Lie theory for symmetric Leibniz algebras

Lie algebras and groups equipped with a multiplication μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} satisfying some compatibility properties are studied. These structures are called symmetric Lie μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-algebras and symmetric μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-groups respectively. An equivalence of categories between symmetric Lie μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-groups and finite dimensional symmetric Leibniz algebras.

Reading these notes after 24 years, we think they still are of some interest. The present paper essentially consists of those notes of ours, with the addition of present Sect. 6.
The aim of this work is to extend the classical correspondence between Lie groups and Lie algebras to a bit wider class, namely between so called Lie μ-groups and symmetric Leibniz algebras.
In Sect. 2 we introduce symmetric Leibniz algebras and state some of their properties that we will need.
In Sect. 3 we define the closely related symmetric Lie μ-algebras, and in Sect. 4 we prove that upon inverting 2 they become equivalent to symmetric Leibniz algebras (Theorem 11).
In Sect. 5, group side counterparts of these structures enter, under the name of symmetric μ-groups ( μ-groups for short). We exhibit there some of their properties parallel to those for symmetric Leibniz algebras.
In Sect. 6 we consider the straightforward analogs of Lie groups for μ-groups and prove the analog of the classical equivalence between Lie algebras and simply connected Lie groups-Corollary 19 describes an equivalence between the category of finite dimensional symmetric Leibniz algebras and simply connected Lie μ-groups.
Section 7 recasts our results in terms of the formalism of [1]. We describe the theory of symmetric Leibniz algebras as a linear extension of the theory of Lie algebras. We also describe theories of symmetric Leibniz algebras and symmetric μ-groups as pullbacks of diagrams of simpler theories.
Let us finish the introduction with acknowledging very useful advice of the referee which helped to improve the paper a lot.

Symmetric Leibniz algebras
We fix a commutative ring K. All modules and tensor products are taken over K. The category of modules is denoted by MOD and the category of Lie algebras by LIE. We identify modules with abelian Lie algebras, so we have the inclusion MOD ⊂ LIE.
Leibniz algebras are a generalization of Lie algebras. Importance of these algebras was realized by Jean-Louis Loday, see [3,Section 10.6]. Recall that a (right) Leibniz algebra is a module L, equipped with an operation [−, −] : L ⊗ L → L such that the right Leibniz identity holds: We refer the reader to [5][6][7]10] which are direct consequences of the right Leibniz identity (1). An exact sequence of Leibniz algebras and Leibniz algebra homomorphisms is called abelian if L 1 is an abelian Leibniz algebra.
hold for all x ∈ L and a ∈ L 1 . Let L be a Leibniz algebra. The submodule of L generated by elements of the form [x, x], x ∈ L is denoted by L ann . It is a two-sided ideal of L [6]. The quotient L/L ann is a Lie algebra, denoted by L Lie and called the Liezation of L. The functor L → L Lie is the left adjoint to the inclusion LIE ⊂ LB [6]. The inclusion MOD ⊂ LB also has a left adjoint functor, which is given by L → L ab := L/ [L, L]. The module L ab is known as the abelization of L. Proof Take x = y in the equality (3) to obtain i). The identity ii) is a formal consequence of i). Relation iii) can be obtained by addition of the identities (1) and (3). To show iv) one uses iii) and then the equality (1):

Lemma 1 Let L be a Leibniz algebra satisfying the left Leibniz identity:
Relation v) can be deduced from iv), if one puts y = [a, b], z = c, x = d and then uses the equality (1): Let L be a symmetric Leibniz algebra. Then by Lemma 1 we have Thus one has a central extension of Leibniz algebras 0 → L ann → L → L Lie → 0.
It follows from the definition that x · y is bilinear and symmetric on x and y. If V is a free module, then 2 V fits in the short exact sequence where i(x [2] Recall also that if 2 is invertible in K, then the map V ⊗2 → 2 (V ) given by x ⊗ y → x · y yields an isomorphism Sym 2 (V ) ∼ = 2 (V ). Here and elsewhere Sym 2 denotes the second symmetric power. Lemma 3 i) Let L be a symmetric Leibniz algebra. Then there is a well-defined linear map σ : 2 (L ab ) → L given by ii) One has an exact sequence of Leibniz algebras and Leibniz algebra homomorphisms, where 2 (L ab ) is considered as an abelian Leibniz algebra. Moreover Im(σ ) is a central subalgebra of L.
Proof i) For any x, y, z ∈ L we have Here we used iii) and vi) of Lemma 1. Thus σ is well-defined. ii) Comparing the definitions we see that Im(σ ) = L ann and hence the result.
Recall that a Leibniz algebra L is perfect if L = [L, L].

Corollary 4 Any perfect symmetric Leibniz algebra is a Lie algebra.
Proof In this case L ab = 0. Hence L → L Lie is an isomorphism thanks to the part ii) of Lemma 3.
Recall that [15] the free Lie algebra Lie(V ) generated by a free module V has a natural grading where Lie n (V ) is spanned by all n-fold commutators of elements of V . For n = 1 and Recall also that [6] the free Leibniz algebra Leib(V ) generated by a module V is also graded: where the bracket on Leib(V ) is uniquely defined by the rule: Here ω ∈ V ⊗n , n ≥ 1, v ∈ V . In particular the map defines a surjective graded homomorphism of Leibniz algebras which clearly induces an isomorphism (Leib(V )) Lie ∼ = Lie(V ). It is also clear that the map π is an isomorphism in degree 1.
Our next goal is to describe the free symmetric Leibniz algebra SymLeib(V ) generated by V . Clearly the homomorphism π has the following decomposition where 1 and 2 are surjective Leibniz algebra homomorphisms.

Proposition 5 Let V be a free module. Then for L = SymLeib(V ), one has a central extension of Leibniz algebras:
Proof Since the defining relations of Leibniz and symmetric Leibniz algebras are of degree 3, the free Leibniz and symmetric Leibniz algebras have the same components in degree 1 and 2, which are respectively V = L ab and V ⊗2 . Thus injectivity of σ follows from the exact sequence (5). The rest follows from Lemma 3.

Moreover, the map
is an isomorphism when n = 1, 2, while the map is an isomorphism, when n ≥ 3. Thus we have This result in the same period (i.e. around 1995) was obtained independently by Ronco [9] when K is a field of characteristic = 2.

Symmetric Lie -algebras
We now introduce the notion of symmetric Lie μ-algebra, or, for short, Lie μalgebra. The strange terminology stems from the fact that we actually aim at general Lie μ-algebras, to be investigated in [2], of which the symmetric Lie μ-algebras are a particular case. (m, μ), where m is a Lie algebra and μ : m ⊗ m → m is a multiplication, such that the following identities hold

Definition 7 A symmetric Lie μ-algebra is a pair
Here {−, −} denotes the Lie bracket on m and x y = μ(x ⊗ y).
The first two identities show that the pair (m, μ) is a commutative, associative nilpotent algebra of class two. Conversely, any such algebra can be seen as a Lie μ-algebra with trivial bracket.
Any Lie algebra can be considered as a Lie μ-algebra with zero multiplication x y = 0. In particular any module has the structure of a Lie μ-algebra with trivial bracket and trivial multiplication. Such Lie μ-algebras are called abelian Lie μalgebras.
A Lie μ-ideal of a Lie μ-algebra m is a submodule a such that [m, a] ⊂ a and ma ⊂ a. Moreover a is a central μ-ideal if [m, a] = 0 = ma.
It follows from iv) of Definition 7 that submodule Im(μ) is a central ideal of m. The quotient algebra is denoted by g = m Lie and is called the Liezation of m. The abelization of the Lie μ-algebra m is the module g ab and is denoted by m AB in order to distinguish it from m ab which is the abelization of the underlying Lie algebra. The last object has a canonical structure of a commutative, associative nilpotent algebra of class two, because by the identity iii) Lie commutator [m, m] is an ideal of the underlying commutative algebra.
where g = m Lie , μ (x ȳ) = x y and Sym 2 (m AB ) is considered as abelian Lie μalgebra. Herex denotes the class of x ∈ m in m AB . Furthermore Im(μ ) is a central μ-ideal of m.
Proof By our construction we have an exact sequence: Let us use identities of Definition 7. By the identity i) the map μ factors through Sym 2 (m). Next, by the identity ii) it further factors through Sym 2 (m Lie ). It follows from the identities iii) that this map factors through the second symmetric power of abelization m AB and hence the result.
A Lie algebra g is called μ-rigid if the only μ-algebra structure on it is the trivial one: x y = 0, x, y ∈ g. Proof i) is a weak form of Lemma 8. ii) is trivial, while iii) is immediate from i).
As an illustration of Corollary 9 we can take m to be the Heisenberg Lie algebra of dimension 2k + 1, with basis x 1 , . . . , x k , y 1 , . . . , y k , z such that {x i , y i } = z for all i = 1, . . . , k and other nontrivial brackets being zero. Choose a subset I ⊂ {1, . . . , k} and define a μ-algebra structure by Proposition 10 Let Lie μ (V ) be the free Lie μ-algebra freely generated by a module V . Then the map μ from the Lemma 8 is injective and hence one has a central extension of Lie μ-algebras: Here u, v ∈ V = m AB and uv ∈ m. Any commutative, associative nilpotent algebra of class two can be considered as a Lie μ-algebra with trivial bracket. In particular, we can take the free such algebra generated by V , that is where for x, y ∈ V , ω, ω 1 ∈ Sym 2 (V ) one sets By universal property of free Lie μ-algebras the identity map Id V has a unique extension c : Thus c • μ = Id Sym 2 (V ) and the result follows.

The case when 2 is invertible in K
In this section we assume that 2 is invertible in K. Our goal is to prove the following result.

Theorem 11 The categories of symmetric Leibniz algebras and Lie μ-algebras are isomorphic.
This a consequence of Propositions 12 and 13 below.

Proposition 12
Let L be a symmetric Leibniz algebra. We put: This shows that the multiplication defines a commutative and associative algebra structure of nilpotence degree two. We

6 -groups
In this section K = Z. The category of abelian groups is denoted by AB, while the category of groups is denoted by GR. It is well known that the inclusion AB ⊂ GR has the left adjoint given by T → T ab = T /[T , T ], T ∈ GR. The group T ab is called the abelization of the group T .
The following is a group-theoretic version of Definition 7. Just as the latter, this is a particular case of the notion of general μ-group, which we will study in [2]. A μ-group is a pair (G, μ), where G is a group, not necessarily commutative, written additively, together with a binary operation μ : G × G → G such that:
Obviously any commutative associative nilpotent algebra of class two is a μgroup. Also any group can be considered as a μ-group with zero multiplication.
Denote the category of μ-groups by GR μ . Since any group has a trivial μgroup structure (i.e. μ = 0), we see that GR ⊂ GR μ . The inclusion has a left adjoint. To describe it, we fix some obvious facts.
For a μ-group G, denote by Z gr (G) the center of the group G, that is Denote by K the subgroup of G generated by all elements of the form x y, x, y ∈ G. By iii) K ⊂ Z gr (G). Thus K is a central subgroup of the underlying group of G and we can consider the corresponding quotient group G gr = G/K . Then the assignment G → G gr defines the functor GR μ → GR, which is the left adjoint to the inclusion GR ⊂ GR μ . We also need the group (G gr ) ab = G gr /[G gr , G gr ], which is denoted by G AB and is called the abelization of a μ-group G. Thanks to the condition iv), the map λ x : G → G given by λ x (y) = x y is a group homomorphism for all x ∈ G. Moreover by ii) it factors through the group G gr . Since Im(λ x ) ⊂ Z gr (G), we see that λ x is a homomorphism into an abelian group Z gr (G) and hence it factors through the abelization G AB . It follows from the condition i) that the map μ yields a well-defined homomorphism which we by abuse of notation will still be denoted by μ. Herex denotes the class of x ∈ G in G AB .
It is clear that K = Im(μ) and hence one has a natural exact sequence of μ-groups Moreover, Im(μ) = K is a central μ-subgroup of G. A group G is called μ-rigid if the only μ-group structure on it is the trivial one: x y = 0, x, y ∈ G. As in the case of Lie μ-algebras we have the following analogue of Corollary 9.

iii) A group M is μ-rigid if either M is perfect, that is M = [M, M] or the center of M is trivial.
We omit the proof of this fact, since it goes along the same lines as the proof of Corollary 9.
Here is some examples based on Part ii) of Corollary 15. Take the dihedral group D 2n of order 4n with generators x and y, such that 2nx = 0 = 2y and y+x +y = (2n−1)x. One easily sees that there exists a unique μ-algebra structure such that x y = nx, x x = 0 = yy. As another example, one can introduce a nontrivial μ-group structure on the Heisenberg group, similar to one which we described for Heisenberg Lie algebra. Details are left to the interested readers.
Denote by Z[S] the free abelian group generated by a set S. Similarly, denote by Gr[S] (resp. Gr μ [S]) the free group (resp. free μ-group) generated by S. In these notations we have Lemma 16 One has the following central extension of μ-groups.
Proof For G = Gr μ [S], one has G gr = Gr[S] and G AB = Z[S], by the adjoint functor argument (see the proof of Proposition 10). Thus we can use the exact sequence (6). We only need to show injectivity of μ.
Any commutative, associative nilpotent ring of class two can be considered as a μ-group. In particular, we can take the free such a ring generated by V = Z[S], that is

Lie -groups
In this section K = R. Denote by L the canonical functor from the category of Lie groups to the category of finite dimensional Lie algebras. Recall that the restriction of L to the subcategory of simply connected Lie groups is an equivalence of categories [15,Theorem 2,p.152].
As Lie groups are smooth analogues of discrete groups, the following is a smooth analogue of μ-groups. Since the left arrow is an isomorphism of Lie groups, we obtain a smooth map H × H → A ⊂ Z . Now we can take the composite map to obtain the multiplication map in M. By Part ii) of Corollary 15 in this way we obtain a μ-group structure on M. This finishes the construction of the functor R. Take a simply connected Lie μ-group M. Recall that K was defined as the closure of the subgroup generated by x y, x, y ∈ M. It follows that K is connected and hence G = M/K is also simply connected. Now, comparing the constructions we see that Conversely, if m is a finite dimensional μ-algebra, then a, which is defined as the subspace of m generated by elements of the form x y, x, y ∈ m coincides with the subgroup generated by the same elements. This follows from the (bi)linearity of the product. Since A is abelian, the exponential map exp : a → A is epimorphism. From this follows that the subgroup K ⊂ M = R(m) and A coincide and LR(m) ∼ = m.

Corollary 19
The category of finite dimensional symmetric Leibniz algebras is equivalent to the category of simply connected Lie μ-groups.
Proof This is a direct consequence of Theorems 11 and 18.

Interpretation in language of algebraic theories
The algebraic theories corresponding to Lie, Leibniz and symmetric Leibniz Kalgebras are denoted respectively by LIE K , LB K and SLB K .
We refer to [1] for the notion of (linear) extension of algebraic theories. Comparing the definitions we see that Proposition 5 can be restated as Lemma 20 One has a linear extension of algebraic theories: Moreover if 2 is invertible in K, this extension splits.
Recall that the functor Rings → Algebraic Theories which sends the ring R to the theory of right R-modules is full and faithful. Because of this we identify rings with their corresponding theories of modules.
Let COM K (2) denote the algebraic theory of nilpotent commutative associative K-algebras of class 2, that is algebras obeying the identities If 2 is invertible in K, then 2 = Sym 2 and we get the following Proposition 21 If 2 is invertible in K, then the diagram is the pull-back diagram in the category of algebraic theories.
Quite similarly, if one denotes by GR (resp. GR μ , AB) the algebraic theory of groups (rep. μ-groups, abelian groups), then as an immediate corollary of Lemma 16 we obtain the following fact