---
title: Kontsevich–Soibelman Lie Algebra
url: https://www.emergentmind.com/topics/kontsevich-soibelman-lie-algebra
type: topic
---

# Kontsevich–Soibelman Lie Algebra

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The Kontsevich–Soibelman Lie algebra is a $\mathbb{Z}$–graded super-Lie algebra of BPS states obtained from a cohomological Hall algebra built on quiver moduli. In the formulation based on quiver representations, one starts from the direct sum $A=\bigoplus_\gamma H^*(M_\gamma)$, equips it with a convolution product $\mu$ coming from extension correspondences, and then passes to the super-commutator bracket. The same object admits a generating-function realization in a quantum torus, where wall-crossing is expressed by quantum dilogarithm identities, and it also coincides with the BPS scattering algebra defined from $S$-matrix residues. In the preprojective setting, the associated BPS Lie algebra $\mathfrak{g}_{\Pi_Q}$ is extracted from the zeroth piece of the less perverse filtration on the preprojective CoHA, which is isomorphic to a universal enveloping algebra [1812.05801; 2007.03289].

## 1. Quiver-theoretic origin and Hall-algebra construction

Let $\Gamma \simeq \mathbb{Z}^{|V|}$ be the lattice of dimension-vectors $\gamma=(\gamma_v)_{v\in V}$ of a finite quiver $Q=(V,A)$. The Dirac–Schwinger–Zwanziger, or Euler, pairing is the bilinear form
$$
\langle \gamma_1,\gamma_2\rangle
=
\sum_{v\in V}\gamma_{1,v}\gamma_{2,v}
-
\sum_{a\in A}\gamma_{1,t(a)}\gamma_{2,h(a)},
$$
where for an arrow $a\colon v\to w$, $t(a)=v$ and $h(a)=w$. For each $\gamma$ one defines the affine representation space
$$
R_\gamma
=
\bigoplus_{a\colon v\to w}\mathrm{Hom}(\mathbb{C}^{\gamma_v},\mathbb{C}^{\gamma_w}),
$$
with natural action of the gauge group
$$
G_\gamma=\prod_{v\in V}\mathrm{GL}(\gamma_v).
$$
The complex moment map is
$$
\mu_{\mathbb{C}}\colon R_\gamma\to \mathrm{Lie}\,G_\gamma,\qquad
\mu_{\mathbb{C}}((\phi_a))_v
=
\sum_{a:\,t(a)=v}\phi_a\phi_a^\dagger
-
\sum_{a:\,h(a)=v}\phi_a^\dagger\phi_a,
$$
and the affine GIT quotient of the zero-locus is the stack, or coarse moduli space,
$$
M_\gamma=\mu_{\mathbb{C}}^{-1}(0)\,//\,G_\gamma.
$$
The cohomological Hall algebra is defined by
$$
A_\gamma:=H^*_{G_\gamma}(R_\gamma)\simeq H^*(M_\gamma),
$$
graded by cohomological degree [1812.05801].

The Hall product is constructed from short exact sequences of quiver representations. For a decomposition $\gamma=\gamma_1+\gamma_2$, one considers the correspondence
$$
R_{\gamma_1}\times R_{\gamma_2}\leftarrow E_{\gamma_1,\gamma_2}\to R_\gamma,
$$
where $E_{\gamma_1,\gamma_2}$ parametrizes tuples $(\phi^{(1)},\phi^{(2)},V')$ with $V'\subset \mathbb{C}^\gamma$ a subspace of dimension $\gamma_2$ on which $\phi$ restricts to $\phi^{(2)}$, and so forth. Writing the projections as
$$
p_{12}\colon E_{\gamma_1,\gamma_2}\to R_{\gamma_1}\times R_{\gamma_2},\qquad
p_{13}\colon E_{\gamma_1,\gamma_2}\to R_\gamma,
$$
the convolution product
$$
\mu_{\gamma_1,\gamma_2}\colon A_{\gamma_1}\otimes A_{\gamma_2}\to A_{\gamma_1+\gamma_2}
$$
is given by
$$
\mu_{\gamma_1,\gamma_2}(\alpha,\beta)
=
(p_{13})_*\Bigl(p_{12}^*(\alpha\boxtimes\beta)\cup [E_{\gamma_1,\gamma_2}]^{\mathrm{vir}}\Bigr).
$$
In concrete terms, if $\{u_i\}$ and $\{v_j\}$ are bases of $H^*(R_{\gamma_1})$ and $H^*(R_{\gamma_2})$, then
$$
\mu(u_i,v_j)=\sum_k c_{i,j}^k\,w_k,
$$
with structure constants
$$
c_{i,j}^k
=
\int_{[E_{\gamma_1,\gamma_2}]}
p_{12}^*(u_i\boxtimes v_j)\cup p_{13}^*(w_k^\vee).
$$

This construction supplies the algebraic substrate of the Kontsevich–Soibelman Lie algebra: the Lie bracket is not imposed independently, but induced from the associative Hall product. A plausible implication is that the relevant Lie-theoretic information is already encoded in the geometry of extension correspondences.

## 2. Super-commutativity, commutator, and the Lie structure

The graded-commutativity property of the Hall product is controlled simultaneously by cohomological degree and the Euler pairing. If $x,y\in A$ have cohomological degrees $|x|,|y|$ and charges $\gamma_x,\gamma_y$, then
$$
\mu(x,y)=(-1)^{|x||y|+\langle \gamma_x,\gamma_y\rangle}\,\mu(y,x).
$$
Thus $\mu$ is graded-commutative up to the Euler pairing [1812.05801].

The Kontsevich–Soibelman Lie bracket is then defined on $A=\bigoplus_\gamma A_\gamma$ by
$$
[x,y]
:=
\mu(x,y)-(-1)^{|x||y|}\mu(y,x).
$$
This bracket satisfies the graded Jacobi identity and endows $A$ with the structure of a $\mathbb{Z}$–graded super-Lie algebra. In the formulation of the comparison with BPS scattering states, the fundamental properties of this object are that it is graded by the charge lattice $\Gamma$, the Euler pairing $\langle\cdot,\cdot\rangle$ controls the signs, the convolution $\mu$ is graded-commutative, and the induced super-Lie bracket satisfies the Jacobi identity [1812.05801].

A common source of confusion is whether the Hall product itself is the Lie algebra operation. It is not: the Lie structure arises from the super-commutator of the Hall product. The associative and Lie-algebraic levels coexist, and later constructions, including wall-crossing and the scattering comparison, use both.

## 3. Quantum torus, generating series, and wall-crossing

The Kontsevich–Soibelman Lie algebra admits a generating-function realization in a quantum torus. Introduce formal symbols $X_\gamma$ for $\gamma\in\Gamma$, with multiplication
$$
X_\gamma X_{\gamma'}
=
(-1)^{\langle \gamma,\gamma'\rangle}
\,y^{\frac12\langle \gamma,\gamma'\rangle}
X_{\gamma+\gamma'}.
$$
The assignment
$$
x\in A_\gamma\longmapsto \Omega_x(y)\,X_\gamma,
$$
where $\Omega_x(y)$ is the Poincaré polynomial of $x$, encodes the algebra in generating-function form [1812.05801].

For each ray $\ell\subset \mathbb{C}$ and each primitive $\gamma$ with Poincaré polynomial $\Omega_\gamma(y)$, one defines the quantum dilogarithm operator
$$
K_\gamma
=
\prod_{k\ge 1}\exp\!\bigl(\Omega_{k\gamma}(y)\,X_{k\gamma}/k\bigr).
$$
Wall-crossing across a change in stability that reorders rays $\{\gamma\}$ is encoded by the identity
$$
\prod_{\ell\;\mathrm{clockwise}}K_\gamma
=
\prod_{\ell\;\mathrm{counterclockwise}}K_\gamma.
$$
This is the Kontsevich–Soibelman wall-crossing formula. In Lie algebra language, it states that the product of exponentials in the quantum torus is unchanged as stability varies, and the Baker–Campbell–Hausdorff formula translates it into Jacobi-type identities in the Lie algebra [1812.05801].

The quantum torus presentation is therefore not merely a bookkeeping device. It packages characters, sign conventions, and wall-crossing constraints into a single formalism. This suggests that the Lie algebra is naturally situated at the interface of representation-theoretic Hall algebra and Donaldson–Thomas-type wall-crossing.

## 4. Equivalence with the Harvey–Moore scattering algebra

A second definition of an algebra of BPS states, proposed by Harvey and Moore, uses scattering data. In quiver quantum mechanics one can build an associative scattering algebra of BPS wave-functions $\Psi_\gamma$ by studying two-particle $S$-matrix residues at the bound-state pole
$$
s=|Z_{\gamma_1+\gamma_2}|^2.
$$
The basic product takes the form
$$
\Psi_{\gamma_1}\star \Psi_{\gamma_2}
=
S_{\gamma_1,\gamma_2}\,\Psi_{\gamma_1+\gamma_2},
$$
where $S_{\gamma_1,\gamma_2}$ is the residue of the $2\to 1$ $S$-matrix. Iteration yields a graded associative product $\star$ on $H^*$ of framed and unframed moduli [1812.05801].

The comparison theorem identifies this physical product with the Hall convolution. A detailed localization and Morse-theoretic analysis shows that the convolution $\mu$ in CoHA coincides identically with the scattering product $\star$ on anti-holomorphic wave-functions. More specifically, both products are $G_\gamma$–equivariant, their structure constants $c_{\gamma_1,\gamma_2}^{\gamma}$ computed by Ext-group integrals agree with the $S$-matrix residues $S_{\gamma_1,\gamma_2}$, and the induced super-Lie bracket
$$
[\Psi_{\gamma_1},\Psi_{\gamma_2}]_{\mathrm{CoHA}}
=
\Psi_{\gamma_1}\star \Psi_{\gamma_2}
-
(-1)^{|\gamma_1||\gamma_2|}
\Psi_{\gamma_2}\star \Psi_{\gamma_1}
$$
reproduces exactly the physical BPS Lie bracket of Harvey–Moore [1812.05801].

Accordingly, the two constructions are not competing definitions. They coincide exactly, term-by-term, at the level of structure constants and brackets. This resolves the apparent discrepancy between a mathematically natural Hall-algebra construction and a physically natural algebra of BPS $S$-matrix residues.

## 5. The preprojective BPS Lie algebra and the less perverse filtration

For a finite quiver $Q$ with vertex set $Q_0$, the preprojective algebra is
$$
\Pi_Q
=
\mathbb{C}\overline{Q}\Big/\Big\langle\sum_{a\in Q_1}[a,a^*]\Big\rangle.
$$
Write $M_d(\Pi_Q)$ for the stack of $d$-dimensional $\Pi_Q$-modules and $M(\Pi_Q)=\bigsqcup_d M_d(\Pi_Q)$. The corresponding cohomological Hall algebra is defined from Borel–Moore homology by
$$
H_d
=
H^{\mathrm{BM}}_*\bigl(M_d(\Pi_Q),\mathbb{Q}\bigr)\otimes L^{-\,Q(d,d)},
$$
where
$$
Q(d,d)=\sum_{i\in Q_0}d_i^2-\sum_{a\in Q_1}d_{s(a)}d_{t(a)},
$$
and $L$ is the Tate twist of weight $-2$. Its Hall product
$$
\star\colon H_{d'}\otimes H_{d''}\to H_{d'+d''}
$$
is defined by the stack of short exact sequences
$$
0\to \rho'\to \rho\to \rho''\to 0,\qquad \underline{\dim}(\rho')=d',\ \underline{\dim}(\rho'')=d'',
$$
through the usual Hall correspondence; associativity follows from the usual 2-step flag argument [2007.03289].

The central additional structure is the less perverse filtration. Let
$$
JH\colon M(\Pi_Q)\to M(\Pi_Q)^{\mathrm{coarse}}
$$
be the affinization, or semisimplification, morphism from the stack of all modules to its good moduli space. The derived direct image with compact support of the dualizing complex,
$$
\mathfrak{rCoha}_{\Pi_Q}
=
\bigoplus_d JH_!\bigl(\mathbb{Q}_{M_d(\Pi_Q)}\bigr)\otimes L^{-\,Q(d,d)},
$$
is pure of weight zero and hence splits, in Saito’s version of the Beilinson–Bernstein–Deligne–Gabber decomposition theorem, into pure summands. This yields a canonical ascending filtration on
$$
\mathfrak{HCoha}_{\Pi_Q}
=
\mathcal{H O}\bigl(\mathfrak{rCoha}_{\Pi_Q}\bigr)
=
\bigoplus_d H_d
$$
by
$$
\mathfrak{P}_p\mathfrak{HCoha}_{\Pi_Q}
=
\mathcal{H O}\bigl(\tau_{\le p}\,\mathfrak{rCoha}_{\Pi_Q}\bigr).
$$
The main structural theorem states that the filtration is compatible with the CoHA product and that
$$
\boxed{\mathfrak{P}_0\,\mathfrak{HCoha}_{\Pi_Q}\cong \mathrm{UEA}\bigl(\mathfrak{g}_{\Pi_Q}\bigr),}
$$
an isomorphism of associative $\mathcal{H O}(\mathrm{pt})$-algebras, up to the usual Koszul sign-twist [2007.03289].

This identifies the preprojective BPS Lie algebra as the Lie-theoretic core of the zeroth less perverse piece. A plausible implication is that the less perverse filtration isolates the primitive BPS sector inside a larger Hall algebra that still remembers extension-theoretic complexity.

## 6. BPS sheaves, cuspidal cohomology, and standard examples

Inside the pure complex $\mathfrak{rCoha}_{\Pi_Q}$, the first nontrivial perverse cohomology defines the BPS sheaf
$$
\mathrm{BPS}_{\Pi_Q}
=
{}^{\mathfrak p}\!H^1\bigl(\mathfrak{rCoha}_{\Pi_Q}\bigr)\otimes L^{-1/2}
\in \mathrm{MHM}\bigl(M(\Pi_Q)^{\mathrm{coarse}}\bigr).
$$
This is a perverse sheaf, in fact a pure Hodge module, and
$$
\mathfrak{g}_{\Pi_Q}
=
\mathcal{H O}\bigl(\mathrm{BPS}_{\Pi_Q}\bigr)
\subset \mathfrak{P}_0\,\mathfrak{HCoha}_{\Pi_Q}
$$
is closed under the commutator
$$
[x,y]=x\star y-(-1)^{\deg x\,\deg y}y\star x.
$$
Equivalently,
$$
\mathfrak{g}_{\Pi_Q}
=
\bigoplus_d H\bigl(M_d(\Pi_Q),\mathrm{BPS}_{\Pi_Q,d}\bigr)\otimes L^{1/2},
$$
with bracket induced from the CoHA product. Moreover, the CoHA product and its commutator lift to morphisms of pure Hodge modules, so $\mathrm{BPS}_{\Pi_Q}$ is itself a Lie algebra object in $\mathrm{Perv}\bigl(M(\Pi_Q)^{\mathrm{coarse}}\bigr)$ [2007.03289].

Intersection cohomology provides distinguished generators. For each $d$ such that there exists a stable $\Pi_Q$-module of dimension $d$, the intersection-cohomology sheaf
$$
\mathrm{IC}_{M_d(\Pi_Q)}(\mathbb{Q})
\in
\mathrm{Perv}\bigl(M(\Pi_Q)^{\mathrm{coarse}}\bigr)
$$
injects into $\mathrm{BPS}_{\Pi_Q,d}$, and under the Lie bracket these summands are primitive in the sense that any bracket of lower-dimensional summands lands in the complement. Passing to global sections produces distinguished subspaces
$$
\mathrm{cu}_{\Pi_Q,d}
=
H\bigl(M_d(\Pi_Q),\mathrm{IC}_{M_d(\Pi_Q)}(\mathbb{Q})\bigr)\otimes L^{\ast}
\subset
\mathfrak{g}_{\Pi_Q,d},
$$
which conjecturally form a minimal set of generators of $\mathfrak{g}_{\Pi_Q}$. The cuspidal conjecture, now a theorem in many cases, asserts that the only simple roots of the BPS Borcherds algebra are the real roots $1_i$ and the imaginary roots $d$ which admit a stable module, and that the corresponding cuspidal cohomology spaces $\mathrm{cu}_{\Pi_Q,d}$ provide all of the fundamental generators [2007.03289].

Several standard examples organize the representation-theoretic range of the construction. If $Q$ has no loops, then
$$
\mathfrak{P}_0\,H^0\!\mathrm{Coha}_{\Pi_Q}\cong \mathrm{UEA}\bigl(\mathfrak n^-_{\mathfrak g_Q}\bigr),
$$
equivalently $\mathfrak g_{\Pi_Q}\cong \mathfrak n^-_{\mathfrak g_Q}$ in cohomological degree zero. If $Q$ is an extended affine Dynkin quiver, then there is one imaginary root $\delta$ and
$$
\mathfrak g_{\Pi_Q}\cong \mathfrak n^-_{\mathrm{aff}}\oplus s\mathbb{Q}[s],
$$
where the extra summand $s\mathbb{Q}[s]$ lives in cohomological degree $-2$. If one replaces the trivial cubic potential on the tripled quiver by $\widetilde W+W_0$ with $W_0$ a generic cycle, then the same construction yields
$$
\mathfrak g_{\Pi_Q,W_0}\cong \mathfrak n^-_{\mathfrak g_Q}
$$
even for quivers with loops. The CoHA also acts on the cohomology of Nakajima quiver varieties $\mathrm{Hilb}_{f,d}$, and restricting to the spherical subalgebra recovers the usual half-Kac–Moody action of raising and lowering operators. Under change of slopes, the isomorphism class of $\mathfrak g_{\Pi_Q}$ is unchanged, and the wall-crossing formulas of Kontsevich–Soibelman are encoded by the change of stability and the factorization of the CoHA in terms of BPS generators [2007.03289].

Source: https://www.emergentmind.com/topics/kontsevich-soibelman-lie-algebra