Associative-Yamaguti Algebras

• Associative-Yamaguti algebras are binary–ternary structures defined on vector spaces with one bilinear and two trilinear maps that satisfy a complex system of associativity-type identities.
• They generalize familiar structures such as associative algebras, reductive associative algebras, and associative triple systems by incorporating controlled failures of strict associativity through auxiliary ternary operations.
• The theory further introduces enveloping associative algebras, a skew-symmetrization functor yielding Lie-Yamaguti algebras, and a (2,3)-cohomology framework for studying deformations and abelian extensions.

Searching arXiv for recent and foundational papers on associative-Yamaguti and related Lie-Yamaguti structures. Associative-Yamaguti algebras are binary–ternary algebraic structures introduced as the associative analogue of Lie-Yamaguti algebras. An associative-Yamaguti algebra, or AY-algebra, is a quadruple

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

on a vector space over a field $\mathbf{k}$ of characteristic $0$, consisting of one bilinear product and two trilinear operations constrained by a system of associativity-type identities. The construction is designed so that associative algebras, reductive associative algebras, and associative triple systems of the first kind appear as subclasses, while any diassociative algebra canonically determines an AY-structure. The theory further includes an enveloping associative algebra, a skew-symmetrization functor to Lie-Yamaguti algebras, a $(2,3)$-cohomology governing deformations and abelian extensions, an operadic reformulation via Yamaguti multiplications, and a splitting theory through dendriform-Yamaguti algebras and relative Rota-Baxter operators (Das, 3 Sep 2025).

1. Definition and basic axioms

An associative-Yamaguti algebra is defined by a binary product $\cdot : A\otimes A\to A$ and two trilinear operations $$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$ satisfying, for all $a,b,c,d,e\in A$,

$$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$

$\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$

$\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$

$\mathbf{k}$0

$\mathbf{k}$1

$\mathbf{k}$2

$\mathbf{k}$3

$\mathbf{k}$4

$\mathbf{k}$5

These identities can be encoded through operator-valued maps

$\mathbf{k}$6

In this form, AY1–AY11 become associativity-type equations for $\mathbf{k}$7 together with compatibility relations between $\mathbf{k}$8 and $\mathbf{k}$9 (Das, 3 Sep 2025).

AY1 identifies the associator $0$0 with the difference of the two ternary operations. AY2–AY6 impose rigid trilinearity over the binary product. AY7–AY11 express the mutual compatibility of the two ternary operations and play the role of associative analogues of the Jacobi-type and fundamental identities in Lie-Yamaguti theory. The paper characterizes the resulting formalism as a non-skew, “Loday-type” version of Lie-Yamaguti algebras because the variables appear in the same order in each term.

2. Subclasses and constructions from known algebraic structures

The class of AY-algebras is explicitly designed to contain several established structures (Das, 3 Sep 2025).

Structure	AY realization	Defining specialization
Associative algebra	AY-algebra	$0$1
Reductive associative algebra $0$2	AY on $0$3	$0$4, ternaries from the $0$5-part
Associative triple system of first kind	AY-algebra	$0$6, $0$7
Diassociative algebra $0$8	Canonical AY-algebra	$0$9 with ternaries from $(2,3)$0

For an associative algebra $(2,3)$1, setting

$(2,3)$2

reduces AY1 to associativity, while AY2–AY11 become tautological because the ternaries are iterated products.

A reductive associative algebra is an associative algebra $(2,3)$3 such that

$(2,3)$4

On $(2,3)$5, one defines

$(2,3)$6

and this yields an AY-algebra. In this construction, the $(2,3)$7-component of the associative product is transferred into the ternary operations.

For an associative triple system of first kind,

$(2,3)$8

the AY-structure is obtained by taking $(2,3)$9 and identifying the two ternary operations. AY7–AY11 then become precisely the triple-system identities, while AY1–AY6 degenerate.

Diassociative algebras provide a more distinctly nonclassical source. If $\cdot : A\otimes A\to A$0 satisfies Loday’s axioms

$\cdot : A\otimes A\to A$1

the associated AY-operations are

$\cdot : A\otimes A\to A$2

$\cdot : A\otimes A\to A$3

$\cdot : A\otimes A\to A$4

The paper proves that these operations satisfy AY1–AY11 and that a homomorphism of diassociative algebras induces a homomorphism of the associated AY-algebras, giving a functor

$\cdot : A\otimes A\to A$5

A common misconception would be to view AY-algebras merely as associative algebras with auxiliary ternary operations. The diassociative and reductive examples show that the ternary structure is not incidental: it records the controlled failure of strict associativity and the transfer of hidden algebraic components into ternary data.

3. Skew-symmetrization and the relation to Lie-Yamaguti theory

The principal structural bridge from AY-theory to Lie-Yamaguti theory is skew-symmetrization. Given an AY-algebra $\cdot : A\otimes A\to A$6, define

$\cdot : A\otimes A\to A$7

and

$\cdot : A\otimes A\to A$8

Theorem 3.8 shows that $\cdot : A\otimes A\to A$9 is a Lie-Yamaguti algebra (Das, 3 Sep 2025).

This statement places AY-algebras in direct analogy with the classical passage from associative algebras to Lie algebras by skew-symmetrization. In the special case where $$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$0 is associative, the bracket becomes the usual commutator and

$$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$1

recovering the standard Lie-algebraic example of a Lie-Yamaguti algebra.

The same passage is compatible with representations. If $$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$2 is a representation of an AY-algebra $$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$3, then the paper defines

$$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$4

$$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$5

and proves that $$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$6 is a representation of the induced Lie-Yamaguti algebra.

This construction clarifies the status of AY-theory relative to earlier work on Lie-Yamaguti algebras. The 2013 paper on symmetric matrices, orthogonal Lie algebras, and Lie-Yamaguti algebras constructs LY-algebras from reductive decompositions

$$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$7

with products

$$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$8

and studies a family arising from the embedding

$$\{\, ,\, ,\,\},\,\{\!\!\{\, ,\, ,\,\}\!\!\}:A^{\otimes 3}\to A$$9

obtained from the action of $a,b,c,d,e\in A$0 on trace-zero symmetric matrices $a,b,c,d,e\in A$1 (Benito et al., 2013). That paper does not define associative-Yamaguti algebras, but it is important context because it exhibits how binary–ternary Yamaguti-type structures arise from associative matrix data after passage through Lie and Jordan constructions. A plausible implication is that AY-theory systematizes, on the associative side, patterns that were already visible in matrix-origin Lie-Yamaguti examples.

4. Enveloping associative algebras and reductive realization

One of the central results of the theory is that every AY-algebra admits an enveloping associative algebra (Das, 3 Sep 2025). The starting point is a general construction. Let $a,b,c,d,e\in A$2 be an AY-algebra, let $a,b,c,d,e\in A$3 be an associative algebra, let $a,b,c,d,e\in A$4 be a $a,b,c,d,e\in A$5-bimodule with actions $a,b,c,d,e\in A$6 and $a,b,c,d,e\in A$7, and let $a,b,c,d,e\in A$8 satisfy

$a,b,c,d,e\in A$9

together with the compatibility identities denoted $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$0, $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$1, and $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$2 in the paper. Then

$$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$3

defines an associative product on $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$4, and $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$5 becomes a reductive associative algebra whose induced AY-structure on $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$6 agrees with the original one.

The canonical realization is obtained internally from the AY-structure itself. Define

$$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$7

and let $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$8 be the subspace spanned by all pairs $$\tag{AY1} (a\cdot b)\cdot c - a\cdot(b\cdot c) + \{a,b,c\} - \{\!\!\{a,b,c\}\!\!\} =0,$$9. On $\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$0, the product

$\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$1

is associative, with AY7–AY11 encoding precisely its associativity. The bimodule actions are

$\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$2

and the map $\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$3 supplies the required defect term.

The resulting enveloping algebra is

$\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$4

with multiplication

$\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$5

The induced AY-structure on the summand $\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$6 is exactly the original one.

This enveloping theorem is structurally parallel to the standard-envelope construction for Lie-Yamaguti algebras. In AY-theory, however, the envelope is associative rather than Lie. That distinction is conceptually significant: the theory is not simply a reformulation of Lie-Yamaguti theory with relaxed skew-symmetry, but a genuinely associative envelope-based analogue.

5. Cohomology, formal deformations, and abelian extensions

The paper introduces a $\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$7-cohomology tailored to the binary–ternary nature of AY-algebras (Das, 3 Sep 2025). A representation of an AY-algebra $\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$8 on a vector space $\tag{AY2} \{a\cdot b,c,d\} = \{a,b\cdot c,d\}, \qquad \tag{AY3} \{a,b,c\cdot d\} = \{a,b,c\}\cdot d,$9 consists of structure maps

$\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$0

satisfying all AY identities with exactly one variable in $\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$1 and the rest in $\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$2. The paper states that this gives 58 identities: 3 from AY1, 4 each from AY2–AY6, 10 each from AY7 and AY9, and 5 each from AY8, AY10, and AY11. Equivalently, $\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$3 is a representation if and only if the semidirect sum $\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$4 with the induced operations is again an AY-algebra.

A $\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$5-cocycle is a triple

$\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$6

satisfying 11 compatibility identities that mirror AY1–AY11. The first of these is

$\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$7

and the remaining relations are AY2-type through AY11-type conditions. The interpretation is that $\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$8-cocycles are precisely infinitesimal corrections to the binary and ternary operations that preserve the AY identities to first order.

Coboundaries arise from linear maps $\tag{AY4} \{\!\!\{a\cdot b,c,d\}\!\!\} = a\cdot\{\!\!\{b,c,d\}\!\!\}, \qquad \tag{AY5} \{\!\!\{a,b\cdot c,d\}\!\!\} = \{\!\!\{a,b,c\cdot d\}\!\!\},$9: $\mathbf{k}$00

$\mathbf{k}$01

$\mathbf{k}$02

The quotient

$\mathbf{k}$03

is the $\mathbf{k}$04-cohomology group.

This cohomology controls formal one-parameter deformations. For deformed operations

$\mathbf{k}$05

$\mathbf{k}$06

$\mathbf{k}$07

Theorem 5.1 states that the first-order term $\mathbf{k}$08 is a $\mathbf{k}$09-cocycle in $\mathbf{k}$10. Equivalent deformations have infinitesimals differing by a $\mathbf{k}$11-coboundary, so the cohomology class $\mathbf{k}$12 is the invariant of the deformation class at first order.

The same cohomology classifies abelian extensions. If

$\mathbf{k}$13

is an extension with $\mathbf{k}$14 an abelian AY-algebra, a linear section $\mathbf{k}$15 determines cocycle data

$\mathbf{k}$16

$\mathbf{k}$17

$\mathbf{k}$18

Theorem 5.4 identifies the set of isomorphism classes of abelian extensions with the cohomology group: $\mathbf{k}$19

A plausible implication is that $\mathbf{k}$20-cohomology plays for AY-algebras a role analogous to Hochschild cohomology for associative algebras and the established cohomology theories for Lie-Yamaguti algebras, but with a grading adapted to simultaneous binary and ternary deformation data.

6. Operadic formulation, dendriform splitting, and further directions

The operadic formulation begins with a nonsymmetric operad $\mathbf{k}$21. A multiplication in the sense of Gerstenhaber–Voronov is an element $\mathbf{k}$22 such that

$\mathbf{k}$23

A Yamaguti multiplication is a triple $\mathbf{k}$24 with $\mathbf{k}$25 and $\mathbf{k}$26 satisfying the operadic identities

$\mathbf{k}$27

$\mathbf{k}$28

together with the five ternary associativity conditions listed in the paper. In the endomorphism operad

$\mathbf{k}$29

the assignment

$\mathbf{k}$30

identifies AY-algebra structures on $\mathbf{k}$31 with Yamaguti multiplications on $\mathbf{k}$32. Theorem 6.3 states that this correspondence is one-to-one (Das, 3 Sep 2025).

This operadic description leads to dendriform-Yamaguti algebras, which split AY-structures in the same way that dendriform algebras split associativity. Such an algebra has binary operations $\mathbf{k}$33 and six ternary operations

$\mathbf{k}$34

subject to a large family of relations denoted DY1A–DY11E. Their totalizations are

$\mathbf{k}$35

$\mathbf{k}$36

$\mathbf{k}$37

Theorem 6.8 asserts that these total operations define an AY-algebra.

Relative Rota-Baxter operators provide the converse mechanism. If $\mathbf{k}$38 is an AY-algebra with representation $\mathbf{k}$39, a linear map $\mathbf{k}$40 is a relative Rota-Baxter operator if

$\mathbf{k}$41

$\mathbf{k}$42

$\mathbf{k}$43

Theorem 6.12 constructs a dendriform-Yamaguti structure on $\mathbf{k}$44 from such an $\mathbf{k}$45, while Theorem 6.13 shows that every dendriform-Yamaguti algebra arises in this way from its total AY-algebra and the identity map.

Several open directions are explicitly proposed. These include constructing a full cochain complex $\mathbf{k}$46 extending the present $\mathbf{k}$47-theory, developing cup products and Gerstenhaber-type operations, studying weak associative triple systems satisfying only AY7 and AY9, investigating group-like objects tentatively described as Lie-Yamaguti groups, defining Poisson-Yamaguti algebras, and building a representation theory and classification theory for AY-algebras. These proposals indicate that the initial 2025 framework is foundational rather than exhaustive.

Within the broader Yamaguti landscape, AY-algebras occupy a precise position: they are not merely associative shadows of Lie-Yamaguti algebras, but a parallel theory with its own envelopes, cohomology, operads, splitting theory, and examples. Earlier matrix-based work on Lie-Yamaguti algebras revealed that associative multiplication can generate rich binary–ternary structures after projection and skew-symmetrization (Benito et al., 2013); the AY formalism turns that associative origin into the primary organizing principle.