Regular Hom-Lie Yamaguti Algebras

• Regular Hom-Lie Yamaguti algebras are binary–ternary Hom-type algebras with a bijective twisting map, unifying Hom-Lie, Hom-Lie triple systems, and classical Lie-Yamaguti structures.
• They exhibit rich structure through quotient constructions, central ideals, and isoclinism formalism, culminating in a finite-dimensional rigidity theorem based on stem decompositions.
• These algebras connect to Hom-Malcev, Hom-Leibniz, and cohomological frameworks, underpinning deformation, extension, and representation theories in Hom-algebra.

Searching arXiv for the cited HLYA papers to ground the article in the specified literature. arXiv search query: (Asif et al., 3 Aug 2025) "Isoclinism in regular Hom-Lie Yamaguti algebras" Regular Hom-Lie Yamaguti algebras are multiplicative Hom-Lie-Yamaguti algebras $(A,[-,-],[-,-,-],\alpha_A)$ for which the twisting map $\alpha_A$ is bijective. They are binary–ternary Hom-type algebras over a field $\mathbb K$ of characteristic zero, and they unify several standard structures: when the ternary bracket vanishes one recovers a Hom-Lie algebra, when the binary bracket vanishes one obtains a Hom-Lie triple system, and when $\alpha=\mathrm{Id}$ one returns to the classical Lie-Yamaguti setting. In the regular case, bijectivity of $\alpha_A$ governs quotient constructions, centrality, extension theory, and the isoclinism formalism, culminating in a finite-dimensional rigidity theorem: for regular Hom-Lie Yamaguti algebras of equal dimension, isoclinism implies isomorphism (Asif et al., 3 Aug 2025).

1. Defining identities and basic reductions

A Hom-Lie-Yamaguti algebra consists of a linear space $A$, a linear self-map $\alpha_A:A\to A$, a bilinear bracket $[-,-]:A\times A\to A$, and a trilinear bracket $[-,-,-]:A\times A\times A\to A$, subject to the identities, for all $x,y,z,w,t\in A$,

$\alpha_A$0

$\alpha_A$1

$\alpha_A$2

$\alpha_A$3

$\alpha_A$4

It is multiplicative when

$\alpha_A$5

and it is regular when $\alpha_A$6 is bijective (Asif et al., 3 Aug 2025).

The same structure appears in the earlier Hom-Lie-Yamaguti literature in equivalent binary–ternary form, usually denoted by $\alpha_A$7 and organized through the twisted Yamaguti identities $\alpha_A$8–$\alpha_A$9 or $\mathbb K$0–$\mathbb K$1. In that formulation, the category of multiplicative Hom-Lie-Yamaguti algebras is closed under twisting by self-morphisms: if $\mathbb K$2 is an endomorphism commuting with $\mathbb K$3, then

$\mathbb K$4

again defines a multiplicative Hom-Lie-Yamaguti algebra (Gaparayi et al., 2010).

Two reductions are structurally decisive. If $\mathbb K$5, the algebra reduces to a Hom-Lie triple system, with the induced ternary Hom-Nambu behavior emphasized in the foundational papers. If $\mathbb K$6, it reduces to a Hom-Lie algebra. These reductions are not peripheral: they identify regular Hom-Lie Yamaguti algebras as a common envelope for binary Hom-Lie and ternary Hom-Lie-type geometries (Asif et al., 3 Aug 2025).

2. Regularity, central structure, and stem algebras

Regularity means precisely that $\mathbb K$7 is bijective. In the isoclinism theory this hypothesis is essential because it guarantees that the induced map on quotients is bijective and that the center is a Hom-ideal. For a subspace $\mathbb K$8, being a Hom-Lie Yamaguti subalgebra means

$\mathbb K$9

whereas $\alpha=\mathrm{Id}$0 is a Hom-ideal when

$\alpha=\mathrm{Id}$1

The center is

$\alpha=\mathrm{Id}$2

and the derived subalgebra is

$\alpha=\mathrm{Id}$3

A regular Hom-Lie Yamaguti algebra is called stem when

$\alpha=\mathrm{Id}$4

In a regular Hom-Lie Yamaguti algebra, $\alpha=\mathrm{Id}$5 is a Hom-ideal (Asif et al., 3 Aug 2025).

If $\alpha=\mathrm{Id}$6 is a Hom-ideal of a regular Hom-Lie Yamaguti algebra $\alpha=\mathrm{Id}$7, the quotient $\alpha=\mathrm{Id}$8 inherits the regular Hom-Lie Yamaguti structure via

$\alpha=\mathrm{Id}$9

and when $\alpha_A$0 is bijective with $\alpha_A$1, the induced $\alpha_A$2 is bijective as well. This is the mechanism by which central quotients enter the isoclinism formalism (Asif et al., 3 Aug 2025).

Within an isoclinism family, stem objects play the same role that stem groups or stem Lie algebras do in the classical theory: they are the central representatives with no “extraneous” abelian direct summand outside the derived part. For finite-dimensional regular Hom-Lie Yamaguti algebras, this role becomes quantitative: a member of an isoclinism family is stem if and only if its dimension is minimal in that family (Asif et al., 3 Aug 2025).

3. Isoclinism and the invariants it preserves

For regular Hom-Lie Yamaguti algebras

$\alpha_A$3

write $\alpha_A$4 and $\alpha_A$5. The binary and ternary commutator maps are

$\alpha_A$6

$\alpha_A$7

and similarly $\alpha_A$8 on $\alpha_A$9 (Asif et al., 3 Aug 2025).

A pair $A$0, with

$A$1

is a homoclinism when the binary and ternary commutator diagrams commute and the twisting maps are respected: $A$2

$A$3

If both $A$4 and $A$5 are isomorphisms, one has an isoclinism, written $A$6 (Asif et al., 3 Aug 2025).

This notion is weaker than isomorphism. By definition, isomorphism implies isoclinism, but the converse fails in general. The central reason is that isoclinism retains the commutator geometry of the quotient by the center and of the derived subalgebra, while allowing central or abelian enlargement. The preserved invariants are therefore

$A$7

and hence also $A$8 and $A$9. In stem families, the center is also preserved up to isomorphism under the derived-subalgebra isomorphism restricted to the center (Asif et al., 3 Aug 2025).

Several structural consequences parallel the classical theory. If $\alpha_A:A\to A$0 is abelian, then

$\alpha_A:A\to A$1

If $\alpha_A:A\to A$2 is a surjective homomorphism with $\alpha_A:A\to A$3, then $\alpha_A:A\to A$4. For quotients, one has

$\alpha_A:A\to A$5

and if $\alpha_A:A\to A$6 is finite-dimensional and $\alpha_A:A\to A$7, then $\alpha_A:A\to A$8 (Asif et al., 3 Aug 2025).

A common misconception is that isoclinism is merely a reformulation of isomorphism. The finite-dimensional theory shows that this is false in general and true only under additional hypotheses. In particular, the equal-dimension condition in the main rigidity theorem is indispensable (Asif et al., 3 Aug 2025).

4. Factor sets and central extension models

A factor set for a regular Hom-Lie Yamaguti algebra $\alpha_A:A\to A$9 is a pair

$[-,-]:A\times A\to A$0

satisfying the identities $[-,-]:A\times A\to A$1–$[-,-]:A\times A\to A$2 together with multiplicativity. These conditions encode $[-,-]:A\times A\to A$3-skew-symmetry, a Hom-Jacobi-type identity, ternary Hom-Jacobi compatibility, binary–ternary compatibility, and higher-order compatibility on the central component (Asif et al., 3 Aug 2025).

Given such a factor set, one defines

$[-,-]:A\times A\to A$4

with operations

$[-,-]:A\times A\to A$5

$[-,-]:A\times A\to A$6

$[-,-]:A\times A\to A$7

Then $[-,-]:A\times A\to A$8 is a regular Hom-Lie Yamaguti algebra, and

$[-,-]:A\times A\to A$9

If $[-,-,-]:A\times A\times A\to A$0 is multiplicative, $[-,-,-]:A\times A\times A\to A$1 is multiplicative regular (Asif et al., 3 Aug 2025).

Every regular Hom-Lie Yamaguti algebra admits such a presentation. Choosing a linear section $[-,-,-]:A\times A\times A\to A$2, one sets

$[-,-,-]:A\times A\times A\to A$3

$[-,-,-]:A\times A\times A\to A$4

and obtains

$[-,-,-]:A\times A\times A\to A$5

Thus regular Hom-Lie Yamaguti algebras can be reconstructed as central extensions of their central quotient by their center (Asif et al., 3 Aug 2025).

Inside an isoclinism family, factor sets organize the passage from one stem algebra to another. If $[-,-,-]:A\times A\times A\to A$6 and $[-,-,-]:A\times A\times A\to A$7 are stem regular Hom-Lie Yamaguti algebras in the same isoclinism family, then there exists a factor set $[-,-,-]:A\times A\times A\to A$8 on $[-,-,-]:A\times A\times A\to A$9 such that

$x,y,z,w,t\in A$0

Moreover, isomorphisms between such central extension models are characterized by induced automorphisms on $x,y,z,w,t\in A$1 and $x,y,z,w,t\in A$2 together with a linear correction term into the center, giving an explicit equivalence criterion for factor sets inside the family (Asif et al., 3 Aug 2025).

5. Stem decomposition and finite-dimensional rigidity

The existence of stem representatives is a basic structural theorem: every isoclinism family of regular Hom-Lie Yamaguti algebras contains at least one stem algebra. In the finite-dimensional case, stem members are exactly the minimal-dimensional members of the family (Asif et al., 3 Aug 2025).

The decomposition theorem sharpens this. If $x,y,z,w,t\in A$3 is an isoclinism family of finite-dimensional regular Hom-Lie Yamaguti algebras and $x,y,z,w,t\in A$4, then

$x,y,z,w,t\in A$5

where $x,y,z,w,t\in A$6 is a stem regular Hom-Lie Yamaguti algebra and $x,y,z,w,t\in A$7 is a finite-dimensional abelian Hom-Lie Yamaguti algebra. The theorem shows that, despite the presence of both a ternary operation and a twisting map, the family still splits into a stem core and an abelian complement (Asif et al., 3 Aug 2025).

This decomposition drives the rigidity theorem. In the stem case, if $x,y,z,w,t\in A$8 and $x,y,z,w,t\in A$9 are finite-dimensional regular stem Hom-Lie Yamaguti algebras, then

$\alpha_A$00

In the general case, if $\alpha_A$01 and $\alpha_A$02 are finite-dimensional regular Hom-Lie Yamaguti algebras with

$\alpha_A$03

then

$\alpha_A$04

The proof proceeds by choosing stem representatives in the common isoclinism family, decomposing each algebra into stem plus abelian summand, using the invariance of $\alpha_A$05 and $\alpha_A$06 under isoclinism to identify the dimensions of the stem parts, and then invoking the stem rigidity theorem (Asif et al., 3 Aug 2025).

The equal-dimension hypothesis is necessary. The paper gives a $\alpha_A$07-dimensional regular Hom-Lie Yamaguti algebra $\alpha_A$08 with $\alpha_A$09 and $\alpha_A$10, and a $\alpha_A$11-dimensional regular Hom-Lie Yamaguti algebra $\alpha_A$12 with $\alpha_A$13 and $\alpha_A$14, such that

$\alpha_A$15

hence $\alpha_A$16, but $\alpha_A$17 because $\alpha_A$18. A second example exhibits a $\alpha_A$19-dimensional stem algebra $\alpha_A$20 and its central enlargement $\alpha_A$21, showing concretely that $\alpha_A$22 while only $\alpha_A$23 is stem (Asif et al., 3 Aug 2025).

6. Relations to Hom-Lie, Hom-Malcev, Hom-Leibniz, and cohomological constructions

Regular Hom-Lie Yamaguti algebras sit at the intersection of several Hom-algebraic theories. Setting $\alpha_A$24 recovers Lie-Yamaguti algebras, so the finite-dimensional equal-dimension rigidity theorem extends the corresponding result for Lie-Yamaguti algebras. Setting the ternary bracket to zero recovers Hom-Lie algebras; setting the binary bracket to zero recovers Hom-Lie triple systems, hence ternary Hom-Nambu-type structures (Asif et al., 3 Aug 2025).

The foundational construction theory has two classical sources. First, every multiplicative Hom-Lie-Yamaguti algebra can be twisted by a compatible endomorphism, and classical Lie-Yamaguti algebras or Malcev algebras produce Hom-Lie-Yamaguti algebras by this Yau-type mechanism. Second, when the ternary product is defined from the binary one by

$\alpha_A$25

a Hom-Lie-Yamaguti algebra becomes a multiplicative Hom-Malcev algebra, and conversely every multiplicative Hom-Malcev algebra carries a natural Hom-Lie-Yamaguti structure with

$\alpha_A$26

These equivalences remain valid in the regular case, although bijectivity of $\alpha_A$27 is not required for the underlying Hom-Malcev correspondence [(Gaparayi et al., 2010); (Gaparayi et al., 2015)].

A distinct source is Hom-Leibniz theory. Every multiplicative left Hom-Leibniz algebra has a natural Hom-Lie-Yamaguti structure obtained from the skew-symmetrized binary bracket

$\alpha_A$28

and the ternary operation

$\alpha_A$29

When the twisting map $\alpha_A$30 is bijective, this yields a regular Hom-Lie-Yamaguti algebra. The construction is the Hom-analogue of the classical passage from left Leibniz algebras to Lie-Yamaguti algebras (Gaparayi et al., 2012).

Representation and cohomology theory for Hom-Lie-Yamaguti algebras is organized by three action maps $\alpha_A$31 and a cohomology whose basic deformation-theoretic component is the $\alpha_A$32-cohomology. A $\alpha_A$33-parameter infinitesimal deformation of a Hom-Lie-Yamaguti algebra is governed by a pair $\alpha_A$34 that simultaneously defines a Hom-Lie-Yamaguti algebra of deformation type and a $\alpha_A$35-cocycle with coefficients in the adjoint representation. Abelian extensions are classified by the cohomology group

$\alpha_A$36

equivalently

$\alpha_A$37

In the regular case, invertibility of $\alpha_A$38 simplifies the transport of identities through powers of $\alpha_A$39 and $\alpha_A$40, but the cohomological formalism itself is not restricted to bijective twists (Zhang, 2015).

Recent operator-theoretic work extends this further. Twisted $\alpha_A$41-operators on a Hom-Lie-Yamaguti algebra, defined relative to a representation and a $\alpha_A$42-cocycle, induce new Hom-Lie-Yamaguti structures on the representation space. Weighted Reynolds operators are special cases of such twisted $\alpha_A$43-operators, and the induced structures are regular when the twisting maps $\alpha_A$44 and $\alpha_A$45 are bijective. The same framework produces Hom-NS-Lie-Yamaguti algebras as underlying structures of twisted $\alpha_A$46-operators (Mabrouk et al., 26 Feb 2025).

The principal limitations of the current structural theory are equally explicit. The isoclinism framework in the 2025 rigidity theorem depends on regularity; without bijectivity of $\alpha_A$47, induced quotient maps may fail to be bijections. Finite-dimensionality is used in the existence and minimality theory of stem algebras, and the equal-dimension assumption is indispensable for the implication “isoclinism $\alpha_A$48 isomorphism.” Infinite-dimensional regular Hom-Lie Yamaguti algebras do not generally satisfy the same rigidity statement. A plausible implication is that future progress will continue to rely on cohomological control of factor sets, classification within individual isoclinism families, and extensions to broader Hom-algebra classes, including Hom-Lie triple systems and superalgebra analogues (Asif et al., 3 Aug 2025).