Lie Algebroids in the Loday-Pirashvili Category

We describe Lie-Rinehart algebras in the tensor category $\mathcal{LM}$ of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras in $\mathcal{LM}$ to Leibniz algebroids.


Introduction
Leibniz algebras were introduced by Loday and Pirashvili [17,18] as a non skew-symmetric generalisation of Lie algebras. Leibniz algebras appear in differential geometry as the algebraic structure on Courant algebroids [16] and in mathematical physics, for example in Chern-Simons theory [5].
Loday and Pirashvili [18] observed that Leibniz algebras can be described as the canonical map π : g → g Lie where g is a Leibniz algebra and g Lie is the Lie algebra which arises as the quotient of g by the Leibniz ideal generated by elements [x, x] g for x ∈ g. This observation leads to their definition [19] of the monoidal category LM of linear maps and to the construction of a pair of adjoint functors between Lie algebras and Leibniz algebras. Note that the category LM can be seen as the category of truncated chain complexes of length one.
In this paper, we focus on the interplay between Lie-Rinehart algebras [8] (Lie algebroids [20] in a differential geometric context) and Leibniz algebroids [1,10] which are generalisations of Lie algebras and Leibniz algebras respectively. Our goal is not to define a differential geometric counterpart of Loday and Pirashvili's category of linear maps but to use their construction to describe (and understand) some interesting relations between Lie-Rinehart algebras and Leibniz algebras. In this sense, we describe Lie-Rinehart algebras in LM and then construct a functor from Lie-Rinehart algebras in LM to Leibniz algebroids.
Our second result consists in the construction of a functor from the category of Lie-Rinehart algebra objects in LM to the category of Leibniz algebroids. and Leibniz bracket on the A-module M ⊕ N given by for all m 1 , m 2 ∈ M and n 1 , n 2 ∈ N .
Since a very rich class of examples of Hopf algebroids [3,13,22] is the enveloping algebra of Lie-Rinehart algebras, following the argument given by Loday and Pirashvili in [19], we would expect a similar relation between the enveloping algebra of a Leibniz algebroid and Hopf algebroids in LM. However this generalisation of [19] requires not only the definition of a correct notion of enveloping algebra of a Leibniz algebroid (not yet defined in the literature), but also the construction of a functor from the category of Leibniz algebroids to the category of Lie-Rinehart algebras objects in LM, which goes beyond the goals of this paper.

Leibniz algebroids
In this section we first recall the definitions of Leibniz algebras as given by Loday and Pirashvili [17,18]. Secondly, we discuss Leibniz algebroids, see [10] for a differential geometric description, and give some motivating examples.

Leibniz algebras
Leibniz algebras were first defined by Blokh [2], later rediscovered and more intensively studied since [4,18]. For motivation, definitions and basic examples see [17,18]. Definition 2.1. A right Leibniz algebra g is an R-module equipped with a bilinear map, called the right Leibniz bracket and denoted by [−, −] g : g ⊗ g −→ g which satisfies the identity for all x, y, z ∈ g.
Since Loday and Pirashvili use right Leibniz algebra structures in their work [17,18,19], we choose right Leibniz algebras as well, which we call from now on Leibniz algebras. Remark 2.2. For a Leibniz algebra g there exists a corresponding Lie algebra, denoted by g Lie and called the reduced Lie algebra of g, which arises by taking the quotient of g by the Leibniz ideal generated by elements [x, x] g ∈ g for x ∈ g. Hence there exists a surjective map π : g −→ g Lie . Example 2.3 (see [15]). Let L be a Lie algebra over R with bracket [−, −] L . The bracket on the second tensor power of L given by for all x 1 , x 2 , y 1 , y 2 ∈ L endows L ⊗ L with a Leibniz algebra structure.
In the following example we reformulate the construction of the hemi-semi-direct product for left Leibniz algebras introduced by Kinyon and Weinstein in [11,Example 2.2] and endow the direct sum of a Lie algebra L and a (left) L-module V with a (right) Leibniz algebra structure.
Example 2.4. Let L be a Lie algebra over R and V be a L-module with left action L ⊗ V → V given by ξ ⊗ a → ξ(a) for all a ∈ V and ξ ∈ L. The direct sum (of R-modules) V ⊕ L together with the bracket becomes a (right) Leibniz algebra since the identity (2.1) is satisfied Definition 2.5 (see [17]). Let g and g be Leibniz algebras. A map of Leibniz algebras ϕ : g → g is a homomorphism of R-modules satisfying ϕ([x, y] g ) = [ϕ(x), ϕ(y)] g for all x, y ∈ g.
Proposition 2.6. Let g be a Leibniz algebra over R and let M be a left module over its reduced Lie algebra g Lie with left action g Lie ⊗ M → M given by π(g) ⊗ m → π(g)(m) for all m ∈ M and g ∈ g. The direct sum (of R-modules) M ⊕ g together with the bracket is a Leibniz algebra.
Proof . Since π : g → g Lie is a map of Leibniz algebras, a straightforward computation identical to the one carried out in Example 2.4 yields that [−, −] M ⊕g satisfies (2.1) and is hence a Leibniz bracket.

Leibniz algebroids
We propose a definition of Leibniz algebroids in purely algebraic terms, following the definitions given by Rinehart [21] and later by Huebschmann [8] for Lie-Rinehart algebras as an algebraic description of Lie algebroids.
and bracket on the direct sum M ⊕ L given by the We now check that the map in (2.5) is an antihomomorphism Lastly, we check that the compatibility condition beween [−, −] M ⊕L and the A-module structure on M ⊕ L given in (2.4) is satisfied Note that the Leibniz rule for [−, −] M ⊕L given by for all a ∈ A and ξ, ζ ∈ L implies that the Leibniz algebroid (A, M ⊕ L) is local in the sense of [1,Definition 3.4].
In general, the relations between Lie algebras and Leibniz algebras will not induce relations between corresponding Lie-Rinehart algebras and Leibniz algebroids. where L ⊗ L is the Leibniz algebra with bracket given by (2.2) will not be a Leibniz algebroid in general. Since [−, −] L satisfies the Leibniz rule (2.4), we have does not admit an anchor map induced by ρ L .
From [19] we know that to each Leibniz algebra g we can canonically associate a Lie algebra g Lie by taking the quotient of g by the two-sided ideal [x, x] for x ∈ g. This relation does not generalise to a canonical relation between Leibniz algebroids and Lie-Rinehart algebras, so that given a Leibniz algebroid (A, E), the reduced Lie algebra E Lie will not be compatible in general with A.
Proposition 2.11. Let (A, E) be a Leibniz algebroid with anchor ρ E . If Ker(π) is an A-submodule of E, then the pair (A, E Lie ) is a Lie-Rinehart algebra with anchor denoted by ρ E Lie and given by −ρ E .
Proof . First note that the anchor ρ E descends to an R-linear map γ E Lie : Since ρ E is a antihomomorphism while the anchor of a Lie-Rinehart algebra is a homomorphism, we set ρ E Lie := −γ E Lie so that (A, E Lie ) is a Lie-Rinehart algebra and the following diagram commutes.
Note that given a Leibniz algebroid (A, E), the A-module structure on E will not descend to an A-module structure on the reduced Lie algebra E Lie in general. • a bilinear form given by • a derivation d : C ∞ (M ) → T * X given by the differential of a function, • a right Leibniz bracket given by [ is the commutator of vector fields and L is the Lie derivative.
A. Rovi Note that we have [X + ξ, X + ξ] = d(i X ξ) so that the reduced Lie algebra E Lie corresponding In particular, we see that the map π : E → E Lie is not C ∞ (M )-linear. Note also that while [X + ξ, df ] = 0, we have Note that the reduced Lie algebra E Lie is not an 3 Lie-Rinehart algebras in the category LM of linear maps Lie-Rinehart algebras [6,8,21] (Lie algebroids [20] in differential geometric context) were introduced by Herz [6] under the name Lie pseudo-algebra (also known as Lie algebroid [20] in a differential geometric context) and has been developed and studied as a generalisation of Lie algebras. The term Lie-Rinehart algebra was coined by Huebschmann [8], a term which acknowledges Rinehart's fundamental contributions [21] to the understanding of this structure. See [7, Section 1] for some historical remarks on this development. We start this section by giving an overview of the category LM as defined by Loday and Pirashvili [19]. In Section 3.2 we give the necessary tools and background to describe the universal algebra of derivations of an algebra in LM (see Proposition 3.12). Lastly, in Section 3.3, we describe Lie-Rinehart algebras in the category LM of linear maps.

The category LM of linear maps
We first recall some fundamental concepts and definitions about the category LM of linear maps, introduced by Loday and Pirashvili in [19], that are relevant for our main constructions later. We refer to [19] for further details. See [14] for results on Hopf algebras in LM.
Furthermore, the monoidal category LM is symmetric, with interchange morphism denoted [19] for more details.
Commutative diagrams in LM can be seen as commutative "cubes" in the category R-Mod.
Example 3.3. The commutative diagram in LM given by corresponds to the commuting "cube" given by We now describe some of the fundamental algebraic structures in LM. For further details and proofs see [19].  • The surjective map π : E → E Lie is a Lie algebra object in LM.
• Let I be a two-sided ideal in an associative algebra A. The identity map id : I → A is an associative algebra in LM. • Similarly, let A be a Jacobi algebra with bracket {−, −} J , and let J 1 (A) be its 1-jet space. Then the pair (A, J 1 (A)) is a Lie-Rinehart algebra (see [22] for more details), and the map j : A → J 1 (A) given by a → j 1 (a) is a Lie algebra object in LM with right J 1 (A)-action on A given by a ⊗ b · j 1 (c) → b · {a, c} J for all a, b, c ∈ A.
Note that a Lie algebra object (N f → L) in LM is a very similar object to a strict 2-term L ∞ algebra.
We now focus on the description of (M g → A)-modules in LM: where µ 1 and µ 0 satisfy some associativity conditions. From (3.4) we see that µ 0 and the restriction of µ 1 to A ⊗ V turn W and V respectively into left A-modules, so that the vertical map u becomes a map of left A-modules. Now, since M is an A-bimodule and W is a left A-module, we can construct the tensor product M ⊗ A W . Moreover, by the associativity of the module action µ, we deduce that µ 1 vanishes on m · a ⊗ w − m ⊗ a · w where m ∈ M , a ∈ A, w ∈ W so that the map µ 1 : M ⊗ W → V descends to a map α V : M ⊗ A W → V yielding the following diagram: The commutativity of (3.5) ensures that the compatibility relation (3.3) is satisfied.

The Lie algebra of derivations in LM
In this section we describe the Lie algebra of derivations of an associative algebra object in LM. We start by describing morphisms between Lie algebra objects and Lie algebra actions.
for a 1 , a 2 ∈ A and m 1 , m 2 ∈ M .
Proof . An identical argument as in the proof for Proposition 3.8 and abusing notation so that µ is the Lie bracket on (N f → L) we obtain the relation in (3.9) which, in this case, yields the relations in (3.10).

11)
and there exists an R-linear map satisfying the following compatibility condition Moreover, the following compatibility conditions between u, f , α 1 , α 2 and α 3 are satisfied that is, a pair of maps (α 1 + α 2 , α 0 ) where α 0 , α 1 , α 2 are given by the maps in (3.11) and (3.12), such that the diagram commutes which can be seen as the following diagram in cube shape Using (3.1) and (3.2), a long but straightforward computation shows that the compatibility relation making (3.14) commute can be expressed as so that the following diagrams commute encoding the compatibility relation in (3.12).
By the adjoint functor property of tensor products, the maps in (3.11) correspond to that we can describe as the following commutative diagram where h ∈ Hom R (W, V ). satisfying the compatibility conditions ρ 2 (ξ)(a · m) = a · ρ 2 (ξ)(m) + ρ 0 (ξ)(a) · m, g (ρ 2 (ξ)(m)) = ρ 0 (ξ)(g(m)) (3.17) and an R-module map Since ρ = ( 1 + 2 , 0 ), by (3.1) and (3.2) we find so the following diagrams commute: • a diagram which encodes the universal action of a Lie algebra L on A by derivations These maps make the following cube commute By the adjoint functor property of tensor products, the maps 1 , 2 and 3 are equivalent to the maps in (3.16). On the one hand we have On the other hand we have   • an A-module map ρ 1 : N → Der R (A) satisfying ρ 1 (n) = ρ 0 (f (n)).

Lie-Rinehart algebra objects in LM
A Lie-Rinehart algebra [8,21] is an algebraic structure which encompasses a Lie algebra and a commutative algebra which act on each other in a way that both actions are compatible. This object can be described in any symmetric monoidal category.
In this Section we focus on the description of Lie-Rinehart algebra objects in the category LM of linear maps. Based on [19, Lemma 3.6] we give a proof of Theorem 1.1. • the pair (A, L) is a Lie-Rinehart algebra with anchor ρ 0 , • the right L-action on the A-module N satisfies [a · n, b · ξ] = a · [n, b · ξ] − b · ρ 0 (ξ)(a) · n, which provides a compatibility relation between the right L-action on N , given by [−, −] and the A-module structure on N .