(0)},\upalpha_{d^{\prime\prime}}^{(0)}]=0.$

∎

By (38) we have the equality of generating series

$\displaystyle\upchi(\hat{\mathfrak{g}}_{\mathbb{A}^{2}})\coloneqq$
$\displaystyle\sum_{i,j\in\mathbb{Z}}\dim(\hat{\mathfrak{g}}_{\mathbb{A}^{2},i}^{j})t^{i}q^{j}$
(39) $\displaystyle=$ $\displaystyle q^{-2}t(1-t)^{-1}(1-q^{2})^{-1}.$

From the isomorphism
$\hat{\mathfrak{g}}_{\mathbb{A}^{2}}^{\mathbb{C}^{*}}\cong\hat{\mathfrak{g}}_{\mathbb{A}^{2}}\otimes\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}$
(Proposition 3.11) we deduce the equality

(40)
$\upchi(\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}})=q^{-2}t(1-t)^{-1}(1-q^{2})^{-2}.$

We denote by $\upalpha_{d}^{(0)}$ a basis element of
$\mathfrak{g}_{\mathbb{A}^{2},d}^{-2}$, and set $\upalpha_{d}^{(m)}\coloneqq
u^{m}\bullet\upalpha_{d}^{(0)}$. Then the elements $\upalpha_{d}^{(m)}$ with
$d\geq 1$ and $m\geq 0$ form a basis for
$\hat{\mathfrak{g}}_{\mathbb{A}^{2}}$.

######  Proposition 5.2.

For every $d\geq 1$ we have the inequality

(41) $[\upalpha_{1}^{(1)},\upalpha_{d}^{(0)}]=\lambda_{d}\upalpha_{d+1}^{(0)}$

for some nonzero $\lambda_{d}\in\mathbb{Q}$.

###### Proof.

We consider the $\operatorname{\mathcal{A}}_{\mathbb{A}^{2}}$-action on
$M_{1,d}(Q)\cong\bigoplus_{d\in\mathbb{Z}_{\geq
0}}\operatorname{\mathbf{H}}(\operatorname{Hilb}_{d}(\mathbb{C}^{2}),\mathbb{Q}_{\operatorname{vir}})$
recalled in §3.12. We denote by
$\rho\colon\hat{\mathfrak{g}}_{\mathbb{A}^{2}}\rightarrow\mathrm{End}(\operatorname{Hilb}(\mathbb{A}^{2}))$
the action restricted to $\hat{\mathfrak{g}}_{\mathbb{A}^{2}}$. Then
$\rho(\upalpha_{0}^{(0)})=\mathfrak{p}_{1}$ is the action of the Nakajima
raising operator and from

$u\bullet(\upalpha\star m)=(u\bullet\upalpha)\star m+\upalpha\star(u\bullet
m)$

we deduce that

$\rho(\upalpha_{1}^{(1)})=[u\bullet,\rho(\upalpha_{1}^{(0)})]=\mathfrak{p}_{1}^{\prime}.$

From Theorem 3.16 we deduce inductively that

(42)
$\rho\left(\mathbf{ad}_{\upalpha_{1}^{(1)}}^{n}(\upalpha_{0}^{(0)})\right)\neq
0.$

Set
$\upalpha=\mathbf{ad}_{\upalpha_{1}^{(1)}}^{n}(\upalpha_{0}^{(0)})\in\widehat{\mathfrak{g}}_{\mathbb{A}^{2},d+1}$.
By (42), $\upalpha\neq 0$. On the other hand, $\upalpha_{d+1}^{(0)}$ is the
unique nonzero element of $\widehat{\mathfrak{g}}_{\mathbb{A}^{2},d+1}$ of
cohomological degree $-2$, which is the cohomological degree of $\upalpha$,
and the result follows by induction. ∎

######  Corollary 5.3.

The affinized BPS Lie algebra $\hat{\mathfrak{g}}_{\mathbb{A}^{2}}$ is
generated by $\upalpha_{1}^{(m)}$ for $m\geq 0$.

###### Proof.

We assume, for an inductive argument, that the Lie subalgebra
$\mathfrak{g}^{\prime}\subset\hat{\mathfrak{g}}_{\mathbb{A}^{2}}$ generated by
$\upalpha_{1}^{(m)}$ for $m\geq 0$ contains all of the elements
$\upalpha_{i^{\prime}}^{(m)}$ for $i^{\prime}<i$ and $m\in\mathbb{Z}_{\geq
0}$. From Proposition 5.2 we deduce that, possibly after replacing
$\upalpha_{d}^{(0)}$ by itself multiplied by some
$\lambda\in\mathbb{Q}\setminus\\{0\\}$, we have the equality

$[\upalpha_{1}^{(1)},\upalpha_{d}^{(0)}]=\upalpha_{d+1}^{(0)}.$

Acting by $u^{m}\bullet$, we find

$\sum_{0\leq r\leq
m}\binom{m}{r}[\upalpha_{1}^{(1+r)},\upalpha_{d}^{(m-r)}]=\upalpha_{d+1}^{(m)}.$

∎

######  Theorem 5.4.

There is an isomorphism
$F\colon\hat{\mathfrak{g}}_{\mathbb{A}^{2}}\cong\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}$.

###### Proof.

This is a corollary of (39), Theorem 4.4, Corollary 5.3, which states that
$\hat{\mathfrak{g}}_{\mathbb{A}^{2}}$ is generated by
$\hat{\mathfrak{g}}_{\mathbb{A}^{2},1}=\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!V_{2}$,
and Corollary 2.5. ∎

######  Corollary 5.5.

There is an isomorphism of algebras

$\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}\cong\operatorname{\mathbf{U}}(\operatorname{\mathbf{Gr}}^{F}_{\bullet}\\!W^{+}_{1+\infty}).$

In particular, $\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}$ is spherically
generated (i.e. it is generated by
$\operatorname{\mathcal{A}}_{\mathbb{C}^{2},1}$).

The second statement in the corollary follows from Lemma 2.2.

Now we reintroduce the torus $T=(\mathbb{C}^{*})^{2}$, acting by independently
rescaling the two coordinates of $\mathbb{A}^{2}$ (i.e. via the weighting
$\tilde{\uptau}$ of (28)), and the fully equivariant CoHA
$\operatorname{\mathcal{A}}^{T}_{\mathbb{C}^{2}}$.

######  Theorem 5.6.

If $T^{\prime}$ is a torus acting on $Q^{(3)}$, leaving $\tilde{W}$ invariant,
then $\operatorname{\mathcal{A}}^{T^{\prime}}_{Q^{(3)},\tilde{W}}$ is
spherically generated. In particular, the algebra
$\operatorname{\mathcal{A}}^{T}_{\mathbb{C}^{2}}$ is spherically generated.

###### Proof.

By (27) there is a (non-canonical) isomorphism of
$\operatorname{\mathbf{H}}_{T^{\prime}}$-modules

$\operatorname{\mathcal{A}}^{T^{\prime}}_{\mathbb{C}^{2}}\cong\operatorname{\mathcal{A}}_{\mathbb{C}^{2}}\otimes\operatorname{\mathbf{H}}_{T^{\prime}}$

and we set $\tilde{\upalpha}_{1}^{(0)}=\upalpha_{1}^{(0)}\otimes 1$ via this
isomorphism999$\tilde{\upalpha}_{1}^{(0)}$ is defined, up to multiplication by
a scalar, by the condition that it lies in cohomological degree $-2$ and lies
in
$\mathfrak{g}^{T}_{\mathbb{A}^{2}}\subset\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2}}$.,
and $\tilde{\upalpha}_{1}^{(m)}=u^{m}\bullet\tilde{\upalpha}_{1}^{(0)}$. For
$d\geq 1$ we define a morphism of
$\operatorname{\mathbf{H}}_{T^{\prime}}$-modules

$\displaystyle\Psi_{d}\colon\mathbb{Q}[u]\otimes\operatorname{\mathbf{H}}_{T^{\prime}}\rightarrow$
$\displaystyle\operatorname{\mathcal{A}}^{T^{\prime}}_{\mathbb{C}^{2}}$
$\displaystyle u^{m}\otimes t^{n}\mapsto$ $\displaystyle
t^{n}\left([\tilde{\upalpha}_{1}^{(1)},-]^{d-1}(\tilde{\upalpha}_{1}^{(m)})\right)$

and define

$\Psi=\oplus_{d\geq 1}\Psi_{d}\colon\bigoplus_{d\geq
1}\mathbb{Q}[u]\otimes\operatorname{\mathbf{H}}_{T^{\prime}}\rightarrow\operatorname{\mathcal{A}}^{T^{\prime}}_{\mathbb{C}^{2}}.$

Finally we define

$\displaystyle\Phi\colon\operatorname{\mathbf{Sym}}_{\operatorname{\mathbf{H}}_{T^{\prime}}}\\!\left(\bigoplus_{d\geq
1}\mathbb{Q}[u]\otimes\operatorname{\mathbf{H}}_{T^{\prime}}\right)\rightarrow\operatorname{\mathcal{A}}^{T}_{\mathbb{C}^{2}}$

via $\Psi$ and the CoHA product on the target. This is a morphism of free
$\operatorname{\mathbf{H}}_{T^{\prime}}$-modules that (via Lemma 2.3,
Corollary 5.5 and Proposition 3.5) becomes an isomorphism after applying
$\otimes_{\operatorname{\mathbf{H}}_{T^{\prime}}}\mathfrak{m}_{T^{\prime}}$.
It follows that $\Phi$ is an isomorphism, and in particular, the algebra
$\operatorname{\mathcal{A}}^{T^{\prime}}_{\mathbb{C}^{2}}$ is spherically
generated. ∎

There is a partial converse to this theorem:

######  Proposition 5.7.

Let $Q$ be a quiver without loops that is not an orientation of a finite type
Dynkin quiver. Then $\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}$ is
not spherically generated, i.e. it is not generated by the subspaces
$\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W},1_{i}}$ for $i\in Q_{0}$.

###### Proof.

This follows for cohomological degree reasons. The conditions on $Q$ imply
that $\mathtt{a}_{Q,\mathbf{d}}(q)$ is not a constant for some
$\mathbf{d}\in\mathbb{N}^{Q_{0}}$. From (27) and (24) we deduce that
$\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W}}$ has a summand in
strictly negative cohomological degree. On the other hand

$\operatorname{\mathcal{A}}^{T}_{\tilde{Q},\tilde{W},1_{i}}\cong\operatorname{\mathbf{H}}_{T}$

as a cohomologically graded vector space, so the collection of such spaces for
$i\in Q_{0}$ generates an algebra lying in positive cohomological degrees. ∎

######  Theorem 5.8.

There is an isomorphism of algebras
$\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+}\rightarrow\operatorname{\mathcal{A}}^{T}_{\mathbb{A}^{2}}$
sending $e_{i}$ to $\tilde{\upalpha}_{1}^{(i)}$.

###### Proof.

This is an immediate corollary of Theorems 3.18 and 5.6. ∎

We can now give a more complete statement regarding the relation between BPS
Lie algebras and $W^{+}_{1+\infty}$.

######  Corollary 5.9.

Let $\mathbb{C}^{*}$ act with weights $(1,0)$ on the two coordinates of
$\mathbb{A}^{2}$. Then there is an isomorphism

$\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}}\cong\operatorname{\mathbf{R}}_{F}[W^{+}_{1+\infty}]$

between the deformed affinized BPS Lie algebra for the pair
$(Q^{(3)},\tilde{W})$ and the Rees Lie algebra of $W^{+}_{1+\infty}$, and thus
an isomorphism of algebras

$\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\mathbb{C}^{2}}\cong\operatorname{\mathbf{U}}_{\mathbb{Q}[t]}(\operatorname{\mathbf{R}}_{F}[W^{+}_{1+\infty}]).$

###### Proof.

Let $T=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb{Z},(\mathbb{C}^{*})^{2})$ act
on $\tilde{Q}$ via the weighting
$\tilde{\uptau}\colon\tilde{Q}_{1}\rightarrow\mathbb{Z}^{2}$. By Proposition
3.5 there is an isomorphism of algebras

(43)
$\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\mathbb{C}^{2}}\cong\operatorname{\mathcal{A}}^{T}_{\mathbb{C}^{2}}\otimes_{\operatorname{\mathbf{H}}_{T}}\mathbb{Q}[t_{1},t_{2}]/(t_{2}).$

By Theorem 5.8 and (43) there is an isomorphism of algebras

(44) $\displaystyle
F\colon(\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+})_{t_{2}=0}\cong\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\mathbb{C}^{2}}$

sending $t_{1}$ to $t$ and $e_{i}$ to $\upalpha_{1}^{(i)}$. By 3.12 there is
an isomorphism of algebras

$G\colon\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}}\cong\operatorname{\mathbf{U}}_{\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}}\\!\\!\left(\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}}\right).$

Composing $F$ and $G$, we obtain an isomorphism

$\operatorname{\mathbf{U}}_{\operatorname{\mathbf{H}}_{\mathbb{C}^{*}}}\\!\\!\left(\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}}\right)\rightarrow(\mathcal{Y}_{t_{1},t_{2},t_{3}}(\widehat{\mathfrak{gl}(1)})^{+})_{t_{2}=0}$

sending $\tilde{\upalpha}_{1}^{(i)}$ to $e_{i}$. Note that, after setting
$t_{2}=0$, $\upsigma_{3}=0$, and both the identities (34) and (35) become Lie
algebra identities. It’s easy to check that the identities

(45)
$\displaystyle[zD^{i+3},zD^{j}]-3[zD^{i+2},zD^{j+1}]+3[zD^{i+1},zD^{j+2}]-[zD^{i},zD^{j+3}]$
$\displaystyle+t^{2}([zD^{i+1},zD^{j}]-[zD^{i},zD^{j+1}])=0$ (46)
$\displaystyle\mathrm{Sym}_{\mathfrak{S}_{3}}[zD^{i_{1}},[zD^{i_{2}},zD^{i_{3}+1}]]=0$

hold in $\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]$, so that there is a
well-defined morphism of Lie algebras

(47)
$F\colon\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}}\rightarrow\operatorname{\mathbf{R}}[W^{+}_{1+\infty}]$

and induced morphism of algebras

(48)
$\operatorname{\mathbf{U}}_{\mathbb{Q}[t]}(\hat{\mathfrak{g}}^{\mathbb{C}^{*}}_{\mathbb{A}^{2}})\cong\operatorname{\mathcal{A}}^{\mathbb{C}^{*}}_{\mathbb{C}^{2}}\rightarrow\operatorname{\mathbf{U}}_{\mathbb{Q}[t]}(\operatorname{\mathbf{R}}[W^{+}_{1+\infty}])$

both sending $\tilde{\upalpha}_{1}^{(i)}$ to $zD^{i}$. The targets of both
morphisms (47) and (48) are spherically generated by Lemma 2.2, and so both
morphisms are surjective. On the other hand, the domain and target have the
same generating series by (40) and we are done. ∎

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