The controlling $L_\infty$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples

In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_\infty$-algebras, and a functor further to that of $L_\infty$-algebras.

theories and further to higher gauge theories (see e.g. [6,7,18,24,51] and references therein for a rich physics literature on this subject, and see [27,28] for a math-friendly introduction on this subject). Recently, this topic has attracted much attention of the mathematical physics world. First of all, a sharp and beautiful observation by Kotov and Strobl in a recent article [24] demonstrates for us a possible mathematical nature behind the various calculations from embedding tensors to their associated tensor hierarchies in the physics literature. An embedding tensor gives rise to a Leibniz algebra, which further gives rise to an L ∞ -algebra, and this corresponds to tensor hierarchies in physics literature. We see later that both procedures are functorial (the first one in Section 2, and the second one in Section 6), moreover the functoriality can be extended with homotopy added in for all objects. In fact, the second procedure is a composition of several very classic results [35,37]. This therefore guarantees us, from a category viewpoint, that the process from embedding tensors to tensor hierarchies, and its corresponding transition from supergravity theories to higher gauge theories, is natural. Then, almost at the same time, appeared several other approaches to encode this process to tensor hierarchies, involving also Leibniz field theory: [50,52] are from the point of view of enhanced Leibniz structures; [29] builds an L ∞ -algebra extension from a Leibniz algebra, which is apparently different from the second functor described above. The functoriality was shown in both procedures.
In our setting, we further conjecture that the above two functors are functorial in an ∞-category sense. We will explore this direction in a future work [45]. Notice that the homotopy nature of L ∞ -algebras suggests homotopy also in the category hosting these objects.
In this article, we provide a rich math tool box for embedding tensors and Lie-Leibniz triples, which seem not yet existing in the mathematical literature. Indeed, as a sort of algebra (or operad), embedding tensors involve not only binary but also unary operations. We develop the theory of controlling algebras, thus further the theory of cohomology and homotopy for embedding tensors and Lie-Leibniz triples.
To establish a good cohomology theory for an object, besides the standard homological algebraic method of projective resolutions, there is also another shorter way through its "controlling algebraic object". Let us explain this idea in the case of a Lie algebra g. We start with a vector space g, then the graded vector space ⊕ +∞ k=0 Hom(∧ k g, g) equipped with the Nijenhuis-Richardson bracket [·, ·] NR becomes a graded Lie algebra (g.l.a.), or a differential graded Lie algebra (d.g.l.a.) with 0 differential [38]. Then a Lie algebra structure on g corresponds exactly to a Maurer-Cartan element π ∈ Hom(∧ 2 g, g). We call this g.l.a. (⊕ +∞ k=0 Hom(∧ k g, g), [·, ·] NR ) the controlling algebra of Lie algebra structures on g. Furthermore, since [π, π] NR = 0, d π := [π, ·] NR satisfies d 2 π = 0, thus d π is a differential. The controlling g.l.a. ⊕ +∞ k=0 Hom(∧ k g, g) together with d π becomes exactly the Chevalley-Eilenberg complex to calculate the cohomology of g with coefficients in its adjoint representation. This is a general phenomenon and works not only for Lie algebras, but also for associative algebras, Leibniz algebras, n-Lie algebras, and pre-Lie algebras. See the review [10,17] for more details. Thus, we use this principal as a guide to develop cohomology theories for embedding tensors (Section 3) and Lie-Leibniz triples (Section 4). Here a Lie-Leibniz triple [28] consists of a Lie algebra representation (g, V), and an embedding tensor T : V → g. The subtle difference between these two concepts shows up, e.g. in deformation theory. To deform an embedding tensor, we fix (g, V) and deform only the operator T , but to deform a Lie-Leibniz triple, we are allowed to deform also the Lie algebra and its representation (g, V) simultaneously. It turns out that the controlling algebraic structure for embedding tensors is a g.l.a. and that for Lie-Leibniz triples is an L ∞ -algebra. Thus indeed the theory of Lie-Leibniz triples is more involved.
We give some immediate applications (also as verifications) of this cohomology theory of Lie-Leibniz triples in Section 5. It does behave as it should: Given a Lie-Leibniz triple, (1) its second cohomology classes in H 2 with coefficients in the adjoint representation correspond exactly to the equivalence classes of its infinitesimal deformations; (2) its second cohomology classes in H 2 with coefficients in the trivial representation correspond exactly to the equivalence classes of central extensions.
Here, we actually need a bit of additional luck in the second application: We need to develop a cohomology theory for Lie-Leibniz triples with arbitrary coefficients, but not just the one from adjoint representation. For this, we find that there is a natural Lie-Leibniz triple structure on the endomorphisms of a 2-term complex of vector spaces. This natural structure comes from the strict Lie 2-algebra structure on them [46], and a strict Lie 2-algebra is a special Lie-Leibniz triple (see Example 2.10 and Example 2.13). Finally, in Section 6, we study how embedding tensors cooperate with homotopy. That is, what a homotopy embedding tensor should be, and how Kotov-Strobl's functor KS behaves with respect to homotopy. Will KS still produces an L ∞ -algebra or something involving even more homotopy? This will test how stable the concept of embedding tensors and the procedure to topological hierarchies are. We still use the tool of controlling algebras to develop the homotopy theory. A standard approach [35,54] to give a homotopy P-algebraic structure is to construct a minimal model P ∞ of the operad P. Along this approach, L ∞ -algebras and A ∞ -algebras are well developed. Moreover, Leibniz ∞ -algebras are defined as the algebras over the minimal model [2,31] of the Leibniz operad. However, apart from this approach, we can also use Maurer-Cartan elements of the aforementioned controlling algebra on a graded vector space to define a homotopy algebraic structure. For example, to define an L ∞ -algebra, one can start with a graded vector space g • and define an L ∞ -algebra to be a Maurer-Cartan element of the g.l.a. (Hom(S(g • ), g • ), [·, ·] NR ). Using this method, we define a homotopy embedding tensor to be a Maurer-Cartan element of a graded version of the controlling algebra that we develop in Section 3. Then we show that a homotopy embedding tensor gives rise to a Leibniz ∞ -algebra, and a Leibniz ∞ -algebra further gives rise to an L ∞ -algebra. We further prove that these two processes are functorial. Thus the functor KS extends to a homotopic version.
We want to emphasis that embedding tensors and Lie-Leibniz triples have been already known in mathematics literature under the name of averaging operators and averaging algebras respectively for a long time. In the last century, many studies on averaging operators were done for various special algebras, such as function spaces, Banach algebras, and the topics and methods were largely analytic [5,9,19,41]. See the well-written introduction in [40] for more details. More recently, people have begun to study averaging operators in double algebras, classical Yang-Baxter equation, conformal algebras, and the procedure of replication in the operad theory [1,15,21,39]. It is not yet clear to us how these aspects of embedding tensors and Lie-Leibniz triples are related. But we wish our article makes some first steps to understand these concepts more deeply. Acknowledgements. Y. Sheng is supported by National Science Foundation of China (11922110). R. Tang is supported by National Science Foundation of China (12001228) and China Postdoctoral Science Foundation (2020M670833). C. Zhu is funded by Deutsche Forschungsgemeinschaft (ZH 274/1-1, ZH 243-3-1, RTG 2491). We thank warmly Florian Naef, Dmitry Roytenberg, Jim Stasheff, and Bruno Vallette for very helpful discussions and suggestions. We also thank ESI, Vienna, for their invitation to present a preliminary version of this work during the Programme on Higher Structures and Field Theory.

Embedding tensors, omni-Lie algebras and Leibniz algebras
In this section, first we establish relations between embedding tensors, omni-Lie algebras and Leibniz algebras. Then we give some interesting examples of embedding tensors. Definition 2.1. A LieRep pair consists of a Lie algebra (g, [·, ·] g ) and a representation ρ : g −→ gl(V) of g on a vector space V.

Definition 2.2.
(i) An embedding tensor on a LieRep pair ((g, [·, ·] g ), (V; ρ)) is a linear map T : V −→ g satisfying the following quadratic constraint: is a LieRep pair and T : V −→ g is an embedding tensor on the LieRep pair (g, V).
consists of a Lie algebra homomorphism φ : g ′ −→ g and a linear map ϕ : In particular, if φ and ϕ are invertible, then (φ, ϕ) is called an isomorphism.
The algebraic structure underlying an embedding tensor is a Leibniz algebra, which is a vector space G together with a bilinear operation [·, ·] G : Then there exists a Leibniz algebra structure [·, ·] T on V given by [u, v] Remark 2.5. This association gives rise to a functor F : ET → Leibniz-Alg from the category of embedding tensors to that of Leibniz algebras. The direction from Leibniz algebras to embedding tensors is less well behaved. It is easy to check that the association in [24] gives rise to a functor G : Leibniz-Alg → ET, and F • G = Id as also noticed therein. But G • F Id, and these two functors do not differ even by a natural transformation. There is another natural association given in [28], namely for a Leibniz algebra (G, [·, ·] G ), the left multiplication L : G → gl(G) given by is an embedding tensor on the Lie algebra gl(G) with respect to the natural representation on the vector space G. Even though with this method, it is more likely to create a natural transformation, it does not give rise to even a functor Leibniz-Alg → ET in the first place.
In the sequel, we give an alternative explanation of Proposition 2.4 using integrable subspaces of omni-Lie algebras. For this purpose, we give an interesting example of embedding tensors. Example 2.6. Let V be a vector space. Then the general linear Lie algebra gl(V) represents on the direct sum gl(V) ⊕ V naturally via: Let P : gl(V) ⊕ V → gl(V) be the projection to the first summand. Then we have for all A, B ∈ gl(V), u, v ∈ V. Thus, P is an embedding tensor on gl(V) with respect to the representation (gl(V) ⊕ V; ρ).
By Proposition 2.4, there is an induced Leibniz algebra structure on gl(V) ⊕ V given by (6) [ The above bracket [·, ·] ol is exactly the omni-Lie bracket on gl(V) ⊕ V introduced by Weinstein in [56]. Recall that an omni-Lie algebra is a triple (gl(V) ⊕ V, [·, ·] ol , (·, ·) + ), where the omni-Lie bracket [·, ·] ol is given by (6), and (·, ·) + is a symmetric V-valued pairing given by Now we are ready to characterize embedding tensors using integrable subspaces of the omni-Lie algebra.

linear map. Then T is an embedding tensor on the general linear Lie algebra gl(V) with respect to the natural representation on V if and only if the graph of T , denoted by G T , is an integrable subspace of the omni-Lie algebra
Proof. For all u, v ∈ V, we have which implies that the graph of T is integrable if and only if [T u, T v] = T ((T u)v), i.e. T is an embedding tensor. Remark 2.9. Since the omni-Lie bracket [·, ·] ol is a Leibniz algebra structure, it follows that an integrable subspace is also a Leibniz algebra. Thus, if T : V → gl(V) is an embedding tensor, then G T is a Leibniz algebra. Since G T and V are isomorphic, so there is an induced Leibniz algebra structure on V. This Leibniz algebra structure on V is exactly the one given in Proposition 2.4. In the rest of this section, we give various interesting examples. Example 2.10 (differential Lie algebras). Let (g, [·, ·] g , d) be a differential Lie algebra, that is a Lie algebra (g, [·, ·] g ) with a derivation d such that d • d = 0. Then we have Thus, d is an embedding tensor on the LieRep pair ((g, [·, ·] g ), (g; ad)). Example 2.11 (an example from supergravity). This example is taken from physics literature [24,42] on supergravity in space-time dimension 4, which is one of the origins where the concept of embedding tensors appear. The vector space V is taken as the fundamental representation 56 = (∧ 2 R 8 ) ⊕ (∧ 2 R 8 ) * , of E 7(7) , the non-compact real form of E 7 . We take g = so (8), the Lie algebra of real skew-symmetric matrices. In fact SO(8) ⊂ E 7 (7) and the naturally induced representation ρ of so(8) on V is simply the sum of a wedge product of the fundamental representation of so (8) and its dual. More precisely, so(8) naturally represents on W := ∧ 2 R 8 viā Letρ * be the dual representation of so(8) on W * . Then ρ =ρ +ρ * is a representation of so(8) on V. Let E i j = R i j − R ji be a basis of g, where R i j denotes the 8 × 8 matrix with the (i, j)-position being 1 and elsewhere being 0. Then we have Let {e 1 , · · · , e 8 } be the basis of R 8 where e i is the vector with the i-th position being 1 and elsewhere being 0. Then {e i ∧e j } i< j forms a basis of W. Let {e * 1 , · · · , e * 8 } be the dual basis. So {e * i ∧e * j } i< j forms a basis of W * .
Define T : V → so(8) by which implies that T is an embedding tensor on the LieRep pair (so (8), V).
The induced Leibniz algebra structure on V is given by [e i ∧ e j + e * p ∧ e * q , e k ∧ e l + e * m ∧ e * n ] T = ρ(T (e i ∧ e j + e * p ∧ e * q ))(e k ∧ e l + e * m ∧ e * n ) =ρ(E i j )(e k ∧ e l ) +ρ * (E i j )(e * m ∧ e * n ) = δ jk e i ∧ e l − δ ik e j ∧ e l + δ jl e k ∧ e i − δ il e k ∧ e j +δ jm e * i ∧ e * n − δ im e * j ∧ e * n + δ jn e * m ∧ e * i − δ in e * m ∧ e * j . Notice that even though the first termρ(E i j )(e k ∧ e l ) = [E i j , E kl ] = −ρ(E k j )(e i ∧ e j ) has antisymmetric property, the second term make the bracket [·, ·] T not antisymmetric. Thus we end up really with a Leibniz algebra, not a Lie algebra. Mathematically, this example can be generalized to all Lie algebra g acts on V = g ⊕ g * via its adjoint and coadjoint representation. That is, the natural projection to the first factor T : V → g is an embedding tensor on g with respect to the action on V = g ⊕ g * . Example 2.12 (endomorphism algebra of a 2-term complex). Given a 2-term complex of vector It is obvious that End(W T → h) with the commutator bracket [·, ·] C is a Lie algebra. Moreover it represents on Hom(h, W) via Then it is straightforward to deduce that (End(W T → h), Hom(h, W), T) is a Lie-Leibniz triple.
This Lie-Leibniz triple plays an important role in the representation theory of Lie-Leibniz triples. See Definition 5.11 for more details. In fact, this embedding tensor comes from a strict 2-Lie algebra structure on the endomorphisms of a 2-term complex described in [46]. This can be generalized to any strict Lie 2-algebra as follows.
As strict Lie 2-algebras are equivalent to crossed modules of Lie algebras, we naturally have the following example.
Example 2.13 and Example 2.14 can be generalized to the following more general case. Example 2.15 (Lie objects in the infinitesimal tensor category of linear maps). Let (g, [·, ·] g ) be a Lie algebra and (V; ρ) a representation. If a linear map T : V → g is g-equivariant, that is, then T is an embedding tensor on the LieRep pair ((g, [·, ·] g ), (V; ρ)). In fact V T → g is a Lie object in the infinitesimal tensor category of linear maps if and only if T : V → g is g-equivariant. Let (G, [·, ·] G ) be a Leibniz algebra and G ann be the two-sided ideal of G generated by [x, x] G for all x ∈ G. Then the natural projection π from G to G/G ann gives a Lie object in the infinitesimal tensor category of linear maps and π is an embedding tensor. See [24,33] for more details.

The controlling graded Lie algebra and cohomology of embedding tensors
In this section first we recall the controlling g.l.a. that characterize Leibniz algebras as Maurer-Cartan elements and the g.l.a. governing a LieRep pair that was originally given in [3]. Then we construct the controlling g.l.a. of embedding tensors. Finally we introduce the cohomologies of embedding tensors.
Let g be a vector space. We consider the graded vector space . It is known that C • (g, g) equipped with the Balavoine bracket [4]: Remark 3.1. In fact, the Balavoine bracket is the commutator of coderivations on the cofree conilpotent coZinbiel coalgebraT(g). See [2,53] for more details. Note that on the same graded vector space C • (g, g) there is the Gerstenhaber bracket [49] which is the commutator of coderivations on the cofree conilpotent coassociative coalgebraT(g).
The following conclusion is straightforward.

Thus, Ω defines a Leibniz algebra structure if and only if [Ω, Ω]
Let g 1 and g 2 be vector spaces and elements in g 1 will be denoted by x, y, z, x i and elements in g 2 will be denoted by u, v, v i . Let c : g ⊗n 2 → g 1 be a linear map. Define a linear mapĉ ∈ C n−1 (g 1 In general, for a given linear map f : We call the linear mapf a horizontal lift of f . . Moreover, let (µ, ρ) ∈ Hom(∧ 2 g, g) ⊕ Hom(g ⊗ V, V) be a Maurer-Cartan element. Then [μ +ρ,μ +ρ] B = 0 implies that µ defines a Lie algebra structure on g and ρ is a representation of the Lie algebra (g, µ) on V.
3.2. The controlling graded Lie algebra of embedding tensors. Let ((g, [·, ·] g ), (V; ρ)) be a LieRep pair. Usually we will also use µ to indicate the Lie bracket [·, ·] g . We have a Leibniz algebra structure µ ⊞ ρ on g ⊕ V, which is given by This Leibniz algebra is called the hemisemidirect product of the Lie algebra (g, [·, ·] g ) and the representation (V; ρ). It first appeared in [20].
is a graded Lie algebra, where the graded Lie bracket ·, · is given in a derived fashion More precisely, it is given by

Moreover, its Maurer-Cartan elements are precisely embedding tensors.
Proof. In short, the graded Lie algebra (⊕ +∞ k=1 Hom(⊗ k V, g), ·, · ) is obtained via the derived bracket [22,23,55]. In fact, the Balavoine bracket [·, ·] B associated to the direct sum vector space g ⊕ V gives rise to a graded Lie algebra ( ). Since µ : ∧ 2 g −→ g is a Lie algebra structure and ρ : g ⊗ V −→ V is a representation of g on V, therefore the hemisemidirect product Leibniz algebra structure µ ⊞ ρ is a Maurer-Cartan element of the graded Lie algebra is abelian under [·, ·] B by degree reasons, the differential d µ⊞ρ gives rise to a graded Lie algebra structure on the graded vector space ⊕ +∞ k=1 Hom(⊗ k V, g) via the derived bracket (10). For T ∈ Hom(V, g), we have which implies that Maurer-Cartan elements are precisely embedding tensors.
The regular representation (G; L, R) is very important. We denote the corresponding cochain complex by (C • G, G), ∂ reg ) and the resulting cohomology by H * reg (G). Let T : V −→ g be an embedding tensor on a LieRep pair ((g, [·, ·] g ), (V; ρ)). By Proposition Proof. It follows from direct verification.
be the corresponding Loday-Pirashvili coboundary operator of the Leibniz algebra (V, [·, ·] T ) with coefficients in (g; ρ L , ρ R ). More precisely, ∂ T : The coboundary operator ∂ T can be alternatively described by the following formula. Proposition 3.9. Let T : V → g be an embedding tensor. Then we have where the bracket ·, · is given by (10).
Proof. It follows from straightforward verification. Now we define a cohomology theory governing deformations of an embedding tensor T : V → g. Define the space of 0-cochains C 0 (T ) to be 0 and of 1-cochains C 1 (T ) to be g. For n ≥ 2, define the space of n-cochains C n (T ) as C n (T ) = Hom(⊗ n−1 V, g). Definition 3.10. Let T be an embedding tensor on a LieRep pair ((g, [·, ·] g ), (V; ρ)). We define the cohomology of the embedding tensor T to be the cohomology of the cochain complex (C At the end of this section, we study the relation between the cohomology of an embedding tensor T : V −→ g and the cohomology of the underlying Leibniz algebra (V, [·, ·] T ) given in Proposition 2.4. Theorem 3.11. Let T : V −→ g be an embedding tensor on a LieRep pair ((g, [·, ·] g ), (V; ρ)). Then Φ, defined by (11), is a homomorphism from the cochain complex (C • (T ), ∂ T ) of the embedding tensor T to the cochain complex (C • (V, V), ∂ reg ) of the underlying Leibniz algebra (V, [·, ·] T ), that is, we have the following commutative diagram: Consequently, Φ induces a homomorphism Φ * : H k (T ) −→ H k reg (V, V) between the corresponding cohomology groups.
Proof. By Proposition 3.5 and Proposition 3.9, for all θ ∈ Hom(⊗ k V, g), we have Thus, Φ is a homomorphism of cochain complexes from (C * (T ), ∂ T ) to (C * (V, V), ∂ reg ) and Φ * is a homomorphism between the corresponding cohomology groups. 4. The controlling L ∞ -algebra of Lie-Leibniz triples In this section, we apply T. Voronov's higher derived bracket to construct an L ∞ -algebra that characterizes Lie-Leibniz triples as Maurer-Cartan elements.
. Moreover, we denote the reduced tensor algebra and reduced symmetric algebra byTV The desuspension operator s −1 changes the grading of V • according to the rule (s The notion of an L ∞ -algebra was introduced by Stasheff in [48]. See [25,26] for more details. Definition 4.1. An L ∞ -algebra is a Z-graded vector space g • = ⊕ k∈Z g k equipped with a collection (k ≥ 1) of linear maps l k : ⊗ k g • → g • of degree 1 with the property that, for any homogeneous elements x 1 , · · · , x n ∈ g • , we have (i) (graded symmetry) for every σ ∈ S n , l n (x σ(1) , · · · , x σ(n) ) = ε(σ; x 1 , · · · , x n )l n (x 1 , · · · , x n ).
(ii) (generalized Jacobi identity) for all n ≥ 1, Remark 4.2. An L ∞ -algebra structure on a graded vector space g • is equivalent to a codifferential on the cofree conilpotent cocommutative coalgebraS(g • ).
Here a, a 1 , · · · , a k are homogeneous elements of h and x, y are homogeneous elements of L. All other L ∞ -algebra products that are not obtained from the ones written above by permutations of arguments, will vanish. Let L ′ be a graded Lie subalgebra of L that satisfies [∆, L ∞ -algebras were constructed using the above method to study simultaneous deformations of morphisms between Lie algebras in [13,14], and to study simultaneous deformations of relative Rota-Baxter Lie algebras in [30].

4.2.
The controlling L ∞ -algebra of Lie-Leibniz triples. Let g and V be two vector spaces. Then we have a graded Lie algebra (⊕ +∞ n=0 C n (g ⊕ V, g ⊕ V), [·, ·] B ). This graded Lie algebra gives rise to V-data therefore also to an L ∞ -algebra.
Proof. Note that h = ⊕ +∞ n=0 Hom(⊗ n+1 V, g) is an abelian subalgebra of (L, [·, ·]). Since P is the projection onto h, it is obvious that P • P = P. It is also straightforward to see that the kernel of P is a graded Lie subalgebra of (L, [·, ·]). Thus (L, h, P, ∆ = 0) are V-data.
The other conclusions follows immediately from Theorem 4.6.
By Theorem 3.3, we obtain that is a graded Lie subalgebra of ⊕ +∞ n=0 C n (g ⊕ V, g ⊕ V), [·, ·] B . Then we have the following result. Proposition 4.8. With above notations, is an L ∞ -algebra, where l i are given by for homogeneous elements θ 1 , · · · , θ k−1 ∈ h, homogeneous elements Q, Q ′ ∈ L ′ , and all the other possible combinations vanish.
Proof. It follows from Theorem 4.6 and Proposition 4.7. Now we are ready to give the controlling L ∞ -algebra of Lie-Leibniz triples, which is the main result in this subsection. Proof. Let (s −1 (µ ⊞ ρ), T ) be a Maurer-Cartan element of (s Thus, we obtain By Theorem 3.3 and Theorem 3.4, (g, µ) is a Lie algebra, (V; ρ) is its representation and T is an embedding tensor on the LieRep pair ((g, µ), (V; ρ)).

Cohomology of Lie-Leibniz triples and applications
In this section, we introduce a cohomology theory of Lie-Leibniz triples and justify it by using it to classify infinitesimal deformations and central extensions.

Regular cohomology of Lie-Leibniz triples and infinitesimal deformations.
In this subsection, first we recall the cohomology of a LieRep pair, and then we introduce a regular cohomology of a Lie-Leibniz triple. Finally, we use the second cohomology group to characterize infinitesimal deformations of a Lie-Leibniz triple.
Definition 5.1. The cohomology of the cochain complex (⊕ n C n (g, ρ), δ) is called the cohomology of the LieRep pair.
Now we give the precise formula of the coboundary operator δ.
where d CE : Hom(∧ n g, g) → Hom(∧ n+1 g, g) is the Chevalley-Eilenberg coboundary operator of the Lie algebra (g, [·, ·] g ), and (δ f ) V is given by for all x 1 , · · · , x n ∈ g and v ∈ V. Now we are ready to define the cohomology of a Lie-Leibniz triple. Let ((g, µ), (V; ρ), T ) be a Lie-Leibniz triple, i.e. ρ : g → gl(V) is a representation of the Lie algebra (g, µ) and T : V → g is an embedding tensor. Define the set of 0-cochains C 0 (g, ρ, T ) to be 0. For n ≥ 1, define the space of n-cochains C n (g, ρ, T ) by Define the coboundary operator D : C n (g, ρ, T ) → C n+1 (g, ρ, T ) by where δ and ∂ T are given by (19) and (13), and Ω T : C n (g, ρ) → C n+1 (T ) is defined by The precise formula for Ω T is given as follows.
Proof. By Remark 3.1, it is convenient to view the elements of ⊕ +∞ n=0 C n (g ⊕ V; g ⊕ V) as coderivations ofT(g ⊕ V). The coderivations corresponding to f and T will be denoted byf andT respectively. Then, by induction, we have which implies that (23) holds.
By (20), we deduce that is a cochain complex. About the relation between the operator δ, ∂ T and Ω T , we have Proof. For all ( f, θ) ∈ C n (g, ρ, T ), by the fact δ which implies that Ω T • δ + ∂ T • Ω T = 0.
Definition 5.5. The cohomology of the cochain complex (⊕ +∞ n=0 C n (g, ρ, T ), D) is called the regular cohomology of the Lie-Leibniz triple ((g, µ), (V; ρ), T ). We denote its n-th cohomology group by H n reg (g, ρ, T ) The formula of the coboundary operator D can be well-explained by the following diagram: Theorem 5.6. Let ((g, µ), (V; ρ), T ) be a Lie-Leibniz triple. Then there is a short exact sequence of cochain complexes: , ρ), δ) −→ 0, where ι and p are the inclusion map and the projection map.
Consequently, there is a long exact sequence of the cohomology groups: where the connecting map c n is defined by c Proof. By (21), we have the short exact sequence of cochain complexes which induces a long exact sequence of cohomology groups. Also by (21), c n is given by At the end of this section, we study infinitesimal deformations of Lie-Leibniz triples. Let for ω ∈ Hom(∧ 2 g, g), ̺ ∈ Hom(g ⊗ V, V) and T : V → g, we say that (ω, ̺, T ) generates an infinitesimal deformation of the Lie-Leibniz triple ((g, µ), (V; ρ), T ).
In the sequel, we define equivalences between infinitesimal deformations of a Lie-Leibniz triple and show that infinitesimal deformations of a Lie-Leibniz triple are classified by its second cohomology group. Definition 5.9. Two infinitesimal deformations of a Lie-Leibniz triple ((g, [·, ·] g ), (V; ρ), T ) generated by (ω, ̺, T ) and (ω ′ , ̺ ′ , T ′ ) are said to be equivalent if there exist N ∈ gl(g), S ∈ gl(V) and x ∈ g such that (Id g + tN + tad x , Since Id g +tN+tad x is a Lie algebra morphism from (g By the equality ( By the equality (

Cohomology with arbitrary coefficients and central extensions.
In this subsection, we introduce the cohomology of a Lie-Leibniz triple with coefficients in an arbitrary representation and classify central extensions of a Lie-Leibniz triple using the second cohomology group.
With the help of the Lie-Leibniz triple given in Example 2.12, we give the notion of a representation of a Lie-Leibniz triple as follows.
Usually we will denote a representation by (W for any x ∈ g, we will always write φ(x) = (φ h (x), φ W (x)) for φ h (x) ∈ gl(h) and φ W (x) ∈ gl(W). The following result follows from straightforward verifications.  (33) and the representation ρ + φ W + ϕ of the Lie algebra (g ⊕ h, [·, ·] φ h ) on V ⊕ W is given by

Homotopy embedding tensors and higher structures
In this section, we define homotopy embedding tensors and establish various relations between homotopy embedding tensors, Leibniz ∞ -algebras, A ∞ -algebras and L ∞ -algebras. In particular, we construct L ∞ -algebras from homotopy embedding tensors via Leibniz ∞ -algebras, which generalizes the construction of L ∞ -algebras from embedding tensors given by Kotov and Strobl in [24]. A hidden aim of us to build up homotopy theory for embedding tensors is to try to find possible equivalence between them to provide equivalence of the corresponding physical models. To build up weak equivalence between homotopy embedding tensors is our next aim which we postpone to the future. If the physical model is topological, then weak equivalence should definitely provide a suitable equivalence. But quite possibly, weak equivalence might not provide non-trivial equivalence between embedding tensors themselves, just like, weak equivalences for Lie algebras viewed as L ∞ -algebras are simply isomorphisms.
Definition 6.1. A Leibniz ∞ -algebra is a Z-graded vector space G • = ⊕ k∈Z G k equipped with a collection (k ≥ 1) of linear maps θ k : ⊗ k G • → G • of degree 1 such that +∞ k=1 θ k is a Maurer-Cartan element of the graded Lie algebra (C • (G • , G • ), [·, ·] B ). More precisely, for any homogeneous elements x 1 , · · · , x n ∈ G • , the following equality holds: It is obvious that an L ∞ -algebra is naturally a Leibniz ∞ -algebra.
Proposition-definition 6.3. Let g • and V • be graded vector spaces. Let l k : S k (g • ) → g • and ρ k : . A representation of an L ∞ -algebra will give rise to V-data.
. Then the following quadruple form V-data: • the graded Lie algebra (L, [·, ·]) is given by • the abelian graded Lie subalgebra h is given by h := ⊕ n∈Z Hom n (T(V • ), g • ); • P : L → L is the projection onto the subspace h; is an L ∞ -algebra, where l k is given by (16). Proof. It is obvious that ImP = h is an abelian graded Lie subalgebra of the g.l.a. (C • (g • ⊕ V • , g • ⊕ V • ), [·, ·] B ). Moreover, ker P is also a graded Lie subalgebra. Since ∆ = +∞ k=1 (l k ⊞ ρ k ) is the hemisemidirect product Leibniz ∞ -algebra structure on g • ⊕ V • , we have [∆, ∆] B = 0 and P(∆) = 0. Thus (L, h, P, ∆) are V-data. Hence by Theorem 4.6, we obtain the higher derived brackets {l k } +∞ k=1 on the abelian graded Lie subalgebra h. Now we are ready to define a homotopy embedding tensor, which is the main object in this section. A homotopy embedding tensor on an L ∞ -algebra is a generalization of an embedding tensor on a Lie algebra.
(iii) the association in (ii) gives rise to a functor S from the category of homotopy embedding tensors to that of Leibniz ∞ -algebras.
which implies that e [·,Θ] B +∞ k=1 (l k ⊞ ρ k ) is a Maurer-Cartan element of the graded Lie algebra (ii) By (38), It is a filtered L ∞ -algebra [11]. The condition of being filtered ensures convergence of the series figuring in the definition of Maurer-Cartan elements.
6.3. Leibniz ∞ -algebras and L ∞ -algebras. There is a procedure to associate an L ∞ -algebra to a Leibniz algebra [24]. In this section, we extend this construction to a functor from the category of Leibniz ∞ -algebras to that of L ∞ -algebras. Thus we arrive at a functor from the category of homotopy embedding tensors to that of L ∞ -algebras.
Let V • be a Z-graded vector space. Then the tensor algebra (T(V • ), ⊗) is a free graded unital associative algebra. The freeness implies the uniqueness of the graded unital algebra morphism More precisely, △ cosh is given by It is immediate to check that it is coassociative and counital. Hence (T(V • ), ⊗, △ cosh ) is a graded bialgebra. We call (T(V • ), ⊗, △ cosh ) the (graded) coshuffle bialgebra.
Let Lie(V • ) be the free graded Lie algebra generated by the graded vector space V • . In fact, Lie(V • ) is the intersection of all the Lie subalgebras of the (graded) commutator Lie algebra T(V • ) Lie = (T(V • ), [·, ·] C ) containing V • . Note that the space of primitive elements of (T(V • ), ⊗, △ cosh ) is Lie(V • ). Thus the grading on T(V • ) induces a natural grading on Lie(V • ) making it a graded Lie algebra. We have an embedding V • ⊂ Lie(V • ) of graded vector spaces. This induces i : T(V • ) → T(Lie(V • )) a grading preserving inclusion of graded coshuffle bialgebras. Recall that the universal enveloping algebra U(Lie(V • )) = T(Lie(V • ))/I U , where the two sided ideal I U is generated by a ⊗ b − b ⊗ a − [a, b]. Moreover, we have △ cosh (I U ) ⊂ I U ⊗ T(Lie(V • )) + T(Lie(V • )) ⊗ I U .
Proof. Since Φ is a homomorphism of graded bialgebras, we only need to prove that Φ is an isomorphism. Let A be a unital associative algebra, A Lie the commutator Lie algebra of A and f : Lie(V • ) → A Lie a Lie algebra homomorphism. Since T(V • ) is a free graded unital associative algebra, we have a unique associative algebra homomorphismf : Lie is the inclusion of graded Lie algebras. Thus T(V • ) satisfies the universal property of the universal enveloping algebra of the free graded Lie algebra Lie(V • ). Set A = U(Lie(V • )), by the universal property, we deduce that p • i : T(V • ) → U(Lie(V • )) is an isomorphism of graded associative algebras. Thus, Φ is an isomorphism of bialgebras.
Remark 6.18. This generalizes Kotov and Strobl's construction of an L ∞ -algebra from a Leibniz algebra [24]. In particular, by truncation, one can obtain a Lie 2-algebra, which was applied to the supergravity theory. See also [44] for a direct construction of a Lie 2-algebra form a Leibniz algebra.
We denote the category of Leibniz ∞ -algebras and the category of L ∞ -algebras by Leibniz ∞ -Alg and Lie ∞ -Alg respectively. We show that the above construction is actually a functor.