3-Algebras: Ternary Algebraic Structures

Updated 29 August 2025

3-Algebras are nonassociative algebraic structures with a trilinear product that generalizes Lie algebras via a totally skew-symmetric bracket and the Filippov identity.
They include variants such as 3-Leibniz, Cartan–Weyl, and semi-associative 3-algebras, each relaxing traditional symmetry or associativity conditions.
Applications span gauge theory, quantum field theory, and noncommutative geometry, underpinning models like BLG theory and M-brane dynamics.

A 3-algebra is a nonassociative algebraic structure equipped with a trilinear product (ternary operation), generalizing the familiar concept of a Lie algebra to the field of higher-arity operations. In the most prominent case, a 3-Lie algebra, the trilinear bracket is totally skew-symmetric and obeys a "fundamental identity" that controls its natural interaction with itself, serving as a direct generalization of the Jacobi identity. 3-algebras underpin a vast landscape of algebraic structures, including Filippov (n-Lie) algebras, 3-Leibniz algebras, Cartan–Weyl 3-algebras, semi-associative 3-algebras, and operators such as twisted Rota–Baxter operators, with applications spanning algebraic geometry, representation theory, mathematical physics, and quantum field theory.

1. Core Definition and Algebraic Structures

The archetypal 3-algebra is the 3-Lie algebra, formally defined on a vector space $V$ by a bracket product $[\cdot,\cdot,\cdot] : V^{\otimes 3} \to V$ that is:

Totally skew-symmetric: $[x_1, x_2, x_3]$ changes sign under any transposition of its arguments.
Satisfies the Filippov (fundamental) identity:

$[x_1, x_2, [y_1, y_2, y_3]] = [[x_1, x_2, y_1], y_2, y_3] + [y_1, [x_1, x_2, y_2], y_3] + [y_1, y_2, [x_1, x_2, y_3]].$

The definition can be extended by relaxing skew-symmetry or the structure of the bracket, giving rise to generalized structures such as 3-Leibniz algebras (which drop total skew-symmetry) and semi-associative 3-algebras (which weaken associativity conditions but retain partial antisymmetry in selected arguments) (Bai et al., 2019, Teng et al., 2023).

Many constructions and classifications of 3-algebras leverage familiar binary structures such as associative commutative algebras, Lie algebras, group algebras, and derivations, but 3-algebras exhibit properties not reducible to binary algebras in general (Ahmed et al., 2021). For example, not every ternary multiplication admits a description in terms of iterated binary multiplications.

2. Cartan–Weyl and Generalized Cartan–Weyl 3-Algebras

Cartan–Weyl 3-algebras are a direct analog of the Cartan–Weyl basis for semisimple Lie algebras. Such 3-algebras possess:

A Cartan subalgebra $\mathcal{H}$ of mutually commuting generators $H_I$ , with $[H_I, H_J, H_K] = 0$ ;
Step generators $E^\alpha$ labeled by two-form roots $\alpha_{IJ}$ , satisfying $[H_I, H_J, E^\alpha] = \alpha_{IJ} E^\alpha$ ;
Nondegenerate invariant metric with $\langle H_I, H_J\rangle = g_{IJ}$ invertible and $\langle E^\alpha, E^\beta\rangle \propto \delta_{\alpha+\beta,0}$ ;
Further structure in the 3-brackets, where mixed brackets close either on the Cartan subalgebra or reproduce new step generators, and closure is controlled by factorization of roots into wedge products of a fixed null one-form and root one-forms of an underlying semisimple Lie algebra (Chu, 2010).

Classification is achieved by analyzing the signature (“index”) of the invariant metric:

Index 0: allows only a single pair of step generators (degenerately decomposable 3-algebra).
Index 1 (Lorentzian case): aligns with 3-algebras relevant for the BLG theory (see below), decomposing as $\mathfrak{g}\oplus\mathbb{C}(u,v)$ , where the “light-cone” $u,v$ correspond to null directions.
Index ≥2: the algebra splits into semisimple, light-cone, and additional step generator sectors, with elaborate interplay among brackets (Chu, 2010).

Generalized Cartan–Weyl 3-algebras relax the requirement that the Cartan subalgebra be abelian, allowing $[H_I, H_J, H_K] \ne 0$ (encoded via nontrivial structure constants $L_{IJK}{}^M$ ) (Chu, 2010). These generalizations are critical in providing algebraic settings suitable for incorporating fuzzy $S^3$ solutions within the BLG theory, as traditional Cartan–Weyl 3-algebras (with abelian Cartan) do not support such embeddings.

3. 3-Algebras in Gauge Theory and Quantum Field Theory

The BLG (Bagger–Lambert–Gustavsson) theory for multiple M2-branes demands a metric 3-Lie algebra structure for the gauge symmetry (Chu, 2010, Chu, 2010). In this context:

The 3-algebra must be metric (possess a nondegenerate invariant inner product) for a ghost-free action.
A notion of strong-semisimplicity is imposed, requiring the existence of a semisimple Lie algebra upon “reduction” by fixing (n–2) Cartan elements, i.e., $[x, y]_h := [x, y, h]$ must be semisimple as a Lie algebra bracket (Chu, 2010).
Many traditional 3-algebras (including Lorentzian 3-algebras) are special cases of Cartan–Weyl 3-algebras (Chu, 2010).
The inability to accommodate a fuzzy $S^3$ solution (key for describing polarized M2–M5 bound states) in the BLG theory with abelian Cartan–Weyl 3-algebras motivates the introduction of generalized Cartan–Weyl 3-algebras with nonabelian Cartan sectors (Chu, 2010).

Metric Lie 3-algebras also appear in nonabelian formulations of higher-dimensional tensor multiplets, such as the (2,0) six-dimensional tensor multiplet, where the nonabelian system can be reduced to five-dimensional super-Yang–Mills theory and free six-dimensional abelian (2,0) multiplets, directly implicating D4 and M5-brane physics (Lambert et al., 2010).

4. Variants and Higher-Categorical Generalizations

At the categorical and homotopical level, 3-Lie $_\infty$ -algebras and 3-Lie 2-algebras provide coherent structures encoding higher homotopies and categorification, respectively. A 3-Lie $_\infty$ -algebra comprises a graded vector space with a series of higher ($2n+1$)-ary operations $l_{2n+1}$ satisfying a hierarchy of homotopy-encoded identities (Zhou et al., 2015). Such structures equate, at the 2-category level, to 2-term 3-Lie $_\infty$ -algebras and 3-Lie 2-algebras, with equivalence established via explicit 2-functorial construction.

Skeletal and strict 3-Lie 2-algebras admit classification via cohomology (quadrupoles incorporating a 3-cocycle for skeletal; crossed modules for strict structures) (Zhou et al., 2015).

Operadic “weak Lie 3-algebras” extend the L $_\infty$ approach and permit skeletal/strict truncations and homotopy transfer (Dehling, 2017), furnishing bridges to applications in n-plectic geometry and higher Courant algebroids.

5. Explicit Constructions and Examples

A variety of algebraic recipes exist for producing 3-algebras:

From commutative associative algebras with derivations and involutions: via formulas such as $[a, b, c] = f(a)[b,c] + f(b)[c,a] + f(c)[a,b]$ where $[a,b]$ is constructed via an involution and derivation (Bai et al., 2013).
Group algebra constructions: For an abelian group $G$ and homomorphism $a:G\to \mathbb{F}_+$ , the group algebra $F[G]$ admits a bracket $[e^g, e^h, e^q]$ determined by natural differences and group multiplication (Bai et al., 2013).
From Laurent polynomial algebras using suitable involutions and derivations (infinite-dimensional simple examples; see formulas in (Bai et al., 2013)).
From binary algebras: recursive constructions generate n-ary brackets, but the resulting multiplication may not, in general, be reducible to a binary operation, and non-isomorphic binary algebras may map to isomorphic 3-algebras (Ahmed et al., 2021).
Semi-associative 3-algebras, with trilinear products skew in their first two arguments and satisfying particular “semi-associative” and mixed-symmetry identities; their “adjacent” algebra yields a true 3-Lie bracket by full antisymmetrization (Bai et al., 2019).
Trigroups, with three compatible associative binary operations and bar-units/inverses, generate pointed 3-racks via a ternary conjugation product, and differentiation at the identity in Lie trigroups produces Leibniz 3-algebra structures (Biyogmam et al., 2019).

6. 3-Algebras in Geometric and Physical Models

3-algebras are deeply embedded in models of noncommutative geometry, quantum gravity, and string/M-theory:

Tensor models for quantum gravity employ real, cyclically symmetric rank-three tensors as dynamical variables; induced 3-ary operations represent symmetries more naturally than binary commutators. In the fuzzy space interpretation, the resulting 3-ary coordinate algebras recover Lie triple systems whose associated Lie algebras coincide with those underlying Snyder’s noncommutative spacetime (Sasakura, 2011).
Cartan–Weyl and generalized Cartan–Weyl 3-algebras serve as the algebraic foundation for the BLG theory of multiple M2-branes, with strict algebraic constraints dictating what physical configurations (e.g., fuzzy $S^3$ ) are dynamically allowed (Chu, 2010, Chu, 2010).
In nonabelian (2,0) tensor multiplet theories, 3-algebraic structures with antisymmetric structure constants satisfying the fundamental identity organize the interactions, and appropriate reductions yield D-brane and, via null reductions, M5-brane systems (Lambert et al., 2010).

7. Operator and Deformation Theory; Cohomology and Extensions

The operator theory of 3-algebras has evolved to incorporate twisted Rota–Baxter, $O$ –operators, and Reynolds operators appropriate to the ternary setting. These notions generate new 3-algebra structures on module spaces (notably, NS–3–Lie algebras), and their deformation theory is governed by generalized Chevalley–Eilenberg cohomology. For instance, infinitesimal and formal deformations of twisted $O$ –operators are controlled by the first and second cohomology of the associated induced 3-Lie algebra representation (Chtioui et al., 2021, Hou et al., 2021).

Cohomological techniques extend to geometric settings, as in 3-Hom-Lie-Rinehart algebras and their A-split abelian extensions, with cohomology groups classifying extensions and deformations (Guo et al., 2019, Bai et al., 2019).

Table: Selected Classes of 3-Algebras and Key Features

Structure	Defining Feature	Applications/Context
3-Lie algebra	Skew-symmetric, fundamental identity	M2-brane models, n-plectic geometry
Cartan–Weyl 3-algebra	Root/step decomposition, abelian Cartan, metric, factorization of roots	BLG theory, symmetry classification
Generalized Cartan–Weyl	Nonabelian Cartan subalgebra, metric, strong-semisimplicity via reduction	Fuzzy S $^3$ in BLG, extended symmetry
Semi-associative 3-algebra	Semi-associativity, partial antisymmetry	Double module/extension theory
3-Leibniz algebra	Ternary bracket, Leibniz-type identity (not fully antisymmetric)	Extensions of Filippov theory
3-Lie $_\infty$ /3-Lie 2-alg.	Coherent higher homotopies / categorification	n-plectic geometry, higher gauge
3-Lie-Rinehart algebra	3-Lie structure + module/anchor over commutative algebra	Lie algebroid generalization
NS–3–Lie / 3-NS-Lie algebra	Splitting into two ternary operations compatible with 3-Lie bracket	Operator/deformation theory
3-post-Lie algebra	3-Lie structure plus a compatible ternary operation	Rota–Baxter theory, deformation
Trigroup (Lie 3-rack)	Three associative binary operations with bar-units/inverses	Leibniz 3-algebra via differentiation

References

Cartan–Weyl 3-algebras and BLG Theory I: (Chu, 2010)
Cartan–Weyl 3-algebras and BLG Theory II: (Chu, 2010)
Nonabelian (2,0) Tensor Multiplets and 3-algebras: (Lambert et al., 2010)
Constructing 3-Lie algebras: (Bai et al., 2013)
$3$- $Lie_\infty$ -algebras and $3$-Lie 2-algebras: (Zhou et al., 2015)
On weak Lie 3-algebras: (Dehling, 2017)
Tensor models and 3-ary algebras: (Sasakura, 2011)
3-Lie-Rinehart Algebras: (Bai et al., 2019)
Semi-Associative $3$-Algebras: (Bai et al., 2019)
On 3-Hom-Lie-Rinehart algebras: (Guo et al., 2019)
From Trigroups To Leibniz 3-Algebras: (Biyogmam et al., 2019)
On certain three algebras generated by binary algebras: (Ahmed et al., 2021)
Twisted O-operators on 3-Lie algebras and 3-NS-Lie algebras: (Chtioui et al., 2021)
Twisted Rota-Baxter operators on 3-Lie algebras and NS-3-Lie algebras: (Hou et al., 2021)
$3$-post-Lie algebras and relative Rota-Baxter operators: (Hou et al., 2022)
Nonabelian embedding tensors on 3-Lie algebras and 3-Leibniz-Lie algebras: (Teng et al., 2023)
Classification of Transposed Poisson 3-Lie algebras of dimension 3: (Yaxi et al., 5 Jan 2024)