---
title: Semistrict Lie 2-Algebra
url: https://www.emergentmind.com/topics/semistrict-lie-2-algebra
type: topic
---

# Semistrict Lie 2-Algebra

A semistrict Lie 2-algebra is a categorified Lie algebra in which skew-symmetry is strict but the Jacobi identity holds only up to a coherent homotopy. In the standard algebraic formulation used across the literature surveyed here, it is a 2-term \(L_\infty\)-algebra: a 2-term complex together with a binary bracket and a ternary Jacobiator, with all higher brackets vanishing [1308.4892]. This notion occupies an intermediate position between strict Lie 2-algebras, where the Jacobiator vanishes, and more general weak or higher \(L_\infty\)-structures. It appears as the natural algebraic language for 2-plectic geometry, semistrict higher gauge theory, higher Chern–Simons theory, and various M-brane-inspired models [1308.4892].

## 1. Algebraic definition and equivalent formulations

In the 2-term \(L_\infty\) presentation, a semistrict Lie 2-algebra is based on a graded vector space
\[
L=V \xrightarrow{\mu_1} W \xrightarrow{0} 0,
\]
with \(\deg V=-1\) and \(\deg W=0\), together with non-vanishing graded antisymmetric structure maps
\[
\mu_1:V\to W,\qquad
\mu_2:W\wedge W\to W,\qquad
\mu_2:W\wedge V\to V,\qquad
\mu_3:W\wedge W\wedge W\to V,
\]
all other \(\mu_n\) being zero [1308.4892]. The map \(\mu_1\) is the differential, \(\mu_2\) is the binary bracket together with the degree-\(0\) action on degree \(-1\), and \(\mu_3\) is the Jacobiator: it measures the failure of \(\mu_2\) on \(W\) to satisfy the ordinary Jacobi identity [1308.4892].

The defining homotopy relations specialize to a finite list. They include the derivation property of \(\mu_1\) with respect to \(\mu_2\), the fact that \(\mu_3\) is non-zero only on \(W^{\wedge 3}\), the Jacobiator equation
\[
\mu_1(\mu_3(w_1,w_2,w_3))
=
-\mu_2(\mu_2(w_1,w_2),w_3)
-\mu_2(\mu_2(w_3,w_1),w_2)
-\mu_2(\mu_2(w_2,w_3),w_1),
\]
and a higher coherence identity for \(\mu_3\) [1308.4892]. In this sense, semistrictness means that the Jacobi identity is not abandoned but replaced by exact control through \(\mu_3\).

Several equivalent descriptions are used in the literature. One is the Baez–Crans equivalence between semistrict Lie 2-algebras and 2-term \(L_\infty\)-algebras, adopted explicitly in semistrict higher gauge theory and related field-theoretic constructions [1403.7185]. Another is the weak Lie 2-algebra viewpoint, where a semistrict Lie 2-algebra is a weak Lie 2-algebra with trivial alternator, so skew-symmetry is strict while the Jacobi identity is weakened by a nontrivial Jacobiator [1403.7185]. In a closely related but different direction, Cai–Liu–Xiang study hemistrict Lie 2-algebras, where Jacobi is strict and skew-symmetry is weakened by an alternator; Roytenberg’s skew-symmetrization then produces a semistrict Lie 2-algebra with strictly skew binary bracket and nontrivial ternary Jacobiator [2003.03553].

A notational variant widely used in higher gauge theory writes the same data as
\[
\mathfrak v=(\mathfrak v_0,\mathfrak v_1,\alpha,[\cdot,\cdot],[\cdot,\cdot],[\cdot,\cdot,\cdot]),
\]
with
\[
\alpha:\mathfrak v_1\to\mathfrak v_0,\qquad
[\cdot,\cdot]:\mathfrak v_0\wedge\mathfrak v_0\to\mathfrak v_0,\qquad
[\cdot,\cdot]:\mathfrak v_0\otimes\mathfrak v_1\to\mathfrak v_1,\qquad
[\cdot,\cdot,\cdot]:\mathfrak v_0^{\wedge 3}\to\mathfrak v_1,
\]
with \(l_1=\alpha\), \(l_2=[\cdot,\cdot]\), \(l_3=[\cdot,\cdot,\cdot]\) [2605.29282]. The Jacobi identity on \(\mathfrak v_0\) then reads
\[
[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=\alpha([x,y,z]),
\]
making the role of the ternary bracket explicit [2605.29282].

## 2. Strict, skeletal, hemistrict, and morphism-theoretic variants

The strict case is obtained by setting \(\mu_3=0\), equivalently \([\cdot,\cdot,\cdot]=0\). Then both the degree-\(0\) bracket and the induced degree-\(-1\) bracket
\[
\kappa_2(v_1,v_2):=\mu_2(\mu_1(v_1),v_2)
\]
satisfy honest Jacobi identities, yielding a differential crossed module \(V\xrightarrow{\mu_1}W\) [1308.4892]. In higher gauge theory this is the crossed-module regime, and there is a one-to-one correspondence between strict 2-term \(L_\infty\)-algebras and differential Lie crossed modules [2605.29282].

A standard class of semistrict Lie 2-algebras is furnished by skeletal models \((\mathfrak g,V,\rho,c)\), where \(\mu_1=0\), \(W=\mathfrak g\) is a Lie algebra, \(V\) is a \(\mathfrak g\)-module, and \(\mu_3\) is a Lie algebra \(3\)-cocycle \(c\in H^3(\mathfrak g,V)\) [1308.4892]. The string Lie 2-algebra is the distinguished special case with \(V=\mathbb R\) and
\[
c(g_1,g_2,g_3)=k\langle g_1,[g_2,g_3]\rangle
\]
for the Killing form \(\langle\cdot,\cdot\rangle\) [1308.4892]. Every semistrict Lie 2-algebra is equivalent to a skeletal one of this form [1308.4892].

Morphisms and automorphisms inherit the same homotopical structure. A 1-morphism between 2-term \(L_\infty\)-algebras consists of linear maps on degrees \(0\) and \(1\) together with a bilinear homotopy \(\phi_2:\mathfrak v_0\wedge\mathfrak v_0\to\mathfrak v_1'\), while a 2-morphism is a chain homotopy \(\Phi:\mathfrak v_0\to\mathfrak v_1'\) satisfying compatibility equations [1112.2819]. The automorphism data form a strict 2-group \(\mathrm{Aut}(\mathfrak v)\), with infinitesimal counterpart \(\mathfrak{aut}(\mathfrak v)\), a strict Lie 2-algebra of derivations and higher derivations [1112.2819]. This strictification at the level of automorphisms is central in semistrict higher gauge theory, because one may formulate gauge transformations entirely in terms of \(\mathfrak v\) and \(\mathrm{Aut}(\mathfrak v)\), without first constructing a global integrating Lie 2-group [1112.2819].

The hemistrict-to-semistrict comparison clarifies a common source of confusion. In the hemistrict setting, the Jacobi identity is strict but skew-symmetry holds only up to a degree \(-1\) alternator \(h_2\); Roytenberg’s skew-symmetrization replaces the alternator by a Jacobiator \(\hat l_3\), yielding a semistrict Lie 2-algebra \(\hat L\) on the same underlying complex [2003.03553]. Thus semistrict and hemistrict are not interchangeable descriptions of the same raw brackets; they are related by a specific skew-symmetrization procedure [2003.03553].

## 3. Canonical constructions and examples

A structurally important source of semistrict Lie 2-algebras is 2-plectic geometry. For a 2-plectic manifold \((M,\varpi)\), with \(\varpi\) a closed nondegenerate \(3\)-form, one sets
\[
V=C^\infty(M),\qquad
W=H(M),
\]
where \(H(M)\) is the space of Hamiltonian \(1\)-forms
\[
H(M):=\{\alpha\in\Omega^1(M)\mid \exists X_\alpha \text{ with } \iota_{X_\alpha}\varpi=-d\alpha\}.
\]
The structure maps are
\[
\pi_1(f)=df,\qquad
\pi_2(\alpha,\beta)=-\iota_{X_\alpha}\iota_{X_\beta}\varpi,\qquad
\pi_3(\alpha,\beta,\gamma)=-\iota_{X_\alpha}\iota_{X_\beta}\iota_{X_\gamma}\varpi,
\]
with all other brackets zero [1308.4892]. Here \(\pi_2\) is a higher Poisson bracket on Hamiltonian \(1\)-forms, while \(\pi_3\) encodes the \(3\)-form itself as a higher Jacobiator [1308.4892].

For \(M=\mathbb R^3\) with volume form
\[
\varpi=\frac{1}{3!}\varepsilon_{ijk}\,dx^i\wedge dx^j\wedge dx^k,
\]
one obtains explicit formulas for \(\pi_1,\pi_2,\pi_3\), and in particular the linear \(1\)-forms
\[
\xi_i:=\frac12\varepsilon_{ijk}x^j\,dx^k
\]
satisfy
\[
\pi_2(\xi_i,\xi_j)=\varepsilon_{ijk}\,dx^k,\qquad
\pi_3(\xi_i,\xi_j,\xi_k)=-\varepsilon_{ijk}.
\]
The finite-dimensional sub-Lie-2-algebra generated by constants, linear functions, these \(\xi_i\), and exact \(1\)-forms is called the Heisenberg Lie 2-algebra [1308.4892].

Another systematic construction arises from a Lie algebra \(\mathfrak g\) acting on a 2-term complex \(V_1\xrightarrow d V_0\) by a 2-term representation up to homotopy \((\mu,\nu)\). The associated semidirect product Lie 2-algebra has underlying complex
\[
V_1 \xrightarrow{0+d} \mathfrak g\oplus V_0
\]
and structure maps
\[
l_2((X,\xi),(Y,\eta))=([X,Y],\mu_0(X)(\eta)-\mu_0(Y)(\xi)),
\]
\[
l_2((X,\xi),m)=\mu_1(X)(m),\qquad l_2(m,n)=0,
\]
\[
l_3((X,\xi),(Y,\eta),(Z,\gamma))=-\nu(X,Y)(\gamma)+c.p.
\]
The Jacobiator of \(l_2\) is then \(d\)-exact and governed by \(l_3\) [1003.1348]. This construction includes examples arising from Courant-type structures and the omni-Lie algebra \(gl(V)\oplus V\), whose failure of Jacobi is corrected by an explicit ternary bracket in the corresponding semistrict Lie 2-algebra [1003.1348].

Enhanced Leibniz algebras furnish another route. An enhanced Leibniz algebra \((t:\mathbb W\to\mathbb V,[\cdot,\cdot],\circ)\) canonically induces a semistrict Lie 2-algebra with
\[
V_1=\mathbb W,\qquad V_0=\mathbb V,\qquad l_1=t,
\]
binary bracket
\[
l_2(v_1,v_2)=\{v_1,v_2\}:=[v_1,v_2]-t(v_1\circ v_2),
\]
action
\[
l_2(v,w)=v*w:=t(w)\circ v,
\]
and ternary bracket
\[
l_3(v_1,v_2,v_3)=y(v_1,v_2,v_3),
\]
where \(y\) is the explicit totally antisymmetric expression built from the Leibniz bracket and circle product [1901.01014]. In the positive quadratic case relevant to Yang–Mills-type higher gauge theories, the underlying Leibniz algebra is forced to be a hemisemidirect product \(\mathfrak g\ltimes \mathfrak i\) of a positive quadratic Lie algebra with a module [1901.01014].

## 4. Higher gauge theory and field-theoretic realizations

Semistrict Lie 2-algebras are the symmetry objects of semistrict higher gauge theory. A \(\mathfrak v\)-connection on a manifold \(M\) is a pair
\[
A\in\Omega^1(M,\mathfrak v_0),\qquad
B\in\Omega^2(M,\mathfrak v_1),
\]
with curvature doublet
\[
\mathcal F=dA+\tfrac12[A,A]-\alpha(B),
\qquad
\mathcal H=dB+[A,B]-\tfrac16[A,A,A].
\]
These satisfy the higher Bianchi identities
\[
d\mathcal F+[A,\mathcal F]+\alpha(\mathcal H)=0,
\]
\[
d\mathcal H+[A,\mathcal H]-[\mathcal F,B]+\tfrac12[A,A,\mathcal F]=0
\]
[2605.29282]. The Jacobiator appears explicitly in \(\mathcal H\), so semistrictness is already visible at the level of curvature.

In the local formulation based on \(\mathfrak v\) and \(\mathrm{Aut}(\mathfrak v)\), a connection doublet \((\omega,\Omega_\omega)\) has curvatures
\[
f=d\omega+\tfrac12[\omega,\omega]-\partial\Omega_\omega,\qquad
F=d\Omega_\omega+[\omega,\Omega_\omega]-\tfrac16[\omega,\omega,\omega],
\]
and a finite 1-gauge transformation \(g=(g,\sigma_g,\Sigma_g,\tau_g)\) acts by
\[
g\omega=g_0(\omega-\sigma_g),
\]
\[
g\Omega_\omega
=
g_1\bigl(\Omega_\omega-\Sigma_g+\tau_g(\omega-\sigma_g)\bigr)
-\tfrac12 g_2(\omega-\sigma_g,\omega-\sigma_g).
\]
The associated gauge transformations form a strict 2-group \(\mathrm{Gau}(M,\mathfrak v)\), while global semistrict principal 2-bundles are described via a strict 2-groupoid of gluing data built from local \(\mathrm{Aut}(\mathfrak v)\)-valued transition functions and 2-morphisms [1112.2819].

Semistrict principal 2-bundles with connective structure admit a Deligne-cohomological description. For a cover \(\{U_a\}\), the local data comprise transition 1-cells \(m_{ab}\), transition 2-cells \(n_{abc}\), local \(1\)- and \(2\)-form potentials \(A_a,B_a\), and overlap \(1\)-forms \(\Lambda_{ab}\), subject to cocycle conditions and the vanishing fake curvature condition
\[
F_a=dA_a+A_a\otimes A_a+s(B_a)=0
\]
[1403.7185]. The finite gauge transformation law for \(B\) contains explicit \(\mu_3\)-terms, making the semistrict character of the connective structure unavoidable [1403.7185].

In AKSZ-BV theory, semistrict Lie 2-algebras determine higher BF and higher Chern–Simons models. The 3-dimensional semistrict higher BF BV action is
\[
S_{\mathrm{BV}}
=
\int_{T[1]M}\varrho\,
\bigl(
(p,dq-\tfrac12[q,q]+\partial Q)
+
(P,dQ-[q,Q]+\tfrac16[q,q,q])
\bigr),
\]
and the 4-dimensional semistrict higher Chern–Simons model is a related AKSZ construction for reduced \(\mathfrak v\) with invariant pairing [1112.2819]. In both cases, the cubic term involving the ternary bracket is the genuinely semistrict contribution.

## 5. Metrics, cohomology, and integration

Field theories based on semistrict Lie 2-algebras typically require invariant bilinear pairings. In Lie 2-algebra matrix models, the minimally invariant inner product \(\langle\cdot,\cdot\rangle_0\) is orthogonal between degrees and satisfies
\[
\langle \mu_2(w_1,x_1),x_2\rangle_0+\langle x_1,\mu_2(w_1,x_2)\rangle_0=0,
\qquad
w_1\in W,\ x_i\in V\oplus W
\]
[1308.4892]. In 4-dimensional semistrict higher Chern–Simons theory one instead uses a non-singular invariant pairing
\[
(\cdot,\cdot):\mathfrak v_0\times\mathfrak v_1\to\mathbb R
\]
satisfying compatibility with \(\partial\), the binary brackets, and the ternary bracket [1406.2197]. Balanced Lie 2-algebras, with \(\dim\mathfrak v_0=\dim\mathfrak v_1\), are singled out because the action pairs degree \(0\) and degree \(1\) variables [1406.2197].

Cohomology behaves differently depending on the model. For hemistrict Lie 2-algebras, Cai–Liu–Xiang define a standard complex \(C^\ast(L,V)\) for a representation \(V\), with cochains satisfying a weak symmetry condition governed by the alternator \(h_2\). For injective hemistrict \(L\) and representations with \(l_a=0\) on \(L^{-1}\), they prove
\[
H^\ast(L,V)\cong H^\ast(\hat L,V)\cong H^\ast_{\mathrm{CE}}(L_{\mathrm{Lie}},V),
\]
where \(\hat L\) is the skew-symmetrized semistrict Lie 2-algebra and \(L_{\mathrm{Lie}}=L^0/dL^{-1}\) is the induced ordinary Lie algebra [2003.03553]. In this injective class, the cohomology of the semistrict structure collapses to ordinary Chevalley–Eilenberg cohomology [2003.03553].

A complementary direction is the theory of weak Lie 2-bialgebras. There, a semistrict Lie 2-algebra is encoded by a degree \(-4\) element \(l\) in a big-bracket Gerstenhaber algebra, with \(\{l,l\}=0\) equivalent to the full set of 2-term \(L_\infty\) identities [1109.2290]. Adding compatible dual coalgebra data produces a weak Lie 2-bialgebra, and in the strict case one recovers a one-to-one correspondence with crossed modules of Lie bialgebras [1109.2290].

Integration is subtle in general but explicit for important classes. Semidirect product semistrict Lie 2-algebras \(\mathfrak g\ltimes V\), arising from 2-term representations up to homotopy, can be integrated to finite-dimensional strict Lie 2-groups by passage through crossed modules and Noohi butterflies [1003.1348]. In that construction, a non-strict \(L_\infty\)-morphism is replaced by a zig-zag of strict morphisms between crossed modules, allowing a strict Lie 2-group to encode the same infinitesimal semistrict data up to equivalence [1003.1348]. A distinct differentiation result goes in the opposite direction: Ševera’s method differentiates semistrict Lie 2-groups to semistrict Lie 2-algebras by analyzing descent data on \( \mathbb R^{0|1}\times X\to X\), recovering a 2-term \(L_\infty\)-algebra \(v\to w\) and its Chevalley–Eilenberg differential [1403.7185].

## 6. Geometric and physical significance

Semistrict Lie 2-algebras generalize matrix-model and gauge-theory constructions based on ordinary Lie algebras. In Lie 2-algebra models, the homogeneous bosonic action
\[
S_1
=
\frac12 m_{ab}\langle X^a,X^b\rangle
+\frac13 c_{abc}\langle X^a,\mu_2(X^b,X^c)\rangle
+\frac14\langle \mu_2(X^a,X^b),\mu_2(X^a,X^b)\rangle
\]
reduces to the bosonic IKKT action when \(V=0\), \(W=\mathfrak g\), and \(\mu_2\) is the Lie bracket [1308.4892]. The same framework contains dimensionally reduced BLG and ABJM-type 3-algebra models as special strict or skeletal cases, because metric Lie 2-algebras naturally induce triple brackets satisfying the fundamental identity [1308.4892].

The semistrict setting is essential for 2-plectic quantization. Lie 2-algebra models admit solutions interpreted as quantized 2-plectic manifolds, including \(\mathbb R^3\), \(S^3\), and a 5-dimensional Hpp-wave [1308.4892]. For a 2-plectic manifold \((M,\varpi)\), the proposed categorified correspondence principle requires a quantized Lie 2-algebra \(\hat\Pi_M\) whose brackets reproduce the classical \(\pi_1,\pi_2,\pi_3\) to lowest order in \(\hbar\) [1308.4892]. Expanding a suitable zero-dimensional model around the quantized \(\mathbb R^3\) background yields a semistrict higher BF theory on the quantized 2-plectic space \(\mathbb R^3_\hbar\), with field strengths
\[
\hat F_{ij}
=
\mu_2(\hat\xi_i,\hat A_j)-\mu_2(\hat\xi_j,\hat A_i)+\mu_2(\hat A_i,\hat A_j)-\mu_1(\hat B_{ij}),
\]
\[
\hat H_{ijk}
=
\tfrac12\mu_2(\hat\xi_{[i},\hat B_{jk]})
+\tfrac12\mu_2(\hat A_{[i},\hat B_{jk]})
+\tfrac16\mu_3(\hat\xi_i+\hat A_i,\hat\xi_j+\hat A_j,\hat\xi_k+\hat A_k)
-i\hbar\,\hat1
\]
[1308.4892].

In higher Chern–Simons theory, semistrictness has a direct topological effect. The 4-dimensional semistrict higher Chern–Simons action
\[
CS_2(\omega,\Omega_\omega)
=
\kappa_2\int_N
\left[
(\Omega_\omega,f)-\frac12(\omega,\partial\Omega_\omega)-\frac16(\omega,[\omega,\omega,\omega])
\right]
\]
is invariant under orthogonal higher gauge transformations up to a higher winding number \(Q_2(g)\), which is encoded by a flat gauge-parameter 2-connection \((\sigma_g,\Sigma_g)\) and a Chevalley–Eilenberg \(4\)-cocycle \(X_2\in CE^4(\mathfrak v)\) [1406.2197]. The semistrict quartic term involving \([\omega,\omega,\omega]\) is absent in strict crossed-module models [1406.2197].

A higher-dimensional refinement makes this dependence on the Jacobiator explicit. In \(2n+2\) dimensions, the higher Chern–Simons form constructed from a semistrict Lie 2-algebra produces, under a finite higher gauge transformation, a higher Wess–Zumino–Witten term
\[
S_{\mathrm{WZW,higher}}[g]
=
-\int_{M^{2n+2}}\mathcal{CS}_{2n+2}(\sigma_g,\Sigma_g),
\]
and the key mechanism is that the non-zero Jacobiator enters both the flatness equation
\[
d\Sigma_g+[\sigma_g,\Sigma_g]-\frac16[\sigma_g,\sigma_g,\sigma_g]=0
\]
and the Chern–Simons density itself [2605.29282]. When the 3-bracket is set to zero, the theory reduces to the strict case and the higher WZW term disappears [2605.29282].

A recurrent misconception is therefore that semistrictness is a mild algebraic decoration with no independent field-theoretic effect. The cited constructions show the opposite in precise terms: the ternary bracket enters curvature, Bianchi identities, finite gauge transformations, matrix-model equations of motion, higher BF theory, higher Chern–Simons densities, and higher WZW terms [1112.2819]. In the six-dimensional twistor construction for the non-Abelian \(\mathcal N=(2,0)\) tensor multiplet, the non-linear constraint equations involve both \(\mu_2\) and \(\mu_3\), with \(\mu_3\) contributing directly to the \(3\)-form curvatures of the tensor multiplet [1403.7185].

Semistrict Lie 2-algebras thus serve as the minimal 2-term \(L_\infty\) structures capable of encoding categorified symmetry with a nontrivial Jacobiator. In the strict limit they recover crossed modules; in the 2-plectic and higher gauge-theoretic settings they capture precisely the homotopical data that strict structures omit [1308.4892].

Source: https://www.emergentmind.com/topics/semistrict-lie-2-algebra