---
title: NS-Lie-Yamaguti Family Algebras
url: https://www.emergentmind.com/topics/ns-lie-yamaguti-family-algebras
type: topic
---

# NS-Lie-Yamaguti Family Algebras

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 [2509.25619]. 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 [2408.10815].

## 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 [1401.1075]. In a parallel notation, the same structure is written as $(T,*,[\, ,\, ,\, ])$ or $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$ in the cited literature [2407.16932].

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 $\nabla R=0$ [2408.10815]. 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 [2309.05508].

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 [2509.25619].

## 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 [2509.25619].

| Operation | Arity and index set | Built-in symmetry |
|---|---|---|
| $\bullet_\alpha$ | $L\otimes L\to L$, $\alpha\in\Omega$ | none stated |
| $\vee_{\alpha,\beta}$ | $L\otimes L\to L$, $(\alpha,\beta)\in\Omega^2$ | $x\vee_{\alpha,\beta}y=-y\vee_{\beta,\alpha}x$ |
| $\{\cdot,\cdot,\cdot\}_{\beta,\gamma}$ | $L^{\otimes 3}\to L$, $(\beta,\gamma)\in\Omega^2$ | none stated |
| $[\cdot,\cdot,\cdot]_{\alpha,\beta,\gamma}$ | $L^{\otimes 3}\to L$, $(\alpha,\beta,\gamma)\in\Omega^3$ | $[x,y,z]_{\alpha,\beta,\gamma}=-[y,x,z]_{\beta,\alpha,\gamma}$ |

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 $\alpha,\beta,\gamma$ [2509.25619]. Rather than treating the four operation families independently, the theory packages them into derived “star” and “double-bracket” expressions:
$$
[x, y]^*_{\alpha, \beta}
= x\bullet_\alpha y - y\bullet_\beta x + x \vee_{\alpha, \beta} y,
$$
$$
\{x, y, z\}^*_{\alpha, \beta}
= \{z, y, x\}_{\beta, \alpha} - \{z, x, y\}_{\alpha, \beta}
+ x\bullet_\alpha (y\bullet_\beta z) - y\bullet_\beta (x\bullet_\alpha z)
- [x, y]^*_{\alpha, \beta} \bullet_{\alpha\beta} z,
$$
$$
[\![x, y, z]\!]_{\alpha, \beta, \gamma}
= \{x, y, z\}^*_{\alpha, \beta}
+ \{x, y, z\}_{\beta, \gamma}
- \{y, x, z\}_{\alpha, \gamma}
+ [x, y, z]_{\alpha, \beta, \gamma}.
$$
These formulas exhibit the intended splitting mechanism: the ordinary skew-symmetric Lie-Yamaguti-type operations are reconstructed from nontrivially interlocking pieces indexed by $\Omega$ [2509.25619].

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, $V\otimes\mathbb K\Omega$ carries a classical NS-Lie-Yamaguti algebra structure induced by the family operations [2509.25619]. 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 $(L,[\cdot,\cdot],\{\cdot,\cdot,\cdot\})$ be a Lie-Yamaguti algebra, let $(V;\rho,\theta)$ be a representation, and let $\Gamma=(\Gamma_1,\Gamma_2)$ be a $(2,3)$-cocycle. A twisted Rota-Baxter family is a family of linear maps $\{T_\alpha:V\to L\}_{\alpha\in\Omega}$ satisfying [2509.25619]
$$
[T_\alpha u, T_\beta v]
=
T_{\alpha\beta}\big(\rho(T_\alpha u)v-\rho(T_\beta v)u+\Gamma_1(T_\alpha u,T_\beta v)\big),
$$
and
$$
\{T_\alpha u, T_\beta v, T_\gamma w\}
=
T_{\alpha\beta\gamma}\Big(
D(T_\alpha u,T_\beta v)w
-\theta(T_\alpha u,T_\gamma w)v
+\theta(T_\beta v,T_\gamma w)u
+\Gamma_2(T_\alpha u,T_\beta v,T_\gamma w)
\Big),
$$
where $D$ 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 $V$ [2509.25619]. The induced operations are explicit:

| Operation on $V$ | Induced formula |
|---|---|
| $u\bullet_\alpha v$ | $\rho(T_\alpha u)v$ |
| $u\vee_{\alpha,\beta}v$ | $\Gamma_1(T_\alpha u,T_\beta v)$ |
| $\{u,v,w\}_{\beta,\gamma}$ | $\theta(T_\beta v,T_\gamma w)u$ |
| $[u,v,w]_{\alpha,\beta,\gamma}$ | $\Gamma_2(T_\alpha u,T_\beta v,T_\gamma w)$ |

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 [2509.25619]. 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 $\Gamma_1=-[\cdot,\cdot]$ and $\Gamma_2=-\{\cdot,\cdot,\cdot\}$, then the induced family operations on $L$ itself are
$$
x \bullet_\alpha y=[T_\alpha x,y],\qquad
x\vee_{\alpha,\beta}y=-[T_\alpha x,T_\beta y],
$$
$$
\{x,y,z\}_{\beta,\gamma}=\{z,T_\beta x,T_\gamma y\},\qquad
[x,y,z]_{\alpha,\beta,\gamma}=-\{T_\alpha x,T_\beta y,T_\gamma z\},
$$
yielding an explicit realization of the family axioms inside an ordinary Lie-Yamaguti algebra endowed with a Reynolds family of endomorphisms [2509.25619].

## 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 $\Omega$-Lie-Yamaguti algebra with coefficients in an appropriate representation [2509.25619]. The cochains are families of multilinear maps indexed by $\Omega$, 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
$$
T^t=T+tT^1+t^2T^2+\cdots.
$$
At first order, the condition for $T^1$ to extend to an actual deformation is exactly that $T^1$ be a $1$-cocycle in the associated complex; equivalence classes of deformations are governed by coboundaries at degree $0$; and vanishing of the first cohomology group $H^1$ yields rigidity, meaning that every deformation is trivial up to equivalence [2509.25619]. 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 $H^2(T,T)\times H^3(T,T)$, while obstructions lie in $H^4(T,T)\times H^5(T,T)$; if the former vanishes the algebra is rigid, and if the latter vanishes every infinitesimal deformation is integrable [1401.1075]. A later deformation cohomology adapted to formal one-parameter deformations introduced a modified complex
$$
C(L;L)\xrightarrow{\delta}C_{2,3}(L;L)\xrightarrow{(\delta^*,\partial)}C_{3,4}(L;L)\times C_{4,5}(L;L),
$$
and proved, in particular, that the free Lie-Yamaguti algebra is rigid [2308.03655]. In another direction, the deformation theory of relative Rota-Baxter operators on Lie-Yamaguti algebras was reformulated via a controlling DGLA and an $L_\infty$-algebra whose Maurer-Cartan elements are precisely the operators of interest [2310.05360]. 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 $\bullet$ family and certain $\{\cdot,\cdot,\cdot\}$-operations are trivial, the structure reduces to a Lie-Yamaguti family algebra; if the $\vee$ family and the higher $[\cdot,\cdot,\cdot]$-family are trivial, it reduces to a pre-Lie-Yamaguti family algebra [2509.25619]. 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
$$
(A,\curlyvee,\diamond,\{\cdot,\cdot,\cdot\},\{\cdot,\cdot,\cdot\}',[\cdot,\cdot,\cdot],\alpha)
$$
with two bilinear operations, two trilinear operations, and a twisting map $\alpha$, satisfying identities listed in equations (6.65)–(6.77) of the corresponding paper [2502.18804]. Its derived operations are
$$
[x,y]^*=x\diamond y-y\diamond x+x\curlyvee y,
$$
$$
\{x,y,z\}^*=\{z,y,x\}-\{z,x,y\}+\alpha(x)\diamond(\alpha(y)\diamond z)-\alpha(y)\diamond(\alpha(x)\diamond z)-[x,y]^*\diamond\alpha(z),
$$
together with a derived trilinear bracket. The paper proves that twisted $\mathcal O$-operators by a $(2,3)$-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 [2502.18804].

The operator-decorated landscape is wider still. Modified Rota-Baxter Lie-Yamaguti algebras introduce a linear map $R$ satisfying
$$
[Rx,Ry]=R([Rx,y]+[x,Ry])-[x,y]
$$
and
$$
\{Rx,Ry,Rz\}
=
R(\{x,Ry,Rz\}+\{Rx,y,Rz\}+\{Rx,Ry,z\}+\{x,y,z\})
-\{Rx,y,z\}-\{x,Ry,z\}-\{x,y,Rz\},
$$
and the resulting cohomology controls both formal deformations and abelian extensions [2401.17726]. In the bialgebraic direction, the classical Lie-Yamaguti Yang-Baxter equations
$$
[r,r]=0,\qquad [r,r,r]=0
$$
characterize relative Rota-Baxter operators for the coadjoint representation, and matched pairs, Manin triples, and double construction Lie-Yamaguti bialgebras are proved equivalent [2210.17066]. 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 $T$ is a Lie-Yamaguti algebra whose standard enveloping Lie algebra $L(T)=T\oplus D(T)$ is semisimple and satisfies $[T,T,T]=T$, then the category of representations of $T$ is equivalent to the category of effective, minimal representations of the reductive triple $(L(T),T,D(T))$ [2407.16932]. The representation of $T$ is a quadruple $(V,\rho,\theta,\delta)$, 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 [2407.16932]. 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
$$
(\mathfrak{so}(N,K),\mathfrak{so}(n,K)),\qquad N=\binom{n+1}{2}-1,
$$
using the trace-zero symmetric matrices $H_n(K)_0$ with products obtained by projection from the bracket of $\mathfrak{so}(N,K)$ [1312.5008]. Their family includes the case $n=4$, where 2-irreducibility occurs, and for $n<5$ they show by computer algebra that, up to degree $6$, all polynomial identities are consequences of the defining Lie-Yamaguti identities [1312.5008]. Another classical model is the 7-dimensional simple non-Lie Malcev algebra, realized as the irreducible $\mathfrak{sl}(2,\mathbb C)$-module $V(6)$ with binary product $[x,y]=\alpha(x\wedge y)$ and ternary product $(x,y,z)=\beta(x\wedge y)\cdot z$; its mixed low-degree polynomial identities were computed explicitly by computer algebra [1108.4202]. 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 [2408.10815]. Lie-Yamaguti algebra bundles generalize this structure fibrewise over manifolds, admit a cohomology with coefficients in a representation bundle, and classify abelian extensions by $H^{2,3}(\xi;\eta)$ [2309.05508]. 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-$n$ dimension is given by the Riordan numbers [2510.03148]. Finally, for abelian extensions of Lie-Yamaguti algebras, the inducibility problem for automorphisms is controlled by the $(2,3)$-cohomology group, and a Wells exact sequence relates derivations, compatible automorphism pairs, and cohomological obstructions [2306.12937]. 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.

Source: https://www.emergentmind.com/topics/ns-lie-yamaguti-family-algebras