Factorizations, classifying complements problem and deformation maps for Lie-Yamaguti algebras

Abstract: A Lie-Yamaguti algebra is a non-associative algebraic structure that generalizes both Lie algebras and Lie triple systems. We first consider the factorization problem for Lie-Yamaguti algebras that essentially related to the bicrossed product of Lie-Yamaguti algebras. Next, given an inclusion $\mathfrak{g} \subset E$ of Lie-Yamaguti algebras and a strong $\mathfrak{g}$-complement $\mathfrak{h}$, we describe and classify all $\mathfrak{g}$-complements in $E$. In particular, we show that any other $\mathfrak{g}$-complement in $E$ is isomorphic to $\mathfrak{h}$ by some deformation map $r: \mathfrak{h} \rightarrow \mathfrak{g}$. Despite this importance, it turns out that a deformation map generalizes homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators on Lie-Yamaguti algebras. We define the cohomology of a deformation map unifying the cohomologies of all the operators mentioned above. Finally, we provide a Maurer-Cartan characterization and construct the governing $L_\infty$-algebra of a deformation map $r$ that controls the linear deformations of $r$.

• The paper proves that strong factorization of a Lie-Yamaguti algebra is equivalent to a matched-pair bicrossed product, showing that ordinary factorization fails because of ternary operations.
• The paper characterizes every strong complement as the graph of a deformation map and classifies complements up to isomorphism through equivalence classes of these maps.
• The paper develops cohomological and truncated L∞ frameworks for deformation maps, unifying homomorphisms, derivations, crossed homomorphisms, and relative Rota–Baxter operators.

This paper by Apurba Das extends three interconnected problems from Lie algebra theory—the factorization problem, the classifying complements problem, and the theory of deformation maps—to the setting of Lie-Yamaguti algebras. A Lie-Yamaguti algebra is a vector space equipped with a binary bracket $[~,~]$ and a ternary operation $\llbracket ~, ~, ~ \rrbracket$ satisfying four identities; the structure simultaneously generalizes Lie algebras and Lie triple systems, and arises naturally from torsion and curvature tensors of reductive homogeneous spaces. The central technical difficulty is that the ternary operation obstructs the naive transfer of results known for Lie algebras: the author is forced to introduce a "strong" factorization condition under which the correspondence with matched pairs and bicrossed products holds.

Factorization via matched pairs

The paper recalls the matched pair of Lie-Yamaguti algebras $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$ introduced by Zhao and Qiao, in which each algebra acts on the other through a representation pair (a linear map and a bilinear map into endomorphisms), subject to eighteen compatibility identities plus derived consequences. The bicrossed product $\mathfrak{g} \Join \mathfrak{h}$ carries both operations built from the individual structures and the mutual actions, and contains both factors as subalgebras.

The key structural result is that a Lie-Yamaguti algebra $(E)$ strongly factorizes through two subalgebras $\mathfrak{g}$ and $\mathfrak{h}$—meaning additionally that $$\llbracket \mathfrak{g},\mathfrak{h},\mathfrak{h}\rrbracket_E \subset \mathfrak{g}$$ and $$\llbracket \mathfrak{h},\mathfrak{g},\mathfrak{g}\rrbracket_E \subset \mathfrak{h}$$—if and only if $E$ is isomorphic to the bicrossed product of some matched pair. The strong condition is not merely cosmetic: the paper exhibits explicit counterexamples showing that ordinary factorization does not suffice. Specifically, any bicrossed product of matched pairs of Lie algebras yields, via the construction $\llbracket ~, ~, ~ \rrbracket$0, a Lie-Yamaguti algebra that factorizes through the corresponding factors without those factors forming a matched pair of Lie-Yamaguti algebras; an analogous example arises from Leibniz bicrossed products. This is a genuine divergence from the Lie algebra case, where no such strengthening is required.

The classification of strongly factorizing structures proceeds as in Agore–Militaru: equivalence classes of matched pairs under simultaneous automorphisms of the two factors biject with isomorphism classes of factorizing structures preserving both factors.

Deformation maps and the classifying complements problem

A deformation map in a matched pair is a linear map $\llbracket ~, ~, ~ \rrbracket$1 satisfying two equations—one involving only the binary operations and actions, one involving the ternary operations—that together assert precisely that the graph of $\llbracket ~, ~, ~ \rrbracket$2 is a Lie-Yamaguti subalgebra of $\llbracket ~, ~, ~ \rrbracket$3. The unification claim is substantiated concretely: depending on the choice of matched pair, a deformation map specializes to a homomorphism, a derivation with values in a representation, a crossed homomorphism (differential operator of weight 1), or a relative Rota-Baxter operator of weight 0 or weight 1 on Lie-Yamaguti algebras. The paper notes a limitation of this unification: when the ternary brackets are induced from Lie algebra brackets via $\llbracket ~, ~, ~ \rrbracket$4, the resulting Lie-Yamaguti structures generally fail to form a matched pair, so the deformation map formalism is unavailable in that case.

Two consequences of the definition are established. First, a deformation map induces a new Lie-Yamaguti structure $\llbracket ~, ~, ~ \rrbracket$5 on $\llbracket ~, ~, ~ \rrbracket$6 by "twisting" both operations by $\llbracket ~, ~, ~ \rrbracket$7, and the space $\llbracket ~, ~, ~ \rrbracket$8 becomes a representation of $\llbracket ~, ~, ~ \rrbracket$9 via modified action maps $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$0. Second, and centrally, given an inclusion $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$1 and a strong complement $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$2 (the canonical matched pair), every deformation map produces a $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$3-complement $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$4 in $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$5, and conversely every $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$6-complement arises this way—explicitly as $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$7 where $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$8 is the projection onto $$(\mathfrak{g}, \mathfrak{h}, (\rho,\mu), (\psi,\nu))$$9 along $\mathfrak{g} \Join \mathfrak{h}$0. The proof is constructive and direct.

Classification up to isomorphism follows from an equivalence relation on deformation maps defined through an intertwining isomorphism $\mathfrak{g} \Join \mathfrak{h}$1 of $\mathfrak{g} \Join \mathfrak{h}$2. The main result states that the quotient $\mathfrak{g} \Join \mathfrak{h}$3 is in bijection with isomorphism classes of $\mathfrak{g} \Join \mathfrak{h}$4-complements, so the factorization index satisfies

$\mathfrak{g} \Join \mathfrak{h}$5

All of these statements—factorization, deformation maps, and complement classification—are also carried out verbatim for Lie triple systems (trivial binary bracket), yielding the corresponding matched pairs, bicrossed products, and classification theorem in that special case.

Cohomology and the governing $\mathfrak{g} \Join \mathfrak{h}$6-algebra

Because a deformation map $\mathfrak{g} \Join \mathfrak{h}$7 induces the Lie-Yamaguti algebra $\mathfrak{g} \Join \mathfrak{h}$8 with representation $\mathfrak{g} \Join \mathfrak{h}$9, Yamaguti cohomology immediately provides cochain groups $(E)$0. The paper augments degree zero by $(E)$1, attaching to each bivector $(E)$2 the 1-cochain $(E)$3, and proves this is always a cocycle, so the augmented complex is valid. This cohomology unifies, by specialization, the existing cohomologies of derivations, crossed homomorphisms, and relative Rota-Baxter operators on Lie-Yamaguti algebras.

The Maurer-Cartan characterization uses the graded Lie algebra $(E)$4 associated to any vector space, whose binary operation $(E)$5 encodes insertion of maps into multicochains; a Lie-Yamaguti structure on $(E)$6 is exactly a Maurer-Cartan element here. For a matched pair with bicrossed product structure element $(E)$7, the quadruple $(E)$8 forms a $(E)$9-data in Voronov's sense, with $\mathfrak{g}$0. The resulting $\mathfrak{g}$1-algebra truncates: $\mathfrak{g}$2 for $\mathfrak{g}$3, and the Maurer-Cartan equation $\mathfrak{g}$4 decomposes componentwise into exactly the two defining identities of a deformation map. As corollaries, the paper recovers the Maurer-Cartan characterization of weight-zero Rota-Baxter operators due to Zhao–Qiao and obtains analogous characterizations of the other unified operators.

Finally, twisting by the Maurer-Cartan element $\mathfrak{g}$5 (via Getzler's nilpotent twisting construction) yields the governing $\mathfrak{g}$6-algebra with operations $\mathfrak{g}$7, characterized by the property that a perturbation $\mathfrak{g}$8 makes $\mathfrak{g}$9 again a deformation map if and only if $\mathfrak{h}$0 is a Maurer-Cartan element of $\mathfrak{h}$1. Its differential $\mathfrak{h}$2 reproduces the deformation-map cohomology in degrees $\mathfrak{h}$3.

Limitations and open questions

The strong factorization and strong-complement hypotheses are essential throughout: the classification of arbitrary (non-strong) complements in $\mathfrak{h}$4 remains open, and the counterexamples involving Lie and Leibniz bicrossed products indicate that extending Theorem on factorization beyond the strong condition would require a genuinely different notion than matched pairs. Additionally, the governing $\mathfrak{h}$5-algebra controls linear deformations only; whether it integrates to a formal deformation functor in the standard sense is not addressed. The paper also leaves open the extension-theoretic interpretation of the degree-0 group $\mathfrak{h}$6 augmentation, whose homological meaning is not discussed.

Conclusion

The paper transfers the factorization/complement/deformation-map triad from Lie algebras to Lie-Yamaguti algebras at the cost of a strongness hypothesis tied to the ternary operation, establishes a complete graph-based parametrization and classification of complements up to isomorphism, and supplies both a unifying cohomology theory and a truncated ($\mathfrak{h}$7 for $\mathfrak{h}$8) $\mathfrak{h}$9-algebraic control theory for deformation maps. The results reduce correctly to the Lie triple system case and recover previously known characterizations of Rota-Baxter operators as specializations.