Lie Algebroids in the Loday-Pirashvili Category

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

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Дата:	2015
Автор:	Rovi, A.
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Опубліковано:	Інститут математики НАН України 2015
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Lie Algebroids in the Loday-Pirashvili Category / A. Rovi // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 24 назв. — англ.

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spelling	irk-123456789-1471462019-02-14T01:25:46Z Lie Algebroids in the Loday-Pirashvili Category Rovi, A. We describe Lie-Rinehart algebras in the tensor category LM of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras in LM to Leibniz algebroids. 2015 Article Lie Algebroids in the Loday-Pirashvili Category / A. Rovi // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17A32; 53D17 DOI:10.3842/SIGMA.2015.079 http://dspace.nbuv.gov.ua/handle/123456789/147146 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We describe Lie-Rinehart algebras in the tensor category LM of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras in LM to Leibniz algebroids.
format	Article
author	Rovi, A.
spellingShingle	Rovi, A. Lie Algebroids in the Loday-Pirashvili Category Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Rovi, A.
author_sort	Rovi, A.
title	Lie Algebroids in the Loday-Pirashvili Category
title_short	Lie Algebroids in the Loday-Pirashvili Category
title_full	Lie Algebroids in the Loday-Pirashvili Category
title_fullStr	Lie Algebroids in the Loday-Pirashvili Category
title_full_unstemmed	Lie Algebroids in the Loday-Pirashvili Category
title_sort	lie algebroids in the loday-pirashvili category
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147146
citation_txt	Lie Algebroids in the Loday-Pirashvili Category / A. Rovi // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 24 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT rovia liealgebroidsinthelodaypirashvilicategory
first_indexed	2025-07-11T01:27:34Z
last_indexed	2025-07-11T01:27:34Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 079, 16 pages Lie Algebroids in the Loday–Pirashvili Category? Ana ROVI †‡ † School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK ‡ School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK E-mail: ana.rovi-garcia@ncl.ac.uk URL: https://www.staff.ncl.ac.uk/ana.rovi-garcia/ Received February 27, 2015, in final form September 28, 2015; Published online October 02, 2015 http://dx.doi.org/10.3842/SIGMA.2015.079 Abstract. We describe Lie–Rinehart algebras in the tensor category LM of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie–Rinehart algebras in LM to Leibniz algebroids. Key words: Lie algebroid; Leibniz algebra; Courant algebroid; Leibniz algebroid 2010 Mathematics Subject Classification: 17A32; 53D17 1 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 → gLie where g is a Leibniz algebra and gLie 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. Throughout, let R be a unital commutative ring and let an unadorned ⊗ denote ⊗R. Theorem 1.1. Let (M g→ A) be a commutative R-algebra object and (N f→ L) be a Lie algebra object in LM. A pair ( (M g→ A), (N f→ L) ) is a Lie–Rinehart algebra object in LM if • (A,L) is a Lie–Rinehart algebra with anchor ρ0 : L→ DerR(A), • the A-bimodule M is a left (A,L)-module with action given by ρ2 : L→ HomR(M,M), ?This paper is a contribution to the Special Issue on Poisson Geometry in Mathematics and Physics. The full collection is available at http://www.emis.de/journals/SIGMA/Poisson2014.html mailto:ana.rovi-garcia@ncl.ac.uk https://www.staff.ncl.ac.uk/ana.rovi-garcia/ http://dx.doi.org/10.3842/SIGMA.2015.079 http://www.emis.de/journals/SIGMA/Poisson2014.html 2 A. Rovi • the right L-module N with action N ⊗L→ N given by n⊗ ξ 7→ [n, ξ] for all n ∈ N , ξ ∈ L is also a left A-module with [−,−] satisfying [a · n, ξ] = a · [n, ξ]− ρ0(ξ)(a) · n, ∀ a ∈ A, • both f and g are L-equivariant and A-linear, and there exist • an A-module map λ : M ⊗A L→ N , • an A-module map ρ1 : N → DerR(A,M) satisfying g(ρ1(n)(a)) = ρ0(f(n))(a), ρ1 ([n, ξ]) = [ρ1(n), (ρ0 + ρ2)(ξ)] for all a ∈ A, m ∈M , n ∈ N , ξ ∈ L. 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. Theorem 1.2. For a Lie–Rinehart algebra ( (M g→ A), (N f→ L) ) in LM, the pair (A,M ⊕N) is a Leibniz algebroid with anchor ρM⊕N := −ρ0 ◦ f : M ⊕N −→ DerR(A) (1.1) and Leibniz bracket on the A-module M ⊕N given by [m1 + n1,m2 + n2]M⊕N := −ρ2(f(n2))(m1) + [n1, f(n2)]. (1.2) for all m1,m2 ∈M and n1, n2 ∈ 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. 2 Leibniz algebroids In this section we first recall the definitions of Leibniz algebras as given by Loday and Pi- rashvili [17, 18]. Secondly, we discuss Leibniz algebroids, see [10] for a differential geometric description, and give some motivating examples. 2.1 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 [x, [y, z]g]g − [[x, y]g, z]g + [[x, z]g, y]g = 0, for all x, y, z ∈ g. (2.1) Lie Algebroids in the Loday–Pirashvili Category 3 Correspondingly, a left Leibniz algebra structure [−,−] on an R-module V is defined by [x, y] := [y, x]g for x, y ∈ V where [−,−]g satisfies (2.1), see [17, 18]. 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 gLie 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 −→ gLie. 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 [x1 ⊗ y1, x2 ⊗ y2]L⊗L = [x1, [x2, y2]L]L ⊗ y1 + x1 ⊗ [y1, [x2, y2]L]L (2.2) for all x1, x2, y1, y2 ∈ 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 7→ ξ(a) for all a ∈ V and ξ ∈ L. The direct sum (of R-modules) V ⊕ L together with the bracket [a+ ξ, b+ ζ]V⊕L := ζ(a)− [ξ, ζ]L, a, b ∈ V, ξ, ζ ∈ L becomes a (right) Leibniz algebra since the identity (2.1) is satisfied [a+ ξ, [b+ ζ, c+ γ]V⊕L]V⊕L− [[a+ ξ, b+ ζ]V⊕L, c+ γ]V⊕L + [[a+ ξ, c+ γ]V⊕L, b+ ζ]V⊕L = [a+ ξ, γ(b)− [ζ, γ]L]V⊕L − [ζ(a)− [ξ, ζ]L, c+ γ]V⊕L + [γ(a)− [ξ, γ]L, b+ ζ]V⊕L = −[ζ, γ]L(a) + [ξ, [ζ, γ]L]L − γ(ζ(a))− [[ξ, ζ]L, γ]L + ζ(γ(a)) + [[ξ, γ]L, ζ]L = 0. � 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 gLie with left action gLie ⊗M → M given by π(g) ⊗m 7→ π(g)(m) for all m ∈ M and g ∈ g. The direct sum (of R-modules) M ⊕ g together with the bracket [m1 + g1,m2 + g2]M⊕g := −π(g2)(m1) + [g1, g2]g, m1,m2 ∈M, g1, g2 ∈ g (2.3) is a Leibniz algebra. Proof. Since π : g→ gLie 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. � 2.2 Leibniz algebroids and related structures While Lie algebras can be generalised to Lie–Rinehart algebras [6, 8, 21] (Lie algebroids [20] in differential geometric context), Leibniz algebras [2, 18] give rise to different algebraic objects: Leibniz algebroids, first defined in a differential geometric context in [10]; Loday algebroids [24]; Courant algebroids [16]; Courant–Dorfman algebras, a term coined by Roytenberg [23] to denote a structure encompassing both Courant algebroids [16] and Dorfman algebras. See [12] for a description of the historic development of these structures. 4 A. Rovi 2.2.1 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. Definition 2.7. Let A be a commutative R-algebra and E be a Leibniz algebra over R with bracket [−,−]E . The pair (A, E) is called a Leibniz algebroid if the Leibniz algebra E has a left A-module structure µ : A⊗E → E given by a⊗e 7→ a ·e for all a ∈ A and e ∈ E , and there exists an A-linear Leibniz algebra antihomomorphism ρE : E → DerR(A), called the anchor, satisfying [a · e1, e2]E = a · [e1, e2]E + ρE(e2)(a) · e1, for e1, e2 ∈ E , a ∈ A. (2.4) Example 2.8. A Lie–Rinehart algebra (A,L), with anchor ρL, is a Leibniz algebroid with anchor −ρL. Proposition 2.9. Let (A,L) be a Lie–Rinehart algebra, with anchor ρL, and let M be a left (A,L)-module with action L ⊗M → M given by ξ ⊗ m 7→ ∇`ξ(m). The pair (A,M ⊕ L) is a Leibniz algebroid with anchor ρM⊕L(m+ ξ) := −ρL(ξ) (2.5) and bracket on the direct sum M⊕L given by the (negative) hemi-semi-direct product [−,−]M⊕L of M by L with action L⊗M →M (or equivalently L→ HomR(M,M)) given by ∇`. Proof. First note that [m1 + ξ,m2]M⊕L = 0 for all m1,m2 ∈ M , ξ ∈ L. Now, since M is an (A,L)-module with action ∇` : L⊗M →M given by m⊗ ξ 7→ ∇`ξ(m), we endow the direct sum M ⊕ L with the Leibniz bracket given in (2.3), that is [m1 + ξ,m2 + ζ]M⊕L = −∇`ζ(m1) + [ξ, ζ]L. We now check that the map in (2.5) is an antihomomorphism ρM⊕L([m1 + ξ,m2 + ζ]M⊕L) = ρM⊕L(−∇`ζ(m1) + [ξ, ζ]L) = −ρL([ξ, ζ]L) = −[ρL(ξ), ρL(ζ)]DerR(A) = [ρL(ζ), ρL(ξ)]DerR(A) = [ρM⊕L(ζ), ρM⊕L(ξ)]DerR(A). Lastly, we check that the compatibility condition beween [−,−]M⊕L and the A-module structure on M ⊕ L given in (2.4) is satisfied [a · (m+ ξ), ζ]M⊕L = −∇`ζ(a ·m) + [a · ξ, ζ]L = −a · ∇`ζ(m)− ρL(ζ)(a) ·m+ a · [ξ, ζ]L − ρL(ζ)(a) · ξ = a · (−∇`ζ(m) + [ξ, ζ]L)− ρL(ζ)(a) · (m+ ξ) = a · [m+ ξ, ζ]M⊕L + ρM⊕L(ζ)(a) · (m+ ξ). � Note that the Leibniz rule for [−,−]M⊕L given by [m+ ξ, a · ζ]M⊕L = −∇`a·ζ(m) + [ξ, a · ζ]L = −a · ∇`ζ(m) + a · [ξ, ζ]L + ρL(ξ)(a) · ζ = a · [m+ ξ, ζ]M⊕L − ρM⊕L(ξ)(a) · ζ 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. Lie Algebroids in the Loday–Pirashvili Category 5 Example 2.10. Let (A,L) be a Lie–Rinehart algebra, with anchor ρL. The pair (A,L ⊗ L) 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 [a · x1 ⊗ y1, x2 ⊗ y2]L⊗L = [a · x1, [x2, y2]L]L ⊗ y1 + a · x1 ⊗ [y1, [x2, y2]L]L = a · [x1, [x2, y2]L]L ⊗ y1 − ρL ([x2, y2]L) (a) · x1 ⊗ y1 + a · x1 ⊗ [y1, [x2, y2]L]L = a · ([x1, [x2, y2]L]L ⊗ y1 + x1 ⊗ [y1, [x2, y2]L]L)− ρL ([x2, y2]L) (a) · x1 ⊗ y1 = a · [x1 ⊗ y1, x2 ⊗ y2]L⊗L − ρL([x2, y2])(a) · x1 ⊗ y1, but the map γ(x2 ⊗ y2) := ρL([x2, y2]L) is not A-linear since [−,−]L is not, hence the pair (A,L⊗ L) does not admit an anchor map induced by ρL. From [19] we know that to each Leibniz algebra g we can canonically associate a Lie alge- bra gLie 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 ELie will not be compatible in general with A. Proposition 2.11. Let (A, E) be a Leibniz algebroid with anchor ρE . If Ker(π) is an A-sub- module of E, then the pair (A, ELie) is a Lie–Rinehart algebra with anchor denoted by ρELie and given by −ρE . Proof. First note that the anchor ρE descends to an R-linear map γELie : ELie → DerR(A) since ρE ([e, e]E) = [ρE(e), ρE(e)]E = 0 for all e ∈ E . Let us assume that π : E → ELie is A-linear. Then we have γELie (a · π(e)) = γELie (π(a · e)) = ρE(a · e) = a · ρE(e) = a · γELie (π(e)), so that γELie : ELie → DerR(A) is A-linear. Since ρE is a antihomomorphism while the anchor of a Lie–Rinehart algebra is a homomorphism, we set ρELie := −γELie so that (A, ELie) is a Lie– Rinehart algebra and the following diagram E ρE // π �� DerA ELie −ρELie ;; 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 ELie in general. Example 2.12. On a manifold M , the bundle E = TM⊕T ∗M has a natural Courant–Dorfman algebra structure with: • a bilinear form given by 〈X + ξ, Y + ζ〉 = ιXζ + ιY ζ, where X,Y ∈ TM , ξ, ζ ∈ T ∗M and ιXξ is the contraction of X with ξ, • a derivation d : C∞(M)→ T ∗X given by the differential of a function, • a right Leibniz bracket given by [X + ξ, Y + ζ] = [X,Y ] − LXζ + LY ξ + d (ιXζ) where [−,−] is the commutator of vector fields and L is the Lie derivative. 6 A. Rovi Note that we have [X + ξ,X + ξ] = d(iXξ) so that the reduced Lie algebra ELie corresponding to E = TM ⊕ T ∗M is ELie = TM ⊕ T ∗M/〈exact forms〉 with Lie bracket given by [X + ξ, Y + ζ] = [X,Y ]− LXζ + LY ξ. In particular, we see that the map π : E → ELie is not C∞(M)-linear. Note also that while [X + ξ, df ] = 0, we have [X + ξ, a · df ] = −〈X, da〉 · df + 〈X, df〉 · da. Note that the reduced Lie algebra ELie is not an C∞(M)-module so that (C∞(M), ELie) is not a Lie–Rinehart algebra. 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 in- troduced 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 ac- knowledges 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. 3.1 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. Definition 3.1. The objects in the category LM are R-module maps (V u→ W ), where u is called the vertical map. The morphisms between objects in LM are pairs of maps h := (h1, h0) such that the following diagram commutes: V h1 // u �� V ′ u′ �� W h0 //W ′ Given two morphisms g := (g1, g0) and h := (h1, h0) in LM, their composition h ◦ g is given by h ◦ g = (h1, h0) ◦ (g1, g0) := (h1 ◦ g1, h0 ◦ g0). (3.1) A morphism φ := (φ1, φ0) is an isomorphism between objects (V u→ W ) and (V ′ u′→ W ′) if and only if φ1 and φ2 are isomorphisms of R-modules. Proposition 3.2. The category LM is monoidal where the tensor product of two objects is( V u→W ) ⊗ ( V ′ u′→W ′ ) := ( V ⊗W ′ ⊕W ⊗ V ′ u⊗1W ′+1W⊗u′−−−−−−−−−−→W ⊗W ′ ) , and the unit object is ({0} 0→ R). Moreover, given two morphisms g := (g1, g0) and h := (h1, h0) in LM, their tensor product g ⊗ h is given by g ⊗ h = (g1, g0)⊗ (h1, h0) := (g1 ⊗ h0 + g0 ⊗ h1, g0 ⊗ h0). (3.2) Lie Algebroids in the Loday–Pirashvili Category 7 Proof. See [19]. � Furthermore, the monoidal category LM is symmetric, with interchange morphism denoted by τ : (V u→ W )⊗ (V ′ u′→ W ′)→ (V ′ u′→ W ′)⊗ (V u→ W ) and given by τ0 : W ⊗W ′ → W ′ ⊗W and τ1 : V ⊗W ′ ⊕W ⊗ V ′ → V ′ ⊗W ⊕W ′ ⊗ V , see [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 (V u→W ) µ′ // ϕ′ �� (V ′ u′→W ′) ϕ �� (M ′ g′→ A′) µ // (M g→ A) corresponds to the commuting “cube” given by V µ′1 // u �� V ′ u′ �� M ′ \|\| ϕ′ 1 µ1 // g′ �� M \|\| ϕ1 g �� W µ′0 //W ′ A′ µ0 // \|\| ϕ′ 0 A \|\| ϕ0 We now describe some of the fundamental algebraic structures in LM. For further details and proofs see [19]. Proposition 3.4. • An associative algebra object (M g→ A) in LM is a triple consisting of an associative R-algebra A, an A-bimodule M and an A-bimodule map g : M → A. Moreover, the algebra object (M g→ A) is commutative, if and only if the A-bimodule M is symmetric and A is commutative. • A Lie algebra object (N f→ L) in LM is equivalent to a Lie algebra L, a right L-modu- le N with right action N ⊗ L → N given by n ⊗ ξ 7→ [n, ξ] for all n ∈ N and ξ ∈ L, and an R-linear L-equivariant map f : N → L, i.e., f([n, ξ]) = [f(n), ξ]L. Example 3.5. We give the following examples of objects in LM: • The surjective map π : E → ELie 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. • Let B be the square-zero extension of an associative algebra A by the A-module M . Then (M ↪→ B) is an algebra object in LM. • Let A be a Poisson algebra with bracket {−,−}, and let Ω1(A) be the A-module of Kähler differentials over A. The pair (A,Ω1(A)) is a Lie–Rinehart algebra with anchor map ρ : Ω1(A)→ Der(A) given by da 7→ {a,−} for all a ∈ A and Lie algebra structure on Ω1(A) given by [da, db]Ω1(A) := d{a, b}. The differential map d : A → Ω1(A) is a Lie algebra object in LM, where A is a right Ω1(A)-module with action A ⊗ Ω1(A) → A given by a⊗ b · dc 7→ b · {a, c} for all a, b, c ∈ A. 8 A. Rovi • 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 7→ j1(a) is a Lie algebra object in LM with right J 1(A)-action on A given by a⊗ b · j1(c) 7→ 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: Proposition 3.6. A left (M g→ A)-module object is a map (V u→W ) of left A-modules such that there exists an A-module map µ1 : W ⊗M → V satisfying g ◦µ1(m⊗w) = µ0(g(m)⊗w) for w ∈W and m ∈M , which descends to an A-module map αV` : M ⊗AW → V (called structure map of the left (M g→ A)-module (V u→W )) satisfying u ◦ αV` (m⊗A w) = µ0 ◦ (g(m)⊗A w). (3.3) Proof. Since (V u→W ) is a left (M g→ A)-module, there exists a morphism µ : ( M g→ A ) ⊗ ( V u→W ) −→ ( V u→W ) , that is, a commuting square M ⊗W ⊕A⊗ V µ1 // g⊗1W +1A⊗u �� V u �� A⊗W µ0 //W (3.4) where µ1 and µ0 satisfy some associativity conditions. From (3.4) we see that µ0 and the restriction of µ1 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 ⊗AW . Moreover, by the associativity of the module action µ, we deduce that µ1 vanishes on m · a ⊗ w −m ⊗ a · w where m ∈ M , a ∈ A, w ∈ W so that the map µ1 : M ⊗W → V descends to a map αV` : M ⊗A W → V yielding the following diagram: M ⊗AW αV ` // g⊗A1W �� V u �� A⊗AW µ0 //W (3.5) The commutativity of (3.5) ensures that the compatibility relation (3.3) is satisfied. � Proposition 3.7. For any Lie algebra object (N f→ L) endowed with an (M g→ A)-module structure, the Leibniz bracket on N given by [n1, n2]N := [n1, f(n2)] satisfies [n1, a · [n2, n2]N ]N = 0, ∀ a ∈ A. (3.6) Proof. By definition, a Lie algebra object (N f→ L) carries a Leibniz algebra structure on the L-module N with bracket given by [n1, n2]N := [n1, f(n2)] so that [n1, n2]N = 0 for all n2 ∈ Ker(f). Now, since (N f→ L) is an (M g→ A)-module, N and L are A-modules and the vertical map f is A-linear, we deduce that Ker(f) is an A-module. Hence [−,−]N satisfies condition (3.6). � Lie Algebroids in the Loday–Pirashvili Category 9 3.2 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. Proposition 3.8. An algebra morphism φ : (M g→ A)→ (M ′ g′→ A′) in LM is given by a pair of maps (φ1, φ0) satisfying φ1(a1m1 +m2a2) = φ0(a1)φ1(m1) + φ1(m2)φ0(a1), (3.7) φ0(a1 · a2) = φ0(a1) · φ0(a2) (3.8) for a1, a2 ∈ A and m1,m2 ∈M . Proof. Assume φ := (φ1, φ0) is an algebra morphism (in LM). Then using (3.1) and (3.2), a straightforward computation shows 0 = φ ◦ µ− µ ◦ (φ⊗ φ) = (φ1 ◦ µ1 − µ1 ◦ (φ1 ⊗ φ0), φ0 ◦ µ0 − µ0 ◦ (φ0 ⊗ φ0)), that is φ1 ◦ µ1 − µ1 ◦ (φ1 ⊗ φ0) = 0, φ0 ◦ µ0 − µ0 ◦ (φ0 ⊗ φ0) = 0 (3.9) which yield the relations in (3.7) and (3.8). � Proposition 3.9. A Lie algebra map a : (N f→ L) → (N ′ f ′→ L′) in LM is given by a pair of maps (a1, a0) satisfying a1([n, ξ]) = [a1(n), a0(ξ)], a0([ξ, ζ]L) = [a0(ξ), a0(ζ)]L′ (3.10) for n ∈ N and ξ, ζ ∈ L. 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). � Proposition 3.10. Given a Lie algebra object (N f→ L), a left (N f→ L)-module (in LM) is an object (V u→W ) such that V and W are left L-modules with actions given by α0 : L⊗W −→W, α2 : L⊗ V −→ V, (3.11) and there exists an R-linear map α1 : N ⊗W −→ V (3.12) satisfying the following compatibility condition α1([n, ξ]⊗ w) = α1(n⊗ α0(ξ ⊗ w))− α2(ξ ⊗ α1(n⊗ w)). (3.13) Moreover, the following compatibility conditions between u, f , α1, α2 and α3 are satisfied u ◦ α1 = α0 ◦ (f ⊗ 1W ), u ◦ α2 = α ◦ (1L ⊗ u). 10 A. Rovi Proof. Since (V u→W ) is a left (N f→ L)-module, we have a morphism α : ( N f→ L ) ⊗ ( V u→W ) −→ ( V u→W ) that is, a pair of maps (α1 +α2, α0) where α0, α1, α2 are given by the maps in (3.11) and (3.12), such that the diagram commutes (N f→ L)⊗ (N f→ L)⊗ (V u→W ) id⊗α−(id⊗α)◦(τ⊗id) // µ⊗id �� ( (N f→ L)⊗ (V u→W ) )⊕2 α �� (N f→ L)⊗ (V u→W ) α // (V u→W ) (3.14) which can be seen as the following diagram in cube shape N ⊗ L⊗W + L⊗N ⊗W + L⊗ L⊗ V β // �� (N ⊗W + L⊗ V )⊕2 f⊗1W +1L⊗u �� N ⊗W + L⊗ V tt µ1⊗1W +µ0⊗1V α1+α2 f⊗1W +1L⊗u �� V tt α1+α2 u �� L⊗ L⊗W id0⊗α0−(id0⊗α0)◦(τ0⊗id0) // (L⊗W )⊕2 L⊗W α0 // tt µ0⊗1W W tt α0 where β := id1 ⊗ α0 − (id0 ⊗ α1) ◦ (τ1 ⊗ id0) + id0 ⊗ α1 − (id1 ⊗ α0) ◦ (τ1 ⊗ id0) + id0 ⊗ α2 − (id0 ⊗ α2) ◦ (τ0 ⊗ id1). Using (3.1) and (3.2), a long but straightforward computation shows that the compatibility relation making (3.14) commute can be expressed as 0 = α ◦ (id⊗ α− (id⊗ α) ◦ (τ⊗ id))− α ◦ (µ⊗ id) = ( α1 ◦ (id1 ⊗ α0)− α2 ◦ (id0 ⊗ α1) ◦ (τ1 ⊗ id0) + α2 ◦ (id0 ⊗ α1) − α1 ◦ (id1 ⊗ α0) ◦ (τ1 ⊗ id0) + α2 ◦ (id0 ⊗ α2)− α2 ◦ (id0 ⊗ α2) ◦ (τ0 ⊗ id1), α0 ◦ (id0 ⊗ α0)− α0 ◦ (id0 ⊗ α0) ◦ (τ0 ⊗ id0) ) − ( α1 ◦ (µ1 ⊗ id0) + α2 ◦ (µ0 ⊗ id1), α0 ◦ (µ0 ⊗ id0) ) = ( α1 ◦ (id1 ⊗ α0)− α2 ◦ (id0 ⊗ α1) ◦ (τ1 ⊗ id0)− α1 ◦ (µ1 ⊗ id0) + ( α2 ◦ (id0 ⊗ α1)− α1 ◦ (id1 ⊗ α0) ◦ (τ1 ⊗ id0)− α1 ◦ (µ1 ⊗ id0) + ( α2 ◦ (id0 ⊗ α2)− α2 ◦ (id0 ⊗ α2) ◦ (τ0 ⊗ id1)− α2 ◦ (µ0 ⊗ id1), α0 ◦ (id0 ⊗ α0)− α0 ◦ (id0 ⊗ α0) ◦ (τ0 ⊗ id0)− α0 ◦ (µ0 ⊗ id0) ) , so that the following diagrams commute • A diagram encoding the L-module action α0 : L⊗W →W L⊗ L⊗W id0⊗α0−(id0⊗α0)◦(τ0⊗id0) // µ0⊗1W �� L⊗W ⊕ L⊗W α0 �� L⊗W α0 //W Lie Algebroids in the Loday–Pirashvili Category 11 • A diagram encoding the L-module action α2 : L⊗ V → V L⊗ L⊗ V id0⊗α2−(id0⊗α2)◦(τ0⊗id1) // µ0⊗1V �� L⊗ V ⊕ L⊗ V α2 �� L⊗ V α2 // V • And lastly N ⊗ L⊗W + L⊗N ⊗W id1⊗α0−(id0⊗α1)◦(τ1⊗id0) +id0⊗α1−(id1⊗α0)◦(τ1⊗id0) // µ1⊗1W �� (N ⊗W ⊕ L⊗ V )⊕2 α1+α2 �� N ⊗W α1 // V encoding the compatibility relation in (3.12). � By the adjoint functor property of tensor products, the maps in (3.11) correspond to α0 : L→ HomR(W,W ), α1 : N → HomR(W,V ), α2 : L→ HomR(V, V ) that we can describe as the following commutative diagram N α1 // f �� HomR(W,V ) g◦h+h◦g �� L α0+α2 // HomR(W,W )⊕HomR(V, V ) (3.15) where h ∈ HomR(W,V ). Proposition 3.11. Let (M g→ A) be a commutative algebra object in LM. A Lie algebra object (N f→ L) is said to act on (M g→ A) by derivations if there exist two Lie algebra maps ρ0 : L→ DerR(A), ρ2 : L→ H := (HomR(M,M), [−,−]H) (3.16) satisfying the compatibility conditions ρ2(ξ)(a ·m) = a · ρ2(ξ)(m) + ρ0(ξ)(a) ·m, g (ρ2(ξ)(m)) = ρ0(ξ)(g(m)) (3.17) and an R-module map ρ1 : N → DerR(A,M) satisfying ρ1([n, ξ]) = [ρ1(n), (ρ0 + ρ2)(ξ)], g(ρ1(n)(a)) = ρ0(f(n))(a) (3.18) for all ξ ∈ L, a ∈ A and m ∈M . 12 A. Rovi Proof. Let a Lie algebra (N f→ L) act on (M g→ A) by derivations, then there exists a left (N f→ L)-module structure on (M g→ A), denoted by ρ : (N f→ L) ⊗ (M g→ A) → (M g→ A), which by Proposition 3.10 endows M and A with left L-actions given by %0 : L ⊗ A → A and %2 : L⊗M →M respectively, and induces a map %1 : N⊗A→M satisfying (3.13). Furthermore, the action of (N f→ L) on (M g→ A) by derivations makes the following commute (N f→ L)⊗ (M g→ A)⊗ (M g→ A) ρ⊗id+(id⊗ρ)◦(τ⊗id) // id⊗µ �� (M g→ A)⊗2 ⊕ (M g→ A)⊗2 µ �� (N f→ L)⊗ (M g→ A) ρ // (M g→ A) Since ρ = (%1 + %2, %0), by (3.1) and (3.2) we find 0 = µ ◦ ( ρ⊗ id + (id⊗ ρ) ◦ (τ⊗ id) ) − ρ ◦ (id⊗ µ) = ( µ1 ◦ (%1 ⊗ id0) + µ1 ◦ (id0 ⊗ %1) ◦ (τ0 ⊗ id1)− ρ1 ◦ (id1 ⊗ µ0) + µ1 ◦ (%2 ⊗ id0) + µ1 ◦ (id1 ⊗ %0) ◦ (τ1 ⊗ id0) + µ1 ◦ (%0 ◦ id1) + µ1 ◦ (id0 ⊗ %2) ◦ (τ1 ⊗ id0) − %2 ◦ (id1 ⊗ µ0), µ0 ◦ (%0 ⊗ id0) + µ0 ◦ (id0 ⊗ %0) ◦ (τ0 ⊗ id0)− %0 ◦ (id0 ⊗ µ0) ) , so the following diagrams commute: • a diagram which encodes the universal action of a Lie algebra L on A by derivations L⊗A⊗A %0⊗id0+(id0⊗%0)◦(τ0⊗id0) // 1L⊗µ0 �� (A⊗A)⊕2 µ0 �� L⊗A %0 // A • a diagram encoding the action of N on A N ⊗A⊗A %1⊗id0+(id1⊗%1)◦(τ0⊗id0) // 1N⊗µ0 �� (M ⊗A⊕A⊗M)⊕2 µ1 �� N ⊗A %1 //M • and lastly, a commutative diagram encoding the action of L on both A and M L⊗M ⊗A⊕ L⊗A⊗M %2⊗id0+(id1⊗%0)◦(τ1⊗id0) +%0⊗id1+(id0⊗%2)◦(τ0⊗id1) // 1L⊗µ1 �� (M ⊗A⊕A⊗M)⊕2 µ1 �� L⊗M %2 //M These maps make the following cube commute N ⊗A⊗A+ L⊗M ⊗A+ L⊗A⊗M β // �� (M ⊗A⊕A⊗M)⊕2 g⊗1A+1M⊗g �� N ⊗A+ L⊗M tt µ1⊗1A+µ0⊗1M %1+%2 f⊗1A+1L⊗g �� M tt µ1 g �� L⊗A⊗A %0⊗id0−(id0⊗%0)◦(τ0⊗id0) // (A⊗A)⊕2 L⊗A %0 //tt 1L⊗µ0 A tt µ0 Lie Algebroids in the Loday–Pirashvili Category 13 By the adjoint functor property of tensor products, the maps %1, %2 and %3 are equivalent to the maps in (3.16). � Proposition 3.12. Recall (M f→ A) is a commutative algebra object and H = HomR(M,M). Let the R-module maps π1 : DerR(A,M)→ DerR(A), π2 : DerR(A,M)→ H be given by π1(∂) := g ◦ ∂ and π2(∂) := ∂ ◦ g respectively. The universal Lie algebra of derivations of the algebra object (M g→ A) is given by DerLM ( (M g→ A) ) = ( DerR(A,M) π1+π2−−−−→ DerR(A)⊕H ) , where the Lie bracket on DerR(A)⊕H is [(α, β), (α′, β′)]DerR(A)⊕H := ( [α, α′]DerR(A),−[β, β′]H ) , and the right DerR(A)⊕H-module structure on DerR(A,M) is given by ∂ ⊗ (α, β) 7−→ [∂, (α, β)] := ∂ ◦ α− β ◦ ∂. (3.19) Proof. We first check that the action [−,−] in (3.19) endows DerR(A,M) with a right Lie algebra module structure over (the Lie algebra) DerR(A)⊕H. On the one hand we have[ ∂, [(α, β), (α′, β′)]DerR(A)⊕H ] = [ ∂, ( [α, α′]DerR(A), [β, β ′]H )] = ∂ ◦ [α, α′]DerR(A) − [β, β′]H ◦ ∂. On the other hand we have[ [∂, (α, β)] , (α′, β′) ] = [(∂ ◦ α− β ◦ ∂) , (α′, β′)] = (∂ ◦ α− β ◦ ∂) ◦ α′ − β′ ◦ (∂ ◦ α− β ◦ ∂) = ∂ ◦ α ◦ α′ − β ◦ ∂ ◦ α′ − β′ ◦ ∂ ◦ α+ β′ ◦ β ◦ ∂, so that[ [∂, (α, β)] , (α′, β′) ] − [[ ∂, (α′, β′) ] , (α, β) ] = ∂ ◦ α ◦ α′ − β ◦ ∂ ◦ α′ − β′ ◦ ∂ ◦ α+ β′ ◦ β ◦ ∂ − ∂ ◦ α′ ◦ α + β′ ◦ ∂ ◦ α+ β ◦ ∂ ◦ α′ − β ◦ β′ ◦ ∂ = ∂ ◦ [α, α′]DerR(A) − [β, β′]H ◦ ∂ = [ ∂, ( [α, α′]DerR(A), [β, β ′]H )] . This shows that the right action of DerR(A) ⊕ H on DerR(A,M) is well defined. Hence, the morphism ρ : (N f→ L) → DerLM ( (M g→ A) ) is given by the following diagram, which follows from (3.15) N ρ1 // f �� DerR(A,M) π1+π2 �� L ρ0+ρ2 // DerR(A)⊕H which commutes. � Remark 3.13. The morphism ρ : (N f→ L) → DerLM ( (M g→ A) ) given by (ρ1, ρ0 + ρ2) is a morphism of Lie algebras in LM. 14 A. Rovi Example 3.14. The universal Lie algebra of derivations of the commutative algebra object (A id→ A) is( DerR(A) −→ DerR(A)⊕DerR(A) ) . Then, the action of a Lie algebra object (N f→ L) by derivations on (A id→ A) is given by • a Lie algebra map ρ0 ≡ ρ2 : L→ DerR(A), • an A-module map ρ1 : N → DerR(A) satisfying ρ1(n) = ρ0(f(n)). 3.3 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. Proof of Theorem 1.1. Assume the pair ( (M g→ A), (N f→ L) ) is a Lie–Rinehart algebra object. Then there exist • a left (M g→ A)-module structure on the Lie algebra object (N f→ L), • an action ρ of the Lie algebra object (N f→ L) on the commutative algebra object (M g→ A) by derivations, and a compatibility condition between these two actions. We now describe what these structures involve. Firstly, by Proposition 3.6, we deduce that a left (M g→ A)-module structure on the Lie algebra object (N f→ L) turns the L-equivariant map f : N → L into an A-module map. Also, it yields an A-module map λ := αN` : M ⊗A L → N . Moreover the Leibniz algebra structure on N given by [−,−]N must satisfy [n1, a · [n2, n2]N ]N = 0 for all a ∈ A and n1, n2 ∈ N . Secondly, by Proposition 3.11, an action ρ of the Lie algebra object (N f→ L) on the commu- tative algebra (M g→ A) by derivations yields • an A-linear Lie algebra map ρ0 : L→ DerR(A), • an A-module map ρ1 : N → DerR(A,M), • an A-linear Lie algebra map ρ2 : L→ HomR(M,M) satisfying conditions (3.17), (3.18), and turns g : M → A into an L-equivariant map. Lastly, the following compatibility conditions between the two module structures are satisfied: • the pair (A,L) is a Lie–Rinehart algebra with anchor ρ0, • the right L-action on the A-module N satisfies [a · n, b · ξ] = a · [n, b · ξ]− b · ρ0(ξ)(a) · n, which provides a compatibility relation between the right L-action on N , given by [−,−] and the A-module structure on N . Note that ρ2 endows M with is a left (A,L)-module structure, see [9] for further details. � Lie Algebroids in the Loday–Pirashvili Category 15 Example 3.15. Let (A,L) be a Lie–Rinehart algebra with anchor ρL : L→ DerR(A). The pair( (A id→ A), (L id→ L) ) is a Lie–Rinehart algebra object in LM with ρ0 ≡ ρ1 ≡ ρ2 = ρL and λ ≡ id. Example 3.16. The pair ( (M g→ A), ( DerR(A,M) π1+π2−−−−→ DerR(A) ⊕ H )) is a Lie–Rinehart algebra object in LM with ρ0 : DerR(A)⊕H → DerR(A), ρ1 : DerR(A,M)→ DerR(A,M), ρ2 : DerR(A)⊕H → H. 4 Leibniz algebroids and Lie–Rinehart algebras in LM In this short section we prove Theorem 1.2. This result provides a functorial relation from Lie–Rinehart algebras to Leibniz algebroids. Proof of Theorem 1.2. First note that N is both a right L-module and a left A-module. Also, recall from [19] that the right L-module N becomes a Leibniz algebra by defining the bracket [n1, n2]N = [n1, f(n2)]. Furthermore, since M is a left (A,L)-module, by Proposition 2.6 we can endow the A-module M ⊕N with a Leibniz algebra structure given by the bracket (1.2). Since f : N → L and ρ0 : L→ DerR(A) are A-linear maps, the map ρM⊕N = −ρ0 ◦ f is also A-linear. We now prove that ρM⊕N is a Leibniz algebra antihomomorphism ρM⊕N ([m1 + n1,m2 + n2]M⊕N ) = ρM⊕N (−ρ2(f(n2))(m1) + [n1, n2]N ) = −ρ0(f([n1, f(n2)])) = −ρ0([f(n1), f(n2)]L) = [ρ0(f(n2)), ρ0(f(n1))]DerR(A) = [ρM⊕N (n2), ρM⊕N (n1)]DerR(A), so the relations in (1.1) hold. We now prove that the Leibniz rule in (2.4) hold [a · (m1 + n1),m2 + n2]M⊕N = −ρ2(f(n2))(a ·m1) + [a · n1, f(n2)] = −a · ρ2(f(n2)) ·m1 − ρ0(f(n2))(a) ·m1 + a · [n1, f(n2)]− ρ0(f(n2))(a) · n1 = a · (−ρ2(f(n2)) ·m1 + [n1, f(n2)]) + ρM⊕N (a) · (m1 + n1) = a · [m1 + n1,m2 + n2]M⊕N + ρM⊕N (m2 + n2)(a) · (m1 + n1). � Example 4.1. Given the Lie–Rinehart algebra( (M g→ A), ( DerR(A,M) π1+π2−−−−→ DerR(A)⊕H )) in LM, the pairs (A,DerR(A,M)) and (A,M ⊕DerR(A,M)) are Leibniz algebroids. Example 4.2. Given a Lie–Rinehart algebra (A,L), the pair(( A id→ A ) , ( L id→ L )) is a Lie–Rinehart algebra object in LM with corresponding Leibniz algebroid given by (A,A⊕L) with structure presented in Proposition 2.9. Acknowledgements The author wishes to thank Uli Krähmer for helpful comments and suggestions. It is also a pleasure to thank Stuart White for his time, José Figueroa O’Farrill for conversations, and the anonymous referees for their comments and suggestions. 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Lie Algebroids in the Loday-Pirashvili Category

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