Georgian Mathematical Journal Volume 9 (2002), Number 4, 671–682

CENTRAL SERIES FOR GROUPS WITH ACTION AND LEIBNIZ ALGEBRAS

T. DATUASHVILI

Abstract. The notion of central series for groups with action on itself is introduced. An analogue of Witt’s construction is given for such groups. A certain condition is found for the action and the corresponding category is defined. It is proved that the above construction defines a functor from this category to the category of Lie–Leibniz algebras and in particular to Leibniz algebras; also the restriction of this functor on the category of groups leads us to Lie algebras and gives the result of Witt.

2000 Mathematics Subject Classification: 17B60.

Key words and phrases: Central series, groups, groups with action on itself, Lie algebras, Leibniz algebras, Lie–Leibniz algebras.

Introduction

The well-known construction of Witt defines a functor from the category of groups to the category of Lie algebras [6], [5]. The aim of this paper is to define a category and to give an analogue of Witt’s construction for its objects, which will lead us to the category of Leibniz algebras. This problem was stated by J.-L. Loday; later an analogous question for the possibly defined partial Leibniz algebras was proposed, which was inspired by the work of Baues and Conduché [1]. Since the main interest lies in the absolute case, the author decided to begin with this one.

In Section 1 we define the category of groups with action on itself 𝔾𝕣 ^{•}, the category of abelian groups with action on itself 𝔸𝕓 ^{•} and the category of groups with bracket operation 𝔾𝕣 ^{[ ]}. This kind of groups are Ω-groups in the sense of [2]. We construct adjoint pairs of functors relating categories 𝔾𝕣 ^{•}, 𝔸𝕓 ^{•}, 𝔾𝕣 ^{[ ]}, 𝔾𝕣 . In Section 2 we define ideals and commutators for the objects of 𝔾𝕣 ^{•} (similarly for 𝔾𝕣 ^{[ ]}) and show that these notions are equivalent to the special case of the known definitions for Ω-groups [2]. In Section 3 we define central series of groups with action on itself and a category of Lie–Leibniz algebras 𝕃𝕃. We consider the category of groups with action on itself satisfying a certain condition 𝔾𝕣 ^{c}. We give an analogue of Witt’s construction [6] and prove that it defines a functor LL : 𝔾𝕣 ^{c} ⟶ 𝕃𝕃, in particular this gives a functor 𝔾𝕣 ^{c} ⟶ 𝕃𝕖𝕚𝕓𝕟𝕚𝕫 . In a similar way one can construct a functor 𝔸𝕓 ^{c} ⟶ 𝕃𝕖𝕚𝕓𝕟𝕚𝕫 , which is actually the restriction of LL on 𝔸𝕓 ^{c}. The functorial relations with the classical situation (𝔾𝕣 ⟶ 𝕃𝕚𝕖) is considered, namely by the restriction of LL on 𝔾𝕣 we obtain the result of Witt [6], [5].

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1. Groups with Action on Itself

Let G be a group which acts on itself from the right side; i.e. we have a map ε : G × G ⟶ G with

\begin{matrix}ε(g, g′ + g′′) = ε(ε(g, g′), g′′), \\ ε(g, 0) = g, \\ ε(g′ + g′′, g) = ε(g′, g) + ε(g′′, g), \\ ε(0, g) = 0, \end{matrix}  (1.1)

for g, g′, g′′ ∈ G. Denote ε(g, h) = g^{h}, for g, h ∈ G. We denote the group operation additively, nevertheless the group is not commutative in general. If (G′, ε′) is another group with action, then a homomorphism (G, ε) ⟶ (G′, ε′) is a group homomorphism φ : G ⟶ G′ for which the diagram

G × G ⟶^{ε} G
(φ,φ)↓  ↓^{φ}
G′ × G′ ⟶^{ε′} G′

commutes. In other words, we have

φ(g^{h}) = φ(g)^{φ(h)}, g, h ∈ G.

If we consider an action as a group homomorphism G ⟶^{ν} Aut G, then a homomorphism between two groups with action means the commutativity of the diagram

\xymatrix{G \ar[r]^{ν} \ar[dd]_{φ} & Aut G ⊂ Hom(G, G) \ar[d]^{Hom(G,φ)} \\  & Hom(G, G′) \\ G′ \ar[r]^{ν′} & Aut G′ ⊂ Hom(G′, G′) \ar[u]_{Hom(φ,G′)}}

so that φ · (ν(h)) = ν′(φ(h)) · φ, h ∈ G.

Recall [2] that an Ω-group is a group with a system of n-ary algebraic operations Ω (n ≥ 1), which satisfies the condition

00 ⋯ 0ω = 0,  (1.2)

CENTRAL SERIES FOR GROUPS WITH ACTION AND LEIBNIZ ALGEBRAS 673

where 0 is the identity element of G, and 0 on the left side occurs n times if ω is an n-ary operation. In special cases Ω-groups give groups, rings and groups with action on itself. In the latter case Ω consists of one binary operation, an action; or Ω consists of only unary operations, elements of G, and this operation is an action again. In both cases the condition (1.2) is satisfied. We shall denote the category of groups with action on itself by 𝔾𝕣 ^{•}. Let 𝔸𝕓 ^{•} denote the category of abelian groups with action on itself. We have functors

𝔸𝕓 ^{•} ⟶^{E} ; ⟵_{A} 𝔾𝕣 ^{•} ⟶^{Q_{1}} ; ⟵^{T} ; ⟶^{Q_{2}} ; ⟵^{C} 𝔾𝕣 ,

where Q_{1}(G), for G ∈ 𝔾𝕣 ^{•}, is the greatest quotient group of G which makes the action trivial; Q_{2}(G) is a quotient of G by the equivalence relation generated by the relation g^{h} ∼ −h + g + h, g, h ∈ G; A is the abelianization functor, thus A(G) = G/(G, G), where (G, G) is the ideal of G generated by the commutator normal subgroup of G (for the definition of an ideal see Section 2). A(G) has the induced operation of action on itself. Each group can be considered as a group with the trivial action or with the action by conjugation, these give functors T and C, respectively. Every object of 𝔸𝕓 ^{•} can be considered as an object of 𝔾𝕣 ^{•}; this functor is denoted by E. It is easy to see that the functors Q_{1}, Q_{2} and A are left adjoints to the functors T, C and E respectively. Let G ∈ 𝔾𝕣 ^{•}. Define the operation of square brackets [ , ] : G × G ⟶ G on G by

[g, h] = −g + g^{h}, g, h ∈ G.

Proposition 1.1. For the operation [ , ] we have the following identities:

(i) [g, h_{1} + h_{2}] = [g, h_{1}] + [g + [g, h_{1}], h_{2}];

(ii) [g + g′, h] = −g′ + [g, h] + g′ + [g′, h];

(iii) [g, 0] = [0, g] = 0.

Proof. These identities follow directly from (1.1). □

Corollary 1.2. For g, h ∈ G

[g^{h}, −h] = −[g, h];
[−g, h] = g − [g, h] − g.

Denote by 𝔾𝕣 ^{[ ]} the category of groups with an additional bracket operation [ , ] satisfying the conditions (i)–(iii) of Proposition 1.1; morphisms of 𝔾𝕣 ^{[ ]} are group homomorphisms preserving the bracket operation. We shall denote the objects of 𝔾𝕣 ^{[ ]} by G^{[ ]}.

Conversely, if G^{[ ]} ∈ 𝔾𝕣 ^{[ ]}, we can define an action of G^{[ ]} on itself due to the bracket operation by

g^{h} = g + [g, h], g, h ∈ G^{[ ]}.

It is easy to prove that these two procedures are converse to each other and actually we have an isomorphism of categories

𝔾𝕣 ^{•} ≈ 𝔾𝕣 ^{[ ]} .

2. Ideals and Commutators in 𝔾𝕣 ^{•}

Let G ∈ 𝔾𝕣 ^{•}.

Definition 2.1. A nonempty subset A of G is called an ideal of G if it satisfies the following conditions:

1. A is a normal subgroup of G as a group;

2. a^{g} ∈ A, for a ∈ A, g ∈ G; 3. −g + g^{a} ∈ A, for a ∈ A and g ∈ G. Definition 2.2 (Kurosh [2]). A nonempty subset A of an Ω-group G is called

an ideal if (a) A is an additive normal subgroup of G;

(b) For any n-any operation ω from Ω, any element a ∈ A and elements x_{1}, x_{2}, …, x_{n} ∈ G

−(x_{1} ⋯ x_{n}ω) + x_{1} ⋯ x_{i−1}(a + x_{i})x_{i+1} ⋯ x_{n}ω ∈ A,

for i = 1, 2, …, n.

This definition in the case of groups is the definition of a normal subgroup of a group, and in the case of rings is the definition of a two-sided ideal of a ring.

Proposition 2.3. For a group G ∈ 𝔾𝕣 ^{•} considered as an Ω-group, where Ω consists of one binary operation of action, Definitions 2.1 and 2.2 are equivalent.

Proof. The condition (b) of Definition 2.2 has the forms:

− x_{1}^{x_{2}} + (a + x_{1})^{x_{2}} ∈ A, for i = 1;  (2.1)
− x_{1}^{x_{2}} + x_{1}^{a+x_{2}} ∈ A, for i = 2.  (2.2)

Taking x_{1} = 0 in (2.1), we obtain a^{x_{2}} ∈ A, which is condition 2 of Definition 2.1. Taking x_{2} = 0 in (2.2), we have −x_{1} + x_{1}^{a} ∈ A, which is condition 3 of Definition 2.1.

Conversely, we shall show that conditions 2 and 3 of Definition 2.1 imply conditions (2.1) and (2.2). From condition 2 we have a^{x_{2}} ∈ A; also

−x_{1}^{x_{2}} + (a + x_{1})^{x_{2}} = −x_{1}^{x_{2}} + a^{x_{2}} + a_{1}^{x_{2}},

and it is an element of A since A is a normal subgroup of G. By condition 3 of Definition 2.1, −x_{1} + x_{1}^{a} ∈ A. We have −x_{1}^{x_{2}} + x_{1}^{a+x_{2}} = (−x_{1} + x_{1}^{a})^{x_{2}} and this is an element of A due to condition 2, which ends the proof. □

Thus an ideal of G is a subobject of G in 𝔾𝕣 ^{•}. It is clear that G itself and the trivial subobject of G are ideals of G. An intersection of any system of ideals of G is an ideal, and therefore we conclude that there exists the ideal generated by a system of elements of G.

Proposition 2.4. Let A be an ideal of G. For a_{1}, a_{2} ∈ A, g_{1}, g_{2} ∈ G we have

(a_{1} + g_{1})^{a_{2}+g_{2}} ∈ g_{1}^{g_{2}} + A.

Proof. Since A is an ideal of G there exist a′_{1}, a′_{2} ∈ A, such that a_{1}+g_{1} = g_{1}+a′_{1}, a_{2} + g_{2} = g_{2} + a′_{2}. Therefore

(a_{1} + g_{1})^{a_{2}+g_{2}} = (g_{1} + a′_{1})^{g_{2}+a′_{2}} = (g_{1}^{g_{2}})^{a′_{2}} + a′_{1}^{g_{2}+a′_{2}}
= g_{1}^{g_{2}} − g_{1}^{g_{2}} + (g_{1}^{g_{2}})^{a′_{2}} + a′_{1}^{g_{2}+a′_{2}} ∈ g_{1}^{g_{2}} + A;

here we apply −g_{1}^{g_{2}} + (g_{1}^{g_{2}})^{a′_{2}} ∈ A. □

Let A and B be subobjects of G. Denote by {A, B} the subobject of G generated by A and B, and let A + B denote the subset of G

A + B = {a + b | a ∈ A, b ∈ B}.

Proposition 2.5. If A is an ideal of G and B is a subobject of G, then

{A, B} = A + B.

Proof. It is obvious that A + B ⊂ {A, B}. Since A is an ideal, it follows that A + B is a subgroup of G. By Proposition 2.4, (a_{1} + b_{1})^{a_{2}+b_{2}} ∈ b_{1}^{b_{2}} + A. Since B is a subobject, b_{1}^{b_{2}} ∈ B, and since A is an ideal, b_{1}^{b_{2}} + A = A + b_{1}^{b_{2}} ∈ A + B which ends the proof. □

For Ω-groups see Propositions 2.4 and 2.5 in [2].

Proposition 2.6. If A and B are ideals of G, then A + B is also an ideal.

Proof. For g ∈ G, a ∈ A and b ∈ B we have

g + (a + b) = (a′ + g) + b = a′ + b′ + g ∈ A + B + g,

for certain a′ ∈ A and b′ ∈ B. Thus g +(A+B) ⊂ (A+B)+g. In the same way we show that (A + B) + g ⊂ g + (A + B) and thus g + (A + B) = (A + B) + g. It is obvious that (a + b)^{g} ∈ A + B. Now we have to show that −g + g^{a+b} ∈ A + B. We have

−g + g^{a+b} = −g + g^{a} − g^{a} + (g^{a})^{b} ∈ A + B

since −g + g^{a} ∈ A, −g^{a} + (g^{a})^{b} ∈ B. □

It is easy to verify that the ideal generated by a system of ideals of G coincides with the additive subgroup of G generated by these ideals. For Ω-groups see [2].

Definition 2.1′. Let G^{[ ]} ∈ 𝔾𝕣 ^{[ ]} and A be a nonempty subset of G^{[ ]}. A is called an ideal of G^{[ ]} if

1′. A is a normal subgroup of G^{[ ]} as of an additive group;

2′. [a, g] ∈ A, for a ∈ A, g ∈ G^{[ ]};

3′. [g, a] ∈ A, for a ∈ A, g ∈ G^{[ ]}.

It is easy to see that the isomorphism of categories 𝔾𝕣 ^{•} ≈ 𝔾𝕣 ^{[ ]} carries ideals to ideals.

Proposition 2.7. If A is an ideal of G, then the quotient group G/A with the induced action on itself is an object of 𝔾𝕣 ^{•}.

Proof. Straightforward verification. □

In what follows, for G ∈ 𝔾𝕣 ^{•} and g, g′ ∈ G, [g, g′] will indicate the element −g + g^{g′} of G and (g, g′) the commutator −g − g′ + g + g′. Let A and B be subobjects of G.

Definition 2.8. A commutator [A, B] of G generated by A and B is the ideal of {A, B} generated by the elements

{[a, b], [b, a], (a, b) | a ∈ A, b ∈ B}.

Definition 2.9 ([2]). Let G be an Ω-group, A, B be Ω-subgroups of G and {A, B}_{Ω} be the Ω-subgroup of G generated by A and B. The commutator [A, B]_{Ω} is the ideal of {A, B}_{Ω} generated by elements of the form

(a, b) = −a − b + a + b, a ∈ A, b ∈ B,

and

[a_{1}, …, a_{n}; b_{1}, …, b_{n}; ω] = −a_{1}a_{2} ⋯ a_{n}ω − b_{1}b_{2} ⋯ b_{n}ω
+ (a_{1} + b_{1})(a_{2} + b_{2}) ⋯ (a_{n} + b_{n})ω,  (2.3)

where ω is an n-any operation from Ω, a_{1}, …, a_{n} ∈ A and b_{1}, …, b_{n} ∈ B.

If G is a group with the trivial action on itself or with the action by conjugation, then [A, B] in Definition 2.8 is the normal subgroup of G generated in {A, B} by commutators (a, b), a ∈ A, b ∈ B, i. e. the usual commutator for the case of groups. The same is true for Definition 2.9; if an Ω-group is a group without multioperations, then the commutator [A, B]_{Ω} is the usual commutator (A, B) of a group [2].

Proposition 2.10. In the case of groups with action on itself Definitions 2.8 and 2.9 are equivalent.

Proof. For groups with action (2.3) has the form

−a^{a_{2}} − b_{1}^{b_{2}} + (a_{1} + b_{1})^{a_{2}+b_{2}}.  (2.4)

Take a_{1} = a, a_{2} = b_{1} = 0, b_{2} = b, then −a + a^{b} ∈ [A, B]_{Ω}. Take in (2.4) a_{1} = b_{2} = 0, a_{2} = a, b_{1} = b, then we obtain

−b + b^{a} ∈ [A, B]_{Ω}.

Thus we have shown that [A, B] ⊂ [A, B]_{Ω}. Conversely, for x = −a_{1}^{a_{2}} − b_{1}^{b_{2}} + (a_{1} + b_{1})^{a_{2}+b_{2}} ∈ [A, B]_{Ω} we have x = −a_{1}^{a_{2}} − b_{1}^{b_{2}} + (a_{1}^{a_{2}})^{b_{2}} + (b_{1}^{a_{2}})^{b_{2}} ∈ {A, B}. Let \overline{\{A, B\}} = \{A, B\}/[A, B] and let \overline{g} be the class of the element g ∈ \{A, B\} in \overline{\{A, B\}}. We have \overline{a^{b}} = \overline{a}, \overline{b^{a}} = \overline{b} in \overline{\{A, B\}}. Thus

\overline{x} = \overline{−a_{1}^{a_{2}} − b_{1}^{b_{2}} + (a_{1}^{a_{2}})^{b_{2}} + (b_{1}^{a_{2}})^{b_{2}}} = \overline{a_{1}^{a_{2}}} − \overline{b_{1}^{b_{2}}} + \overline{a_{1}^{a_{2}}} + \overline{b_{1}^{a_{2}}} \overline{b_{2}}
= −\overline{a_{1}^{a_{2}}} − \overline{b_{1}^{b_{2}}} + \overline{a_{1}^{a_{2}}} + \overline{b_{1}^{b_{2}}} = \overline{−a_{1}^{a_{2}} − b_{1}^{b_{2}} + a_{1}^{a_{2}} + b_{1}^{b_{2}}} = 0,

which means that x ∈ [A, B]. □

Below we formulate without proofs two statements for Ω-groups from [2], which in the case of groups with action give the corresponding results.

Proposition 2.11. For any Ω-subgroups A and B in G we have

[A, B]_{Ω} = [B, A]_{Ω}.

Proposition 2.12. An Ω-subgroup A is an ideal of G if and only if

[A, G]_{Ω} ⊆ A.

Corollary 2.13. Any Ω-subgroup A of an Ω-group G which contains the commutator [G, G]_{Ω} is an ideal of G.

Proof. It follows from the inclusions

[A, G]_{Ω} ⊂ [G, G]_{Ω} ⊂ A. □

3. Central Series in 𝔾𝕣 ^{•} and the Main Result

Let G ∈ 𝔾𝕣 ^{•}.

Definition 3.1. The (lower) central series

G = G_{1} ⊃ G_{2} ⊃ ⋯ ⊃ G_{n} ⊃ G_{n+1} ⊃ ⋯

of the object G is defined inductively by

G_{n} = [G_{1}, G_{n−1}] + [G_{2}, G_{n−2}] + ⋯ + [G_{n−1}, G_{1}].

By definition, we have [G_{n}, G_{m}] ⊂ G_{n+m}.

Proposition 3.2. For each n ≥ 1, G_{n+1} is an ideal of G_{n}.

Proof. We have G_{2} = [G_{1}, G_{1}], which is an ideal of G_{1}, by definition. G_{3} = [G_{1}, G_{2}] + [G_{2}, G_{1}]. By Proposition 2.11, [G_{1}, G_{2}] = [G_{2}, G_{1}]. We have

[G_{1}, G_{2}] ⊂ [G_{1}, G_{1}] = G_{2} ⊂ {G_{1}, G_{2}}

and [G_{1}, G_{2}] is an ideal of {G_{1}, G_{2}}; from this it follows that [G_{1}, G_{2}] is an ideal of G_{2} and therefore, by Proposition 2.6, G_{3} is an ideal of G_{2}. We have

G_{n+1} = [G_{1}, G_{n}] + [G_{2}, G_{n−1}] + … [G_{n−1}, G_{2}] + [G_{n}, G_{1}].

For 1 ≤ k ≤ n, [G_{k}, G_{n−k+1}] is an ideal of {G_{k}, G_{n−k+1}}; G_{n} ⊆ G_{k} from which it follows that G_{n} ⊆ {G_{k}, G_{n−k+1}}. At the same time

[G_{k}, G_{n−k+1}] ⊂ [G_{k}, G_{n−k}] ⊂ G_{n}.

Therefore [G_{k}, G_{n−k+1}] is an ideal of G_{n} for each 1 ≤ k ≤ n. Thus each summand of G_{n+1} is an ideal of G_{n}. By Propositions 2.6 and 2.11 we conclude that G_{n+1} is an ideal of G_{n}. □

Since (G_{i}, G_{i}) ⊂ G_{2i} ⊂ G_{i+1}, each G_{i}/G_{i+1} has an abelian group structure. Let

LL_{G} = G_{1}/G_{2} ⊕ G_{2}/G_{3} ⊕ ⋯ ⊕ G_{n}/G_{n+1} ⊕ ⋯,  (3.1)

where ⊕ denotes the direct sum of abelian groups.

Let k be a commutative ring with the unit, and A a k-module. We recall the definitions of Lie and Leibniz algebras.

Definition 3.3. A Lie algebra (A, ( , )) over k is given by a k-module A and a k-module homomorphism ( , ) : A ⊗_{k} A ⟶ A called a round bracket such that the equation

(x, x) = 0

and the Jacobi identity

(x, (y, z)) + (y, (z, x)) + (z, (x, y)) = 0  (3.2)

hold for x, y, z ∈ A.

Let 𝕃𝕚𝕖 be the category of Lie algebras. Morphisms in 𝕃𝕚𝕖 are k-module homomorphisms φ with

φ(x, y) = (φ(x), φ(y)).

Definition 3.4 ([3]). A Leibniz algebra A over k is a k-module A equipped with a k-module homomorphism called a square bracket

[ , ] : A ⊗_{k} A ⟶ A,

satisfying the Leibniz identity

[x, [y, z]] = [[x, y], z] − [[x, z], y]  (3.3)

for x, y, z ∈ A.

This is in fact a right Leibniz algebra. The dual notion of a left Leibniz algebra is made out of the dual relation

[x, [y, z]] = [[x, y], z] + [y, [x, z]],

for x, y, z ∈ A.

A morphism of Leibniz algebras is a k-module homomorphism f : A ⟶ A′ with φ[x, y] = [φ(x), φ(y)].

In this paper we deal with right Leibniz algebras. Denote this category by 𝕃𝕖𝕚𝕓𝕟𝕚𝕫 .

Definition 3.5. A Lie–Leibniz algebra is a k-module A together with two k-module homomorphisms

( , ), [ , ] : A ⊗_{k} A ⟶ A

called round and square brackets, respectively, such that (x, x) = 0 for x ∈ A and both Jacobi and Leibniz identities ((3.2) and (3.3)) hold.

A morphism of Lie–Leibniz algebras is a k-module homomorphism φ : A ⟶ A′ with

φ(x, y) = (φ(x), φ(y)),
φ[x, y] = [φ(x), φ(y)].

We denote the corresponding category by 𝕃𝕃.

Condition 1. For each x, y, z ∈ G, G ∈ 𝔾𝕣 ^{•}

x − x^{(z^{x})} + x^{y+z^{x}} − x + x^{z} − x^{z+y^{z}} = 0.

It is straightforward to verify that if G satisfies Condition 1, then the group G^{[ ]}, which corresponds to G (i.e. [ , ] is defined by [g, h] = −g + g^{h}, g, h ∈ G) satisfies the following condition.

Condition 1′.

[x^{y}, [y, z]] = [[x, y], z^{x}] + [−[x, z], y^{z}], x, y, z ∈ G^{[ ]}.

Let G be a group. Consider G as a group with the (right) action by conjugation, i.e. g^{g′} = −g′ + g + g′. Then G satisfies Condition 1 and in this case Condition 1′ is equivalent to the Witt–Hall identity for groups. Each group with the trivial action on itself (i.e. g^{g′} = g, g, g′ ∈ G) also satisfies Condition 1. For an arbitrary set X let ℱ_{X} be the free group with action on itself generated by X. The quotient ℱ_{X}/_{∼} of ℱ_{X} by the equivalence relation generated by the relation corresponding to Condition 1 is obviously a group which satisfies Condition 1. See also an example at the end of the proof of Theorem 3.6.

Denote by 𝔾𝕣 ^{c} a category of groups with action on itself satisfying Condition 1. In an analogous way we define the category 𝔸𝕓 ^{c}. It is easy to see that the functors E, A, T, C, Q_{1}, Q_{2}, defined in Section 1, give the functors between categories 𝔸𝕓 ^{c}, 𝔾𝕣 ^{c} and 𝔾𝕣 . We shall denote below these functors by the same letters. ℱ_{X}/_{∼} is a free object in 𝔾𝕣 ^{c} and consequently the action in it is neither the trivial one nor the conjugation.

Let G ∈ 𝔾𝕣 ^{c}. Denote \overline{G}_{m} = G_{m}/G_{m+1}, then LL_{G} = ∑_{m≥1} \overline{G}_{m}.

Consider maps ( , )_{mn}, [ , ]_{mn} : G_{m} × G_{n} ⟶ G_{m+n} defined by round and square brackets in G, respectively:

x, y ⟼ (x, y),
x, y ⟼ [x, y].

By the definition of G_{i}, it is clear that if x ∈ G_{m}, y ∈ G_{n}, then (x, y), [x, y] ∈ G_{m+n}. For x ∈ G_{m}, denote by \overline{x} the corresponding class in \overline{G}_{m}.

Theorem 3.6. Let G be a group with action on itself satisfying Condition 1. Then we have:

(a) \overline{x^{y}} = \overline{x}, \overline{−y + x + y} = \overline{x}, for each x ∈ G_{m}, y ∈ G_{n};

(b) The maps ( , )_{mn} and [ , ]_{mn} : G_{m} × G_{n} ⟶ G_{m+n} induce bilinear maps α_{mn}, β_{mn} : \overline{G}_{m} × \overline{G}_{n} ⟶ \overline{G}_{m+n};

(c) The maps α_{mn}, β_{mn}, m, n ≥ 1 define bilinear maps ( , ), [ , ] : LL_{G} × LL_{G} ⟶ LL_{G}, which give a Lie–Leibniz structure on LL_{G}.

Proof. (a) Let x ∈ G_{m}, y ∈ G_{n}, m, n ≥ 1. Then [x, y] = −x + x^{y} ∈ G_{m+n} ⊂ G_{m} and since x ∈ G_{m} we obtain that x^{y} ∈ G_{m}. In \overline{G}_{m} we have \overline{[x, y]} = −\overline{x} + \overline{x^{y}}, but since [x, y] ⊂ G_{m+n} ⊂ G_{m+1} we have \overline{[x, y]} = 0 in \overline{G}_{m} and thus in \overline{G}_{m} we have \overline{x} = \overline{x^{y}}. In the same way we show for the action with conjugation that \overline{−y + x + y} = \overline{x}. (see also [5]).

(b) We shall check this condition for a square bracket; for a round bracket the proof is similar [5]. First we shall show that the map β_{mn} : \overline{G}_{m} × \overline{G}_{n} ⟶ \overline{G}_{m+n} is defined correctly. Let \overline{x} ∈ \overline{G}_{m}, \overline{y} ∈ \overline{G}_{n}, where x ∈ G_{m}, y ∈ G_{n}. By definition, β_{mn}(\overline{x}, \overline{y}) = [\overline{x}, \overline{y}] = \overline{[x, y]}, where [x, y] ∈ G_{n+m}. Let \overline{x} = \overline{x′} for x′ ∈ G_{m}, thus x − x′ ∈ G_{m+1}. For simplicity, suppose that x − x′ ∈ [G_{i+1}, G_{m−i}] ⊂ G_{m+1} (a more general case is treated similarly). Then x = [a, b] + x′, where a ∈ G_{i+1}, b ∈ G_{m−i}. From this we have in \overline{G}_{m+n}:

\overline{[x, y]} = \overline{[[a, b] + x′, y]} = \overline{−x′ + [[a, b], y] + x′ + [x′, y]}
= \overline{−x′ + [[a, b], y] + x′} + \overline{[x′, y]}.  (3.4)

[[a, b], y] ∈ G_{m+n+1} ⊂ G_{m+n}. Applying the condition (a), we obtain

\overline{−x′ + [[a, b], y] + x′} = \overline{[[a, b], y]} = 0 in \overline{G}_{m+n}.

Thus from (3.4) we have \overline{[x, y]} = \overline{[x′, y]}. If x−x′ = (a, b) ∈ [G_{i+1}, G_{m−i}] ⊂ G_{m+1}, then by the same argument we have

\overline{[x, y]} = \overline{[x′ + (a, b), y]} = \overline{−x′ + [(a, b), y] + x′} + \overline{[x′, y]}
= \overline{[(a, b), y]} + \overline{[x′, y]} = \overline{[x′, y]}, since \overline{[(a, b), y]} = 0 in \overline{G}_{m+n}.

The correctness of β_{mn} for the second argument is proved in an analogous way.

Now we shall show that the maps β_{mn} are bilinear. Let \overline{x}_{1}, \overline{x}_{2} ∈ \overline{G}_{m} and \overline{y} ∈ \overline{G}_{n}. We have in \overline{G}_{m+n}

[\overline{x}_{1} + \overline{x}_{2}, \overline{y}] = \overline{[x_{1} + x_{2}, y]} = \overline{−x_{2} + [x_{1}, y] + x_{2}}
+ \overline{[x_{2}, y]} = \overline{[x_{1}, y]} + \overline{[x_{2}, y]};

here we again apply the condition (a). Let \overline{x} ∈ \overline{G}_{m} and \overline{y}_{1}, \overline{y}_{2} ∈ \overline{G}_{n}. We have in \overline{G}_{m+n}

[\overline{x}, \overline{y}_{1} + \overline{y}_{2}] = \overline{[x, y_{1} + y_{2}]} = \overline{[x, y_{1}] + [x^{y_{1}}, y_{2}]}
= \overline{[x, y_{1}]} + \overline{[x^{y_{1}}, y_{2}]} = [\overline{x}, \overline{y}_{1}] + [\overline{x^{y_{1}}}, \overline{y}_{2}] = [\overline{x}, \overline{y}_{1}] + [\overline{x}, \overline{y}_{2}],

since, by the condition (a) \overline{x^{y_{1}}} = \overline{x}. This proves that maps β_{mn} are bilinear.

(c) The maps α_{mn}, β_{mn} can be continued linearly in a natural way up to the bilinear maps ( , ), [ , ] : LL_{G} × LL_{G} ⟶ LL_{G}. The proof of the fact that ( , ) satisfies the condition (3.2) and (l, l) = 0 for any l ∈ LL_{G} is similar to the proof of the corresponding statement in Witt’s theorem (see [5], Proposition 2.3; [6]). It remains to show that the square bracket operation [ , ] satisfies the Leibniz identity (3.3).

The object G satisfies Condition 1, therefore we have Condition 1′ for the square bracket in G. Since the square bracket operation in LL_{G} is linear for both arguments, we can limit ourself to the case where \overline{x} ∈ G_{m}, \overline{y} ∈ \overline{G}_{n}, \overline{z} ∈ \overline{G}_{t}. Applying the conditions (a) and (b) of the theorem we have

[\overline{x}, [\overline{y}, \overline{z}]] = [\overline{x^{y}}, [\overline{y}, \overline{z}]] = \overline{[x^{y}, [y, z]]};
[[\overline{x}, \overline{y}], \overline{z}] = [[\overline{x}, \overline{y}], \overline{z^{x}}] = \overline{[[x, y], z^{x}]};
−[[\overline{x}, \overline{z}], \overline{y}] = [−[\overline{x}, \overline{z}], \overline{y^{z}}] = \overline{[−[x, z], y^{z}]}.

By Condition 1′ we obtain

[\overline{x}, [\overline{y}, \overline{z}]] = [[\overline{x}, \overline{y}], \overline{z}] − [[\overline{x}, \overline{z}], \overline{y}] in \overline{G_{m+n+t}},

which completes the proof of the theorem. □

The following example is due to the referee.

Example. Let G be the abelian group of integers ℤ^{•}, which acts on itself in the following way: x^{y} = (−1)^{y}x. We have [x, y] = 0 for y even, [x, y] = −2x for y odd and G_{n} = 2^{n−1}ℤ^{•}. It is easy to see that ℤ^{•} ∈ 𝔾𝕣 ^{c} and LL_{ℤ^{•}} is a free Leibniz algebra generated by a single element over a two element field (see also [4]).

It is easy to see that by Theorem 3.6 we have actually constructed the functor LL : 𝔾𝕣 ^{c} ⟶ 𝕃𝕃. In an analogous way one can construct the functor L : 𝔸𝕓 ^{c} ⟶ 𝕃𝕖𝕚𝕓𝕟𝕚𝕫 . For A ∈ 𝕃𝕃 let S_{1}(A) denote the greatest quotient algebra of A which makes square bracket in A trivial. Then S_{1}(A) ∈ 𝕃𝕚𝕖 and we have a functor S_{1} : 𝕃𝕃 ⟶ 𝕃𝕚𝕖. Similarly, we construct the functor S_{2} : 𝕃𝕃 ⟶ 𝕃𝕖𝕚𝕓𝕟𝕚𝕫 . S_{1} and S_{2} are left adjoints to the embedding functors E_{1} and E_{2} respectively. Denote by W : 𝔾𝕣 ⟶ 𝕃𝕚𝕖 the functor defined by Witt’s theorem [6], [5]. Thus we have the following functors between the well defined categories:

𝔸𝕓 ^{c}  ⟶^{E} ; ⟵_{A} 𝔾𝕣 ^{c}  ⟶^{Q_{1}} ; ⟵^{T} ; ⟶^{Q_{2}} ; ⟵^{C} 𝔾𝕣
L↓  ↓^{LL}  ↓^{W}
𝕃𝕖𝕚𝕓𝕟𝕚𝕫 ⟶^{E_{2}} ; ⟵_{S_{2}} 𝕃𝕃 ⟵^{E_{1}} ; ⟶_{S_{1}} 𝕃𝕚𝕖,

where LL ∘ C = E_{1} ∘ W, E_{2} ∘ L = LL ∘ E. A more detailed account of this diagram will be given in the forthcoming paper, where free objects in 𝔾𝕣 ^{•} and free Leibniz algebras are studied.

Acknowledgment

I would like to express my gratitude to J.-L. Loday who suggested the consideration of this problem. The work was partially supported by the grants: PICS CNRS Franco-Georgien de Mathematiques 542, TMR-Network “K-Theory and algebraic groups” ERB FMRX CT-97-0107 and INTAS-97-31961.

References

1. H. J. Baues and D. Conduché, The central series for Peiffer commutators in groups with operators. J. Algebra 133(1990), No. 1, 1–34.

2. A. G. Kurosh, Lectures in general algebra. (Russian) Nauka, Moscow, 1973.

3. J.-L. Loday, Cyclic homology. Grundlehren der mathematischen Wissenschaften, 301, Springer, 1992.

4. J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296(1993), 139–158.

5. J.-P. Serre, Lie algebras and Lie groups. W. A. Benjamin, INC. London, Amsterdam, Tokyo, 1965.

6. E. Witt, Treue Darstellung Liescher Ringe. J. Reine Angew. Math. 1(1937), 152–160.

(Received 4.06.2002)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: tamar@rmi.acnet.ge