Dendriform-Yamaguti Algebras

• Dendriform–Yamaguti algebras are split, Loday-type refinements of associative–Yamaguti algebras defined by two binary and six directional ternary operations satisfying 58 compatibility identities.
• They decompose the total associative–Yamaguti product into sectorial binary and ternary components, enabling a canonical passage to Lie–Yamaguti algebras via skew-symmetrization.
• Operadic and relative Rota–Baxter formulations underpin these structures, unifying dendriform, associative, and triple-system theories with practical symmetry and cohomology applications.

Dendriform–Yamaguti algebras are split, Loday-type refinements of associative–Yamaguti algebras. They are defined on a vector space $D$ by two bilinear operations $\prec,\succ$ and six trilinear operations, organized into three directional components for each of two ternary families, subject to a large system of compatibility identities. Their basic structural role is that of a splitting object: when the binary operations are summed to a total product and the ternary operations are summed sectorwise, one recovers an associative–Yamaguti algebra; from there, a canonical skew-symmetrization yields a Lie–Yamaguti algebra. In operadic terms, a dendriform–Yamaguti algebra is a Yamaguti multiplication on the dendriform operad, and in Rota–Baxter terms it is equivalent to a suitable relative Rota–Baxter operator on an associative–Yamaguti algebra (Das, 3 Sep 2025).

1. Position within Yamaguti-type algebraic structures

The immediate antecedent of the theory is the Lie–Yamaguti algebra, a vector space $\mathfrak g$ with a skew-symmetric bilinear bracket $[x,y]$ and a ternary operation $\llbracket x,y,z\rrbracket$ skew-symmetric in the first two variables, satisfying generalized Jacobi-type identities. Lie–Yamaguti algebras simultaneously generalize Lie algebras and Lie triple systems and arise from reductive decompositions and from Nomizu’s work on invariant affine connections (Das, 3 Sep 2025).

Associative–Yamaguti algebras were introduced as the associative analogues of Lie–Yamaguti algebras. An associative–Yamaguti algebra is a quadruple

$(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$

with one bilinear product and two ternary operations satisfying $11$ identities $(\mathrm{AY}1)$–$(\mathrm{AY}11)$. The first identity is a generalized associativity relation,

$$(a\cdot b)\cdot c-a\cdot(b\cdot c)+\{a,b,c\}-\{\!\!\{a,b,c\}\!\!\}=0,$$

while the remaining identities constrain the interaction of the binary and ternary operations in an ordered, Loday-type fashion (Das, 3 Sep 2025).

Dendriform–Yamaguti algebras occupy the next level in this hierarchy. They play for associative–Yamaguti algebras the same splitting role that dendriform algebras play for associative algebras. This placement is essential: the theory is not merely an enrichment of dendriform algebras by arbitrary ternary maps, but a structured decomposition of associative–Yamaguti data into sectorial binary and ternary pieces. A common misconception is to view the ternary operations as optional adornments; the defining identities instead make them intrinsic corrections to associativity and its higher compatibilities.

2. Defining operations and identity system

A dendriform–Yamaguti algebra is a tuple

$\prec,\succ$0

where $\prec,\succ$1 are bilinear and

$\prec,\succ$2

are trilinear, satisfying the system $\prec,\succ$3–$\prec,\succ$4 (Das, 3 Sep 2025).

The sectorwise total ternary operations are

$\prec,\succ$5

$\prec,\succ$6

The first block of identities is the dendriform splitting of generalized associativity: $\prec,\succ$7

$\prec,\succ$8

$\prec,\succ$9

These formulas show how each sector $\mathfrak g$0 contributes a correction term to the associativity defect of the corresponding dendriform component. The remaining blocks, $\mathfrak g$1–$\mathfrak g$2 and $\mathfrak g$3–$\mathfrak g$4, refine the associative–Yamaguti identities for binary–ternary compatibility and ternary associativity-type behavior. Structurally, all associative–Yamaguti identities hold componentwise after the total binary product is split into $\mathfrak g$5 and each ternary operation is split into three directional pieces (Das, 3 Sep 2025).

This makes the theory more rigid than the notation initially suggests. The paper identifies $\mathfrak g$6 representation conditions corresponding exactly to the $\mathfrak g$7 dendriform–Yamaguti identities. A plausible shorthand is to regard the three indices $\mathfrak g$8 as “directional sectors,” but the sector decomposition is not superficial bookkeeping: it is the mechanism by which ordered associativity data are resolved into dendriform form.

3. Operadic formulation and the splitting theorem

The theory is formulated systematically on nonsymmetric operads. For a vector space $\mathfrak g$9, the endomorphism operad is

$[x,y]$0

while the dendriform operad is

$[x,y]$1

with $[x,y]$2. A binary multiplication $[x,y]$3 encodes

$[x,y]$4

Given any nonsymmetric operad $[x,y]$5, a Yamaguti multiplication is a triple

$[x,y]$6

satisfying operadic analogues of the associative–Yamaguti identities, beginning with

$[x,y]$7

Yamaguti multiplications on $[x,y]$8 are in bijection with associative–Yamaguti structures on $[x,y]$9 (Theorem $\llbracket x,y,z\rrbracket$0), and a dendriform–Yamaguti algebra is, by Definition $\llbracket x,y,z\rrbracket$1, precisely a Yamaguti multiplication on $\llbracket x,y,z\rrbracket$2 (Das, 3 Sep 2025).

The resulting splitting theorem is the central structural statement. If

$\llbracket x,y,z\rrbracket$3

together with the total ternary operations defined above, then

$\llbracket x,y,z\rrbracket$4

is an associative–Yamaguti algebra (Theorem $\llbracket x,y,z\rrbracket$5). The proof is additive: $\llbracket x,y,z\rrbracket$6 is obtained by summing $\llbracket x,y,z\rrbracket$7, $\llbracket x,y,z\rrbracket$8 by summing $\llbracket x,y,z\rrbracket$9–$(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$0, and similarly for the remaining identities (Das, 3 Sep 2025).

This has an immediate consequence. Since any associative–Yamaguti algebra admits the canonical skew-symmetrization

$(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$1

$(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$2

the total structure of a dendriform–Yamaguti algebra canonically determines a Lie–Yamaguti algebra. This implies that dendriform–Yamaguti algebras sit simultaneously in the dendriform, associative–Yamaguti, and Lie–Yamaguti layers of the same operadic passage.

4. Examples, subclasses, and unifying scope

Several constructions show that dendriform–Yamaguti algebras unify pre-existing binary and ternary splitting theories (Das, 3 Sep 2025).

Source structure	Construction	Result
Dendriform algebra $(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$3	Set each $(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$4 and $(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$5 equal to the corresponding iterated sectorial product	Dendriform–Yamaguti algebra
Dendriform triple system	Take $(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$6	Dendriform–Yamaguti algebra with trivial binary part
Associative–Yamaguti algebra with relative Rota–Baxter operator	Transfer operations to a representation space via the operator	Dendriform–Yamaguti algebra

For a dendriform algebra $(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$7, the ternary operations are defined by

$(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$8

$(A,\cdot,\{-, -,-\},\{\!\!\{-,-,-\}\!\!\})$9

$11$0

The total associative–Yamaguti structure is then the associativization of the dendriform algebra, with ternary operations given by iterated products. This realizes ordinary dendriform algebras as a special case of the new notion, but not the generic one: in a general dendriform–Yamaguti algebra the two ternary families are independent, constrained only by the DY-identities (Das, 3 Sep 2025).

At the opposite extreme, dendriform triple systems appear by taking the binary operations trivial. In this limit the DY-identities collapse to the identities of dendriform triple systems. Accordingly, dendriform–Yamaguti algebras unify binary splitting only, ternary splitting only, and the full binary-plus-ternary associative–Yamaguti setting.

A nearby but distinct line of work studies dendriform D-bialgebras, quasi-triangularity, factorizability, and relative Rota–Baxter operators of nonzero weight on dendriform algebras (Wang, 3 Jul 2025). That theory does not define dendriform–Yamaguti algebras, but it supplies a parallel dendriform environment in which three-tensor identities, doubles, and Rota–Baxter mechanisms already play a central role. This suggests that bialgebraic and factorization methods may become useful for explicit families of dendriform–Yamaguti structures.

5. Relative Rota–Baxter operators and reconstruction

The tightest conceptual characterization of dendriform–Yamaguti algebras is through relative Rota–Baxter operators. For an associative–Yamaguti algebra $11$1, a representation $11$2 consists of linear maps for the binary product and the two ternary products with exactly one input in $11$3, satisfying all associative–Yamaguti identities in that mixed setting; equivalently, the semidirect sum $11$4 is again an associative–Yamaguti algebra (Das, 3 Sep 2025).

A relative Rota–Baxter operator $11$5 is a linear map satisfying

$11$6

$11$7

$11$8

It is characterized equivalently by the condition that its graph

$11$9

is a subalgebra of the semidirect associative–Yamaguti algebra.

From such an operator one defines

$(\mathrm{AY}1)$0

$(\mathrm{AY}1)$1

and similarly for $(\mathrm{AY}1)$2. Theorem $(\mathrm{AY}1)$3 states that these operations make $(\mathrm{AY}1)$4 into a dendriform–Yamaguti algebra. Conversely, given a dendriform–Yamaguti algebra $(\mathrm{AY}1)$5, one constructs a representation of its total associative–Yamaguti algebra $(\mathrm{AY}1)$6 on the same underlying space, and Theorem $(\mathrm{AY}1)$7 shows that the identity map

$(\mathrm{AY}1)$8

is a relative Rota–Baxter operator whose induced dendriform–Yamaguti structure is the original one (Das, 3 Sep 2025).

Together, Theorems $(\mathrm{AY}1)$9 and $(\mathrm{AY}11)$0 establish a bijection between dendriform–Yamaguti structures and suitable relative Rota–Baxter operators. This is more than a construction device. It identifies dendriform–Yamaguti algebras as the Rota–Baxter splittings of associative–Yamaguti algebras, exactly paralleling the classical relation between associative algebras, dendriform algebras, and $(\mathrm{AY}11)$1-operators.

6. Cohomology, deformation theory, and broader structural directions

The cohomological theory developed directly in the associative–Yamaguti setting is a $(\mathrm{AY}11)$2-cohomology with cochains $(\mathrm{AY}11)$3, where

$(\mathrm{AY}11)$4

Its cocycle conditions are precisely those needed for the twisted semidirect product to remain an associative–Yamaguti algebra. Formal one-parameter deformations

$(\mathrm{AY}11)$5

have infinitesimal term $(\mathrm{AY}11)$6 a $(\mathrm{AY}11)$7-cocycle in the adjoint representation, while abelian extensions satisfy

$(\mathrm{AY}11)$8

(Theorems $(\mathrm{AY}11)$9 and $$(a\cdot b)\cdot c-a\cdot(b\cdot c)+\{a,b,c\}-\{\!\!\{a,b,c\}\!\!\}=0,$$0) (Das, 3 Sep 2025).

No separate cohomology theory for dendriform–Yamaguti algebras is explicitly developed there. Two points are nevertheless recorded. First, deformations and extensions of associative–Yamaguti algebras often lift to deformations of dendriform–Yamaguti splittings via the correspondence with relative Rota–Baxter operators. Second, the splitting philosophy and the operadic setup suggest that a dendriform–Yamaguti cohomology should refine associative–Yamaguti cohomology, analogously to the relation between dendriform and Hochschild theories. These are prospective rather than completed results.

A further structural direction comes from Lie–Yamaguti symmetry theory. The universal coacting bialgebra and universal coacting Hopf algebra of a finite-dimensional Lie–Yamaguti algebra have been constructed, together with applications to automorphism groups and abelian group gradings (Goswami et al., 2 Jun 2025). Since every dendriform–Yamaguti algebra canonically determines an underlying Lie–Yamaguti algebra through its total associative–Yamaguti structure and skew-symmetrization, this suggests that any future universal coacting Hopf algebra for dendriform–Yamaguti algebras should factor through the corresponding Lie–Yamaguti universal object, with additional relations expressing preservation of the split operations.

Taken together, these results place dendriform–Yamaguti algebras at the intersection of associative, diassociative, dendriform, triple-system, Lie–Yamaguti, and Rota–Baxter theories. Their defining content is the controlled splitting of associative–Yamaguti operations; their principal characterization is operadic and Rota–Baxter-theoretic; and their most immediate open directions concern refined cohomology, deformation theory, and symmetry objects adapted to the split rather than merely total structure.