NS-Lie-Yamaguti Family Algebras

• NS-Lie-Yamaguti family algebras are semigroup-indexed generalizations that extend classical Lie-Yamaguti binary–ternary structures.
• They are derived from twisted Rota-Baxter families, providing explicit formulas for multiple operations and establishing links with operator theory and deformation cohomology.
• Their associated cohomology and deformation theory offer practical tools for classifying extensions and proving rigidity in diverse geometric and algebraic contexts.

NS-Lie-Yamaguti family algebras are semigroup-indexed generalizations of Lie-Yamaguti-type binary–ternary structures. In the formulation introduced through twisted Rota-Baxter families, a vector space is equipped not with one bilinear product and one ternary product, but with several families of binary and ternary operations indexed by a commutative semigroup $\Omega$; these operations satisfy a coupled system of identities that recovers ordinary NS-Lie-Yamaguti algebras when the index dependence is collapsed (Teng, 30 Sep 2025). The theory is positioned at the intersection of Lie-Yamaguti algebra, operator theory, and cohomological deformation theory, and it is best understood against the broader background of Lie-Yamaguti structures arising in differential geometry, representation theory, and nonassociative operad theory (Stava, 2024).

1. Classical Lie-Yamaguti background

A Lie-Yamaguti algebra is a vector space with a skew-symmetric bilinear product and a trilinear product skew-symmetric in the first two variables, subject to six standard compatibility identities. These identities generalize both Lie algebras and Lie triple systems: the ternary operation may vanish, yielding a Lie algebra, while the binary operation may vanish, yielding a Lie triple system (Lin et al., 2014). In a parallel notation, the same structure is written as $(T,*,[\, ,\, ,\, ])$ or $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$ in the cited literature (Takahashi, 2024).

The geometric origin of the theory is central. Lie-Yamaguti algebras appear on reductive homogeneous spaces when torsion and curvature are interpreted algebraically. In one formulation, for vector fields on a reductive homogeneous space with canonical connection, the binary and ternary operations are

$X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$

so the Lie-Yamaguti identities encode the algebraic content of the Bianchi identities together with $\nabla T=0$ and $\nabla R=0$ (Stava, 2024). In the bundle-theoretic setting motivated by M. Kikkawa’s work on homogeneous Lie loops, the tangent bundle carries fibrewise brackets

$[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$

producing Lie-Yamaguti algebra bundles whose fibres vary smoothly across the base manifold (Goswami et al., 2023).

This classical substrate matters for the family theory because NS-Lie-Yamaguti family algebras do not replace Lie-Yamaguti algebras; they refine them by distributing the operations over an index semigroup. The 2025 family construction is explicitly described as a “family” version of classical algebraic structures, motivated by developments in quantum field theory and noncommutative geometry (Teng, 30 Sep 2025).

2. Defining data of an NS-Lie-Yamaguti family algebra

An NS-Lie-Yamaguti family algebra over a commutative semigroup $\Omega$ consists of a vector space $L$ together with four indexed families of multilinear operations (Teng, 30 Sep 2025).

Operation	Arity and index set	Built-in symmetry
$\bullet_\alpha$	$(T,*,[\, ,\, ,\, ])$0, $(T,*,[\, ,\, ,\, ])$1	none stated
$(T,*,[\, ,\, ,\, ])$2	$(T,*,[\, ,\, ,\, ])$3, $(T,*,[\, ,\, ,\, ])$4	$(T,*,[\, ,\, ,\, ])$5
$(T,*,[\, ,\, ,\, ])$6	$(T,*,[\, ,\, ,\, ])$7, $(T,*,[\, ,\, ,\, ])$8	none stated
$(T,*,[\, ,\, ,\, ])$9	$(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$0, $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$1	$(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$2

The structure is governed by a system of identities recorded in equations (4.16)–(4.27) of the defining paper. These identities are described there as generalized associativity, Jacobi, and compatibility relations mixing the indices $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$3 (Teng, 30 Sep 2025). Rather than treating the four operation families independently, the theory packages them into derived “star” and “double-bracket” expressions:

$(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$4

$(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$5

$(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$6

These formulas exhibit the intended splitting mechanism: the ordinary skew-symmetric Lie-Yamaguti-type operations are reconstructed from nontrivially interlocking pieces indexed by $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$7 (Teng, 30 Sep 2025).

A basic structural fact is that tensoring with the semigroup algebra collapses the family indexing into an ordinary, non-family NS-Lie-Yamaguti algebra. More precisely, $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$8 carries a classical NS-Lie-Yamaguti algebra structure induced by the family operations (Teng, 30 Sep 2025). This provides a bridge from semigroup-indexed data to an untwisted algebraic object.

3. Twisted Rota-Baxter families as the operator-theoretic source

The defining source of NS-Lie-Yamaguti family algebras is the notion of a twisted Rota-Baxter family on a Lie-Yamaguti algebra. Let $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$9 be a Lie-Yamaguti algebra, let $X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$0 be a representation, and let $X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$1 be a $X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$2-cocycle. A twisted Rota-Baxter family is a family of linear maps $X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$3 satisfying (Teng, 30 Sep 2025)

$X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$4

and

$X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$5

where $X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$6 is induced by the representation data.

The crucial structural theorem is that every such twisted Rota-Baxter family induces an NS-Lie-Yamaguti family algebra structure on the source space $X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$7 (Teng, 30 Sep 2025). The induced operations are explicit:

Operation on $X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$8	Induced formula
$X*Y=-T(X,Y),\qquad [\![X,Y,Z]\!]=-R(X,Y)Z,$9	$\nabla T=0$0
$\nabla T=0$1	$\nabla T=0$2
$\nabla T=0$3	$\nabla T=0$4
$\nabla T=0$5	$\nabla T=0$6

This theorem is the conceptual core of the subject: the NS-Lie-Yamaguti family algebra is the underlying structure, or algebraic shadow, carried by the domain of the twisted Rota-Baxter family (Teng, 30 Sep 2025). The construction is not merely formal; it explains why the four operation families occur with exactly their observed index pattern.

A concrete special case is the Reynolds family. If $\nabla T=0$7 and $\nabla T=0$8, then the induced family operations on $\nabla T=0$9 itself are

$\nabla R=0$0

$\nabla R=0$1

yielding an explicit realization of the family axioms inside an ordinary Lie-Yamaguti algebra endowed with a Reynolds family of endomorphisms (Teng, 30 Sep 2025).

4. Cohomology, deformations, and extension theory

The family theory comes equipped with a cohomology that is simultaneously a cohomology of twisted Rota-Baxter families and a cohomology of a certain $\nabla R=0$2-Lie-Yamaguti algebra with coefficients in an appropriate representation (Teng, 30 Sep 2025). The cochains are families of multilinear maps indexed by $\nabla R=0$3, subject to symmetry conditions, and the differentials generalize the classical Yamaguti-type differentials while retaining the semigroup-indexed operator data.

This cohomology controls deformation theory. Formal deformations of twisted Rota-Baxter families are written as

$\nabla R=0$4

At first order, the condition for $\nabla R=0$5 to extend to an actual deformation is exactly that $\nabla R=0$6 be a $\nabla R=0$7-cocycle in the associated complex; equivalence classes of deformations are governed by coboundaries at degree $\nabla R=0$8; and vanishing of the first cohomology group $\nabla R=0$9 yields rigidity, meaning that every deformation is trivial up to equivalence (Teng, 30 Sep 2025). The same paper states that low-degree cohomology classes classify infinitesimal deformations as well as abelian extensions.

These statements belong to a broader Lie-Yamaguti cohomological program. For ordinary Lie-Yamaguti algebras, Yamaguti cohomology was used to show that infinitesimal deformations are classified by $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$0, while obstructions lie in $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$1; if the former vanishes the algebra is rigid, and if the latter vanishes every infinitesimal deformation is integrable (Lin et al., 2014). A later deformation cohomology adapted to formal one-parameter deformations introduced a modified complex

$[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$2

and proved, in particular, that the free Lie-Yamaguti algebra is rigid (Goswami, 2023). In another direction, the deformation theory of relative Rota-Baxter operators on Lie-Yamaguti algebras was reformulated via a controlling DGLA and an $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$3-algebra whose Maurer-Cartan elements are precisely the operators of interest (Zhao et al., 2023). Together, these results supply the ordinary, non-family deformation-theoretic background from which the family cohomology naturally extends.

5. Reductions, variants, and adjacent NS-type structures

The family formalism contains several important reductions. If the family operations are constant, an NS-Lie-Yamaguti family algebra reduces to an ordinary NS-Lie-Yamaguti algebra. If the $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$4 family and certain $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$5-operations are trivial, the structure reduces to a Lie-Yamaguti family algebra; if the $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$6 family and the higher $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$7-family are trivial, it reduces to a pre-Lie-Yamaguti family algebra (Teng, 30 Sep 2025). The subject therefore functions as a unifying binary–ternary splitting framework rather than as a single isolated construction.

A Hom-type analogue has also been developed. A Hom-NS-Lie-Yamaguti algebra is a tuple

$[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$8

with two bilinear operations, two trilinear operations, and a twisting map $[a,b]_m=S_m(a,b),\qquad \{a,b,c\}_m=R_m(a,b)c,$9, satisfying identities listed in equations (6.65)–(6.77) of the corresponding paper (Mabrouk et al., 26 Feb 2025). Its derived operations are

$\Omega$0

$\Omega$1

together with a derived trilinear bracket. The paper proves that twisted $\Omega$2-operators by a $\Omega$3-cocycle induce Hom-NS-Lie-Yamaguti structures on the source space, exactly paralleling the role of twisted Rota-Baxter families in the non-Hom family setting (Mabrouk et al., 26 Feb 2025).

The operator-decorated landscape is wider still. Modified Rota-Baxter Lie-Yamaguti algebras introduce a linear map $\Omega$4 satisfying

$\Omega$5

and

$\Omega$6

and the resulting cohomology controls both formal deformations and abelian extensions (Teng et al., 2024). In the bialgebraic direction, the classical Lie-Yamaguti Yang-Baxter equations

$\Omega$7

characterize relative Rota-Baxter operators for the coadjoint representation, and matched pairs, Manin triples, and double construction Lie-Yamaguti bialgebras are proved equivalent (Zhao et al., 2022). These developments are not themselves family constructions, but they clarify the operator-theoretic environment in which NS-Lie-Yamaguti family algebras sit.

6. Representations, examples, and broader mathematical landscape

Classical representation theory supplies one of the most concrete backdrops for the family setting. If $\Omega$8 is a Lie-Yamaguti algebra whose standard enveloping Lie algebra $\Omega$9 is semisimple and satisfies $L$0, then the category of representations of $L$1 is equivalent to the category of effective, minimal representations of the reductive triple $L$2 (Takahashi, 2024). The representation of $L$3 is a quadruple $L$4, and the paper shows that, under the semisimplicity hypothesis, every representation is tight and comes from a representation of the enveloping Lie algebra with additional decomposition data (Takahashi, 2024). A plausible implication is that semisimple-envelope representation theory offers a natural module-theoretic framework for constructing or analyzing family-indexed operator data.

Concrete Lie-Yamaguti models are plentiful. Benito, Bremner, and Madariaga construct irreducible Lie-Yamaguti algebras from reductive pairs

$L$5

using the trace-zero symmetric matrices $L$6 with products obtained by projection from the bracket of $L$7 (Benito et al., 2013). Their family includes the case $L$8, where 2-irreducibility occurs, and for $L$9 they show by computer algebra that, up to degree $\bullet_\alpha$0, all polynomial identities are consequences of the defining Lie-Yamaguti identities (Benito et al., 2013). Another classical model is the 7-dimensional simple non-Lie Malcev algebra, realized as the irreducible $\bullet_\alpha$1-module $\bullet_\alpha$2 with binary product $\bullet_\alpha$3 and ternary product $\bullet_\alpha$4; its mixed low-degree polynomial identities were computed explicitly by computer algebra (Bremner et al., 2011). These examples do not define NS-Lie-Yamaguti family algebras directly, but they supply canonical Lie-Yamaguti substrates on which family constructions may be imposed.

Several surrounding theories broaden the conceptual frame. The free Lie-Yamaguti algebra has been described explicitly as a quotient of a free multioperator algebra by the Lie-Yamaguti identities, together with a Hall-type basis and the usual universal property (Stava, 2024). Lie-Yamaguti algebra bundles generalize this structure fibrewise over manifolds, admit a cohomology with coefficients in a representation bundle, and classify abelian extensions by $\bullet_\alpha$5 (Goswami et al., 2023). In a different direction, Das’s Yamaguti algebras were introduced as envelopes of Lie-Yamaguti algebras arising in differential geometry, and their nonsymmetric operad was identified with noncrossing partitions without singleton blocks; the arity-$\bullet_\alpha$6 dimension is given by the Riordan numbers (Chapoton et al., 3 Oct 2025). Finally, for abelian extensions of Lie-Yamaguti algebras, the inducibility problem for automorphisms is controlled by the $\bullet_\alpha$7-cohomology group, and a Wells exact sequence relates derivations, compatible automorphism pairs, and cohomological obstructions (Goswami et al., 2023). These results collectively show that the NS-Lie-Yamaguti family algebra belongs to a broad and active algebraic ecosystem linking geometry, operads, representation theory, cohomology, and operator-based deformation mechanisms.