Lie Dialgebras: From Loday to Leibniz

• Lie dialgebras are defined via Loday's dual operations where the dicommutator replaces the classical Lie bracket, yielding Leibniz algebras.
• The operadic formulation diLie = Lie ⊗ Perm unifies the structure by explaining the transition from antisymmetric Lie brackets to Leibniz identities.
• Embedding strategies using conformal algebras enable representation theorems and extend the theory to triple systems and higher algebraic structures.

“Lie dialgebra” most commonly denotes the dialgebraic form of Lie theory in which a Loday dialgebra with two binary operations $\dashv$ and $\vdash$ produces a Leibniz algebra rather than an antisymmetric Lie algebra. In operadic terms, $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$, and this operad is precisely the Leibniz operad; accordingly, Lie dialgebras in the standard sense are equivalent to Leibniz algebras (Kolesnikov et al., 2012). The terminology is not completely uniform, however. Some later works use “Lie dialgebra” for other two-bracket compatibility structures, notably totally compatible Lie dialgebras, so the subject has both a standard Loday–Leibniz meaning and several secondary uses (Zhang et al., 2015).

1. Standard meaning: Lie dialgebras as Leibniz algebras

In the standard Loday framework, a dialgebra is a $k$-module with two bilinear products

$\dashv,\ \vdash : A\times A\to A,$

and a $0$-dialgebra satisfies the bar identities

$$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$

An associative dialgebra adds the three associativity-type identities

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$

The associated Lie-type operation is the dicommutator. In one convention,

$[a,b]=a\vdash b-b\dashv a,$

while in another survey convention it is written

$\langle a,b\rangle=a\dashv b-b\vdash a.$

These conventions differ by left–right placement, but both encode the same structural fact: the dialgebraic analogue of a Lie algebra is a Leibniz algebra, not an antisymmetric Lie algebra (Kolesnikov, 2011, Bremner, 2012).

A left Leibniz algebra is a vector space with bilinear bracket satisfying

$\vdash$0

This is the identity governing Lie dialgebras in the standard sense. If the bracket is skew-symmetric, one recovers an ordinary Lie algebra. The passage from dialgebras to Leibniz algebras is therefore the precise replacement of the classical associative-to-Lie commutator construction by a two-product noncommutative version (Kolesnikov et al., 2012).

The standard equivalence is often summarized as

$\vdash$1

This equivalence is not merely terminological. It organizes the theory around the idea that dialgebras are to Leibniz algebras what associative algebras are to Lie algebras, with the loss of antisymmetry compensated by the Leibniz derivation law (Bremner, 2012).

2. Operadic formulation and the passage from Lie identities to dialgebra identities

The operadic formulation makes the standard meaning precise. For any operad $\vdash$2, the corresponding dialgebra operad is

$\vdash$3

the Hadamard product with the operad $\vdash$4. In particular,

$\vdash$5

and this is exactly the operad governing Leibniz algebras. This gives a uniform conceptual explanation of why Lie dialgebras are Leibniz algebras and places them inside the general passage

$\vdash$6

from algebra varieties to dialgebra varieties (Kolesnikov et al., 2012).

A central mechanism here is the Kolesnikov–Pozhidaev algorithm. Applied to the Lie identities—anticommutativity and Jacobi—it introduces two binary operations and then eliminates one through the degree-two relations, yielding a single binary operation $\vdash$7 satisfying the Leibniz identity

$\vdash$8

In the same framework, right anticommutativity appears as a consequence rather than an independent axiom. The same algorithm applied to associativity gives associative dialgebras, and applied to higher identities it produces the corresponding triple-system and disystem structures (Bremner, 2012).

The complementary Bremner–Sánchez-Ortega algorithm acts on operations rather than identities. Applied to the ordinary commutator $\vdash$9, it produces dialgebra operations whose essential content reduces to the dicommutator. This is the operation-level counterpart of the KP passage from Lie identities to Leibniz identities. The two procedures are conjecturally equivalent in the sense that “identify first, then dialgebraize” and “dialgebraize first, then identify” yield the same multilinear theory; the survey literature presents Lie/Leibniz as the cleanest example of this equivalence (Bremner, 2012).

Special identities behave equally rigidly. All polylinear special identities for dialgebras are obtained from special identities for the corresponding ordinary algebras by the procedure $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$0. In the Lie case this means that Leibniz algebras obtained from associative dialgebras satisfy no unexpected new polylinear special identities beyond those inherited from the ordinary Lie-from-associative situation (Kolesnikov et al., 2012).

3. Embeddings, representations, and crossed modules

One of the strongest structural features of Lie dialgebras is their realization inside conformal algebra theory. If $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$1 is a dialgebra satisfying the $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$2-identities, then

$\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$3

is an injective homomorphism of dialgebras

$\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$4

More generally, a dialgebra is a di-$\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$5-algebra if and only if its split null extension $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$6 belongs to $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$7, and equivalently if and only if it embeds into $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$8 for an appropriate $\mathrm{diLie}=\mathrm{Lie}\otimes \mathrm{Perm}$9-algebra $k$0. For Lie dialgebras this identifies Leibniz algebras with objects that can be studied through current conformal constructions (Kolesnikov, 2011).

This conformal embedding leads to a concrete representation theory. If $k$1 is a Leibniz algebra and $k$2 is its associated Lie algebra, then for any $k$3-module $k$4 there is an injective homomorphism

$k$5

This representation-theoretic construction yields analogues of several classical Lie theorems. The paper derives an Engel theorem for Leibniz algebras, a PBW theorem

$k$6

and an Ado-type result: every finite-dimensional Leibniz algebra embeds into a finite-dimensional diassociative algebra, and an $k$7-dimensional Leibniz algebra embeds into some diassociative algebra $k$8 with

$k$9

These results place Lie dialgebras in a representation-theoretic position closely parallel to ordinary Lie algebras, but mediated by dialgebras and conformal algebras rather than by associative envelopes alone (Kolesnikov, 2011).

The crossed-module level preserves the same pattern. Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are constructed, and the familiar relations among Lie, Leibniz, associative algebras, and dialgebras extend to the corresponding crossed-module categories (Casas et al., 2015). This shows that the standard Lie/Leibniz/dialgebra correspondences are stable under one categorical level of internalization.

4. Triple systems and higher structures

The ternary analogue of the Lie dialgebra concept is the Leibniz triple system. Its role relative to Leibniz algebras mirrors the role of Lie triple systems relative to Lie algebras. A Leibniz triple system is a vector space with a trilinear operation $\dashv,\ \vdash : A\times A\to A,$0 satisfying two defining identities,

$\dashv,\ \vdash : A\times A\to A,$1

and

$\dashv,\ \vdash : A\times A\to A,$2

These identities arise functorially from the identities of Lie triple systems via the Kolesnikov–Pozhidaev algorithm (Bremner et al., 2011).

If $\dashv,\ \vdash : A\times A\to A,$3 is a Leibniz algebra and one defines

$\dashv,\ \vdash : A\times A\to A,$4

then any subspace of $\dashv,\ \vdash : A\times A\to A,$5 closed under this ternary product is a Leibniz triple system. This is the direct ternary analogue of the iterated Lie bracket $\dashv,\ \vdash : A\times A\to A,$6. The theory also has a universal envelope: for a Leibniz triple system $\dashv,\ \vdash : A\times A\to A,$7,

$\dashv,\ \vdash : A\times A\to A,$8

and if $\dashv,\ \vdash : A\times A\to A,$9, then

$0$0

Moreover, every polynomial identity satisfied by the iterated Leibniz bracket $0$1 in every Leibniz algebra is a consequence of the defining identities of Leibniz triple systems. In this sense, Leibniz triple systems are the exact equational abstraction of the ternary operation naturally attached to Lie dialgebras (Bremner et al., 2011).

A further higher development appears in Poisson dialgebras. There the Lie-like part is again Leibniz rather than Lie, and the paper emphasizes that the correct noncommutative Poisson structure is a dialgebra together with a compatible Leibniz bracket. From any Leibniz algebra one obtains a Lie $0$2-algebra on the graded space $0$3, with

$0$4

and

$0$5

For a Poisson dialgebra, a graded space $0$6 inherits both a Lie $0$7-algebra and an associative $0$8-algebra structure, and in the reduced case one obtains a $0$9-term homotopy Poisson algebra of degree $$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$0. This indicates that the higher-categorical afterlife of Lie dialgebras is naturally phrased in terms of Leibniz-to-Lie $$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$1-algebra constructions rather than in terms of ordinary Lie brackets alone (Das et al., 2023).

5. Alternative meanings of the term

A persistent source of ambiguity is that some later papers use “dialgebra” in a sense different from Loday’s $$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$2-framework. In particular, a crucial point in the work of Zhang–Bai–Guo is terminological: their “dialgebra” is not the Loday dialgebra with operations $$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$3. Instead, a totally compatible Lie dialgebra is a vector space with two Lie brackets $$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$4 and $$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$5 satisfying

$$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$6

and

$$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$7

This is a two-bracket compatibility structure, not a Leibniz algebra. It is the Lie-side companion of totally compatible associative dialgebras, and every totally compatible associative dialgebra $$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$8 yields such a structure by the commutators

$$(x \dashv y)\vdash z=(x\vdash y)\vdash z,\qquad x\dashv (y\vdash z)=x\dashv (y\dashv z).$$9

Under a Rota–Baxter-type identity,

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$0

one obtains a PostLie algebra with

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$1

This use of “Lie dialgebra” is therefore secondary but well established in the compatibility-operad literature (Zhang et al., 2015).

A different secondary notion is the initial Lie dialgebra. For $$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$2, the initial Lie dialgebra coincides with a Leibniz algebra satisfying the additional identity

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$3

In that setting the combined product

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$4

recovers an ordinary Lie algebra. The free initial Lie dialgebra has an explicit basis, and the paper proves the vector-space decomposition

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$5

together with the arity formula

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$6

This suggests a subvariety of Leibniz algebras specifically tailored to recover ordinary Lie structure under the $$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$7-operation (Dauletiyarova et al., 23 Jan 2026).

These secondary meanings do not replace the standard identification of Lie dialgebras with Leibniz algebras. They show instead that the phrase “Lie dialgebra” has broadened into a family of two-operation Lie-type structures, and that precise convention is indispensable.

6. Refinements and broader categorical setting

One refinement of the standard picture is the theory of symmetric Leibniz algebras. A symmetric Leibniz algebra is a vector space with a product satisfying both left and right Leibniz identities,

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$8

and

$$x \vdash (y\vdash z) = (x \vdash y) \vdash z,\qquad x \dashv (y\dashv z) = (x \dashv y) \dashv z,\qquad x \vdash (y\dashv z) = (x \vdash y) \dashv z.$$9

In such an algebra the commutator

$[a,b]=a\vdash b-b\dashv a,$0

is a genuine Lie bracket. The corresponding symmetric dialgebras satisfy the condition that

$[a,b]=a\vdash b-b\dashv a,$1

for all $[a,b]=a\vdash b-b\dashv a,$2, and they map to symmetric Leibniz algebras by

$[a,b]=a\vdash b-b\dashv a,$3

Operadically, these structures fit into the chain

$[a,b]=a\vdash b-b\dashv a,$4

with dual side

$[a,b]=a\vdash b-b\dashv a,$5

and all the operads involved are Koszul. This gives a symmetric refinement of the ordinary Lie-dialgebra/Leibniz picture in which the commutator remains Lie even without antisymmetry of the underlying product (Hurle, 2019).

A broader categorical generalization appears in the study of Lie algebras in symmetric monoidal categories. There a Lie algebra is an object $[a,b]=a\vdash b-b\dashv a,$6 with a morphism

$[a,b]=a\vdash b-b\dashv a,$7

satisfying categorical antisymmetry

$[a,b]=a\vdash b-b\dashv a,$8

and the corresponding Jacobi identity formulated through the symmetry and associativity constraints. Besides ordinary Lie algebras, this framework includes examples from Jacobi-diagram categories related to knot invariants and from Rozansky–Witten theory. This suggests that dialgebraic Lie theory belongs to a wider reorganization of Lie structures in which the traditional vector-space setting is only one special case (Rumynin, 2012).

The modern landscape is therefore layered. In the standard sense, Lie dialgebras are Leibniz algebras arising from Loday dialgebras and governed by $[a,b]=a\vdash b-b\dashv a,$9. Around this core lie symmetric refinements, triple systems, Poisson and higher-categorical extensions, and alternative two-bracket usages. What remains constant across these variants is the central structural insight: the dialgebraic replacement of antisymmetric Lie brackets is controlled by Leibniz-type identities, and the passage from associative-type data to Lie-type data continues to be organized by operads, envelopes, and categorical adjunctions.