LEIBNIZ n-ALGEBRAS

J. M. CASAS,^{(1)} J.-L. LODAY ^{(2)} and T. PIRASHVILI^{(3)}

(1) Dpto Matematica Aplicada, Universidad de Vigo, 36005 Pontevedra, Spain

(2) IRMA, ULP et CNRS, 7, rue R. Descartes, 67084 Strasbourg, France (3) Math. Inst., Alexidze str. 1, 380093 Tbilisi, Republic of Georgia

Abstract. A Leibniz n-algebra is a vector space equipped with an n-ary operation which has the property of being a derivation for itself. This property is crucial in Nambu mechanics. For n = 2 this is the notion of Leibniz algebra. In this paper we prove that the free Leibniz (n+1)-algebra can be described in terms of the n-magma, that is the set of n-ary planar trees. Then it is shown that the n-tensor power functor, which makes a Leibniz (n + 1)-algebra into a Leibniz algebra, sends a free object to a free object. This result is used in the last section, together with former results of Loday and Pirashvili, to construct a small complex which computes Quillen cohomology with coefficients for any Leibniz n-algebra .

Mathematics Subject Classification (2000): 17Axx, 70H05.

1. Introduction

Leibniz algebras were introduced by the second author in [4]. They play an important role in Hochschild homology theory [4], [5] as well as in Nambu mechanics ([6], see also [1]). Let us recall that a Leibniz algebra is a vector space g equipped with a bilinear map [−, −] : g ⊗ g → g satisfying the identity :

(1.1)  [x, [y, z]] = [[x, y], z] − [[x, z], y].

One easily sees that Lie algebras are exactly Leibniz algebras satisfying the relation [x, x] = 0. Hence Leibniz algebras are a non-commutative version of Lie algebras.

Recently there have been several works dealing with various generalization of Lie structures by extending the binary bracket to an n-bracket (see [1], [2], [9]).

In this paper we introduce the notion of a Leibniz n-algebra — a natural generalization of both concepts. For n = 2 one recovers Leibniz algebras. Any Leibniz algebra g is also a Leibniz n-algebra under the following nbracket:

[x_{1}, x_{2}, ⋯, x_{n}] := [x_{1}, [x_{2}, ⋯, [x_{n−1}, x_{n}] ⋯]].

Conversely, if L is a Leibniz (n + 1)-algebra, then on D_{n}(L) = L^{⊗n} the following bracket

[a_{1} ⊗ ⋯ ⊗ a_{n}, b_{1} ⊗ ⋯ ⊗ b_{n}] := ∑_{i=1}^{n} a_{1} ⊗ ⋯ ⊗ [a_{i}, b_{1}, ⋯, b_{n}] ⊗ ⋯ ⊗ a_{n}

makes D_{n}(L) into a Leibniz algebra. This construction goes back to Gautheron [2] and plays an important role in [1].

The main result of this paper is to show that if L is a free Leibniz nalgebra , then D_{n−1}(L) is a free Leibniz algebra too (on a different vector space). This result plays an essential role in the cohomological investigation of Leibniz n-algebras, which we consider in the last section.

We first introduce a notion of representation of a Leibniz n-algebra L. This notion for n = 2 was already considered in [5]. One observes that if M is a representation of a Leibniz n-algebra L, then Hom(L, M) can be considered as a representation of the Leibniz algebra D_{n−1}(L). The work of [2] and [1] suggests to define the cohomology of L with coefficients in M to be HL^{∗}(D_{n−1}(L), Hom(L, M)). We deduce from our main theorem that this theory is exactly the Quillen cohomology for Leibniz n-algebras.

2. Derivations

In the whole paper K is a field. All tensor products are taken over K. Let A be a vector space equipped with an n-linear operation ω : A^{⊗n} → A. A map f : A → A is called a derivation with respect to ω if

f(ω(a_{1}, …, a_{n})) = ∑_{i=1}^{n} ω(a_{1}, …, f(a_{i}), ⋯, a_{n}).

In this case we also say that f is an ω-derivation. We let Der_{ω}(A) be the set of all ω-derivations. The following is well-known.

Proposition 2.1 i) The subset Der_{ω}(A) of the Lie algebra of endomorphisms End(A) is a Lie subalgebra. ii) If f ∈ Der_{ω}(A) and f ∈ Der_{σ}(A), then f ∈ Der_{ω+σ}(A). Here σ : A^{⊗n} → A is also an n-linear operation.

Proposition 2.2 Let [−, −] : A^{⊗2} → A be a bilinear operation and let ω : A^{⊗n} → A be given by

ω(x_{1}, ⋯, x_{n}) := [x_{1}, [x_{2}, ⋯, [x_{n−1}, x_{n}] ⋯]].

If f is a derivation with respect to [−, −], then f ∈ Der_{ω}(A).

Proof. One has

f(ω(x_{1}, ⋯, x_{n})) = f([x_{1}, [x_{2}, ⋯, [x_{n−1}, x_{n}] ⋯]])

= [f(x_{1}), [x_{2}, ⋯, [x_{n−1}, x_{n}] ⋯]]) + [x_{1}, f([x_{2}, ⋯, [x_{n−1}, x_{n}] ⋯])] =

= ∑_{i} [x_{1}, ⋯, [f(x_{i}), ⋯, x_{n}]] = ∑_{i} ω(x_{1}, ⋯, f(x_{i}), ⋯, x_{n}).

The following is an immediate generalization of Proposition 2.2. Since it has the same proof, we omit it here.

Proposition 2.3 Let ω_{i} : A^{⊗n_{i}} → A be n_{i}-ary operations for i = 1, …, k and let ω : A^{⊗k} → A be a k-ary operation. If f is a derivation with respect to ω_{1}, …, ω_{n}, ω, then it is also a derivation with respect to the composite σ : A^{⊗n} → A. Here n = n_{1} + … + n_{k},

σ(a_{1}, …, a_{n}) := ω(ω_{1}(a_{1}, …, a_{n_{1}}), …, ω_{k}(a_{s}, …, a_{n}))

and s = n − n_{k} + 1 = n_{1} + … + n_{k−1} + 1.

Proposition 2.4 Let ω : A^{⊗(n+1)} → A be an (n + 1)-linear map and let μ_{i} : g ⊗ g → g be the bilinear map given by

μ_{i}(a_{1} ⊗ … ⊗ a_{n}, b_{1} ⊗ … ⊗ b_{n}) = a_{1} ⊗ ⋯ ⊗ ω(a_{i}, b_{1} ⊗ … ⊗ b_{n}) ⊗ ⋯ ⊗ a_{n}.

Here g = A^{⊗n} and 1 ≤ i ≤ n. Suppose that f : A → A is an ω-derivation and φ : g → g is given by

φ(a_{1} ⊗ … ⊗ a_{n}) = ∑_{j=1}^{n} a_{1} ⊗ … ⊗ f(a_{j}) ⊗ ⋯ ⊗ a_{n}.

Then φ is a derivation with respect to μ_{i}, for any 1 ≤ i ≤ n.

Proof. One has

φ(μ_{i}(a_{1} ⊗…⊗a_{n}, b_{1} ⊗…⊗b_{n})) = φ(a_{1} ⊗⋯⊗ω(a_{i}, b_{1} ⊗…⊗b_{n})⊗⋯⊗a_{n}).

Since f is an ω-derivation, we see that this expression is equal to

f(a_{1}) ⊗ ⋯ ⊗ ω(a_{i}, b_{1} ⊗ … ⊗ b_{n}) ⊗ ⋯ ⊗ a_{n} + ⋯ +
+a_{1} ⊗ ⋯ ⊗ ω(f(a_{i}), b_{1} ⊗ … ⊗ b_{n}) ⊗ ⋯ ⊗ a_{n}+

+a_{1} ⊗ ⋯ ⊗ ω(a_{i}, f(b_{1}) ⊗ … ⊗ b_{n}) ⊗ ⋯ ⊗ a_{n} + ⋯ +
+a_{1} ⊗ ⋯ ⊗ ω(a_{i}, b_{1} ⊗ … ⊗ f(b_{n})) ⊗ ⋯ ⊗ a_{n} + ⋯ +
+a_{1} ⊗ ⋯ ⊗ ω(a_{i}, b_{1} ⊗ … ⊗ b_{n}) ⊗ ⋯ ⊗ f(a_{n}).

On the other hand the expression

μ_{i}(φ(a_{1} ⊗ … ⊗ a_{n}), b_{1} ⊗ … ⊗ b_{n}) + μ_{i}(a_{1} ⊗ … ⊗ a_{n}, φ(b_{1} ⊗ … ⊗ b_{n})),

is clearly equal to the previous expression thanks to the definition of μ_{i} and φ. This proves that φ is a derivation with respect to μ_{i}.

3. Leibniz n-algebras

A Leibniz algebra of order n, or simply a Leibniz n-algebra, is a vector space L equipped with an n-linear operation [−, …, −] : L^{⊗n} → L such that for all x_{1}, …, x_{n−1} the map ad(x_{1}, …, x_{n−1}) : L → L given by ad(x_{1}, ⋯, x_{n−1})(x) = [x, x_{1}, …, x_{n−1}] is a derivation with respect to [−, …, −]. This means that the following Leibniz n-identity holds:

(3.1)  [[x_{1}, x_{2}, …, x_{n}], y_{1}, y_{2}, …, y_{n−1}] =

∑_{i=1}^{n} [x_{1}, …, x_{i−1}, [x_{i}, y_{1}, y_{2}, …, y_{n−1}], x_{i+1}, ⋯, x_{n}].

We let _{n}Lb be the category of Leibniz n-algebras. Let us observe that for n = 2 the identity (3.1) is equivalent to (1.1). So a Leibniz 2-algebra is simply a Leibniz algebra in the sense of [4], and so Leibniz 2-algebras are called just Leibniz algebras, and we use Lb instead of _{2}Lb.

Clearly a Lie algebra is a Leibniz algebra such that [x, x] = 0 holds. Similarly for n ≥ 3 an n-Lie or an n-Nambu-Lie algebra is a Leibniz nalgebra such that [x_{1}, …, x_{i}, x_{i+1}, …, x_{n}] = 0 as soon as x_{i} = x_{i+1} for 1 ≤ i ≤ n − 1. Such algebras appear in the so called Nambu mechanics and there exists several interesting papers about them (see [1], [2] and references given there).

Another big class of Leibniz 3-algebras which were considered in the literature are the so called Lie triple systems. Let us recall that a Lie triple system [3] is a vector space equipped with a bracket [−, −, −] that satisfies the same identity (3.1) and, instead of skew-symmetry, satisfies the conditions

[x, y, z] + [y, z, x] + [z, x, y] = 0

and

[x, y, y] = 0.

Proposition 3.2 Let g be a Leibniz algebra. Then g is also a Leibniz (n + 1)-algebra with respect to the operation ω : g^{⊗(n+1)} → g given by

ω(x_{0}, x_{1}, ⋯, x_{n}) := [x_{0}, [x_{1}, ⋯ [x_{n−1}, x_{n}]].

Proof. From the definition of Leibniz algebra we know that

ad(x) = [−, x] : g → g

is a derivation with respect to 2-bracket. By Proposition 2.2 we know that it is also a derivation with respect to ω. Since for all x_{1}, x_{2}, ⋯, x_{n} ∈ g one has ad(x_{1}, x_{2}, ⋯, x_{n}) = [x_{1}, ⋯ [x_{n−1}, x_{n}] ⋯] the Proposition follows. Here ad(x_{1}, x_{2}, ⋯, x_{n}) = ω(−, x_{1}, x_{2}, ⋯, x_{n}) : g → g.

Proposition 3.2 shows that there exists a “forgetful” functor

U_{n} : Lb → _{n}Lb.

Here are more examples of Leibniz 3-algebras. Examples 3.3 i) Let g be a Leibniz algebra with involution σ. This means that σ is an automorphism of g and σ^{2} = id. Then

L := {x ∈ g | x + σ(x) = 0}

is a Leibniz 3-algebra with respect to the bracket

[x, y, z] := [x, [y, z]].

ii) Let V be a 4-dimensional vector space with basis i, j, k, l. Then we define [x, y, z] := det(A), where A is the following matrix

( \begin{matrix}i & j & k & l \\ x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \\ z_{1} & z_{2} & z_{3} & z_{4} \end{matrix} )

One sees that this gives rise to a Leibniz 3-algebra. Moreover it is a Nambu-Lie algebra. Here x = x_{1}i + x_{2}j + x_{3}k + x_{4}l and so on. One easily generalizes this example to obtain an n-Nambu-Lie algebra starting with an (n + 1)-dimensional vector space. This example was a starting point for investigating n-Lie (or Nambu-Lie) algebras.

Let L be a Leibniz n-algebra. Thanks to Proposition 2.1 i) we know that

Der(L) = {f : L → L | f is a derivation}

is a Lie algebra.

Proposition 3.4. Let L be a Leibniz (n + 1)-algebra. Then D_{n}(L) = L^{⊗n} is a Leibniz algebra with respect to the bracket

[a_{1} ⊗ ⋯ ⊗ a_{n}, b_{1} ⊗ ⋯ ⊗ b_{n}] := ∑_{i=1}^{n} a_{1} ⊗ ⋯ ⊗ [a_{i}, b_{1}, ⋯, b_{n}] ⊗ ⋯ ⊗ a_{n}

Moreover

ad : L^{⊗n} → Der(L), x_{1} ⊗ ⋯ ⊗ x_{n} ↦ ad(x_{1}, x_{2}, ⋯, x_{n})

is a homomorphism of Leibniz algebras.

Proof. Fix x_{1}, ⋯, x_{n} ∈ L. We have to prove that

φ : D_{n}(L) → D_{n}(L)

given by

φ(a_{1} ⊗ … ⊗ a_{n}) = ∑_{j=1}^{n} a_{1} ⊗ … ⊗ f(a_{j}) ⊗ ⋯ ⊗ a_{n},

is a derivation with respect to [−, −]. Here f = [−, x_{1}, ⋯, x_{n}] : L → L. Thanks to Proposition 2.4 we know that φ is a derivation with respect of all μ_{i}, 1 ≤ i ≤ n, where

μ_{i}(a_{1} ⊗ … ⊗ a_{n}, b_{1} ⊗ … ⊗ b_{n}) = a_{1} ⊗ ⋯ ⊗ ω(a_{i}, b_{1} ⊗ … ⊗ b_{n}) ⊗ ⋯ ⊗ a_{n}.

Then φ is also a derivation with respect to [−, −] = ∑_{i=1}^{n} μ_{i} thanks to Proposition 2.1 ii) and the first part of the Proposition follows. Let us show that ad is a homomorphism of Leibniz algebras. Indeed, one sees that

ad([a_{1} ⊗ ⋯ ⊗ a_{n}, b_{1} ⊗ ⋯ ⊗ b_{n}])(a) = ∑_{i=1}^{n}[a, [a_{1}, ⋯, [a_{i}, b_{1}, ⋯, b_{n}], ⋯, a_{n}]].

On the other hand

[ad(a_{1} ⊗ ⋯ ⊗ a_{n}), ad(b_{1} ⊗ ⋯ ⊗ b_{n})](a) = = ad(a_{1} ⊗⋯⊗a_{n})ad(b_{1} ⊗⋯⊗b_{n})(a)−ad(b_{1} ⊗⋯⊗b_{n})ad(a_{1} ⊗⋯⊗a_{n})(a)

= [[a, [b_{1}, ⋯, b_{n}]], a_{1} ⊗ ⋯ ⊗ a_{n}] − [[a, a_{1} ⊗ ⋯ ⊗ a_{n}], b_{1}, ⋯, b_{n}].

Therefore (3.1) shows that

ad([a_{1} ⊗ ⋯ ⊗ a_{n}, b_{1} ⊗ ⋯ ⊗ b_{n}]) = [ad(a_{1} ⊗ ⋯ ⊗ a_{n}), ad(b_{1} ⊗ ⋯ ⊗ b_{n})].

Hence ad : D_{n}L → Der(L) is a homomorphism of Leibniz algebras.

Remark 3.5 One can prove that if A is a Leibniz (kn + 1)-algebra, then A^{⊗k} is a Leibniz (n + 1)-algebra with respect to the following bracket

[x_{01} ⊗ x_{02} … ⊗ x_{0k}, …, x_{n1} ⊗ x_{n2} … ⊗ x_{nk}] :=
[x_{01}, x_{11}, …, x_{1k}, …, x_{n1}, …, x_{nk}] ⊗ x_{02} ⊗ … ⊗ x_{0k}+
⋯ + x_{01} ⊗ … ⊗ x_{0k−1} ⊗ [x_{0k}, x_{11}, …, x_{nk}].

By Proposition 3.4 the map L ↦ D_{n}(L) from Leibniz (n + 1)-algebras to Leibniz algebras is a functor that we denote by D_{n}. More generally, by Remark 3.5, there exist functors D_{kn}^{n} : _{kn+1}Lb → _{n+1}Lb (so D_{n} = D_{n}^{1}) and we have D_{n}^{1} ∘ D_{kn}^{n} = D_{kn}^{1}.

4. The main theorem

The goal of this section is to prove that the functor D_{n} : _{n+1}Lb → Lb sends free objects to free objects. For more specific statements see Theorem 4.4 and Theorem 4.8 below. Since D_{1} is nothing but the identity functor, we have to consider the case n ≥ 2. To avoid long formulas we will first restrict ourself to the case n = 2 and, second, we indicate how to modify the argument for n ≥ 3.

Let us recall (see [8]) that a magma M is a set together with a map (binary operation)

M × M → M , (x, y) ↦ x ⋆ y.

Let Y be the free magma with one generator e. We recall from [8] the construction of Y. First one defines the sequence of sets (Y_{m})_{m≥1} as follows:

Y_{1} = {e}, Y_{m} = ∐_{p+q=m} Y_{p} × Y_{q}, (m ≥ 2; p, q ≥ 1).

We let Y be the disjoint union

Y = ∐_{m≥1} Y_{m}.

One defines ⋆ : Y × Y → Y by means of

Y_{p} × Y_{q} → Y_{p+q} ⊆ Y.

Then Y is a magma, which is freely generated by e. Let C_{m} be the number of elements of Y_{m+1}. Clearly C_{0} = 1, C_{1} = 1 and

C_{m+1} = ∑_{i+j=m} C_{i}C_{j}.

Hence the function f(t) = ∑_{m=0}^{∞} C_{m}t^{m} satisfies the functional equation

(4.1)  f(t) − 1 = tf^{2}(t).

Of course this equation is well known, as well as the fact that C_{m} is equal to the Catalan number, that is

C_{m} = \frac{(2m)!}{m!(m + 1)!}.

So one has

C_{0} = 1, C_{1} = 1, C_{2} = 2, C_{3} = 5, C_{4} = 14, C_{5} = 42, C_{6} = 132, …

If ω ∈ Y_{m}, then we say that ω is of length m and we write l(ω) = m. Clearly if ω ∈ Y, then ω = e or ω = ω_{1}⋆ω_{2}, with unique (ω_{1}, ω_{2}) ∈ Y ×Y. Moreover l(ω) = l(ω_{1}) + l(ω_{2}). Recall that the elements of Y_{n} can be interpreted as planar binary trees with n leaves. Under this interpretation the operation ⋆ is simply the grafting operation (join the roots to a new vertex and add a new root).

The following proposition is the analogue for Leibniz 3-algebras of Lemma 1.3 in [5] concerning Leibniz algebras.

Proposition 4.2 Let K[Y] be the vector space spanned by Y. Then there exists a unique structure of Leibniz 3-algebra on K[Y] such that

[ω_{1}, ω_{2}, e] = ω_{1} ⋆ ω_{2}.

Moreover K[Y] with this structure is a free Leibniz 3-algebra generated by e.

Proof. We use the method devised in [5] for the case of Leibniz algebras. Let us observe that (3.1) for Leibniz 3-algebras is equivalent to

(4.2.1)  [a, b, [c, x, y]] = [[a, b, c], x, y] − [[a, x, y], b, c] − [a, [b, x, y], c].

The 3-bracket [ω_{1}, ω_{2}, ω_{3}] has been already defined for ω_{3} = e. If ω_{3} ≠ e, then it is of the form ω ⋆ ω′ for some elements ω and ω′. Hence

[ω_{1}, ω_{2}, ω_{3}] = [ω_{1}, ω_{2}, ω ⋆ ω′] = [ω_{1}, ω_{2}, [ω, ω′, e]],

and one can use (4.2.1) to rewrite it with 3-brackets whose last variable is either ω or ω′. Since l(ω) and l(ω′) are less than l(ω_{3}), we get, by recursivity, the element [ω_{1}, ω_{2}, ω_{3}] as a unique algebraic sum of elements in Y.

We now have to prove that, with this definition, the 3-bracket satisfies the Leibniz 3-identity (4.2.1). Clearly it holds when y = e, since it is precisely this formula which was used to compute the left part. So we can work by induction with respect to l(y). If l(y) ≥ 2 then y = y_{1} ⋆ y_{2} and therefore

[a, b, [c, x, y]] = [a, b, [c, x, [y_{1}, y_{2}, e]]] =

[a, b, [[c, x, y_{1}], y_{2}, e]] − [a, b, [[c, y_{2}, e], x, y_{1}]] − [a, b, [c, [x, y_{2}, e], y_{1}]]

= [[a, b, [c, x, y_{1}]], y_{2}, e] − [[a, y_{2}, e], b, [c, x, y_{1}]] − [a, [b, y_{2}, e], [c, x, y_{1}]]−

[[a, b, [c, y_{2}, e]], x, y_{1}] + [[a, x, y_{1}], b, [c, y_{2}, e]] + [a, [b, x, y_{1}], [c, y_{2}, e]]−

[[[a, b, c], [x, y_{2}, e], y_{1}] + [[a, [x, y_{2}, e], y_{1}], b, c] − [a, [b, [x, y_{2}, c], y_{1}], c] =

[[a, b, c], x, y_{1}], y_{2}, e] − [[[a, x, y_{1}], b, c], y_{2}, e] − [[a, [b, x, y_{1}], c], y_{2}, e]

−[[[a, y_{2}, e], b, c], x, y_{1}] + [[[a, y_{2}, e], x, y_{1}], b, c] + [[a, y_{2}, e], [b, x, y_{1}], c]

−[[a, [b, y_{2}, e], c], x, y_{1}] + [[a, x, y_{1}], [b, y_{2}, e], c] + [a, [[b, y_{2}, e], x, y_{1}], c]

−[[[a, b, c], y_{2}, e], x, y_{1}] + [[[a, y_{2}, e], b, c], x, y_{1}] + [[a, [b, y_{2}, e], c], x, y_{1}]

+[[[a, x, y_{1}], b, c], y_{2}, e] − [[[a, x, y_{1}], y_{2}, e], b, c] − [[a, x, y_{1}], [b, y_{2}, e], c]

+[[a, [b, x, y_{1}], c], y_{2}, e] − [[a, y_{2}, e], [b, x, y_{1}], c] − [a, [[b, x, y_{1}], y_{2}, e], c]

−[[a, b, c], [x, y_{2}, e], y_{1}] + [[a, [x, y_{2}, e], y_{1}], b, c] + [a, [b, [x, y_{2}, e], y_{1}], c].

One sees that 2^{nd} and 13^{th}, as well as 3^{rd} and 16^{th}, 4^{th} and 11^{th}, 6^{th} and 17^{th}, 7^{th} and 12^{th}, 8^{th} and 15^{th} terms cancel. Hence we have

[a, b, [c, x, y]] =

[[[a, b, c], x, y_{1}], y_{2}, e] + [[[a, y_{2}, e], x, y_{1}], b, c] + [a, [[b, y_{2}, e], x, y_{1}], c]

−[[[a, b, c], y_{2}, e], x, y_{1}] − [[[a, x, y_{1}], y_{2}, e], b, c] − [a, [[b, x, y_{1}], y_{2}, e], c]

−[[a, b, c], [x, y_{2}, e], y_{1}] + [[a, [x, y_{2}, e], y_{1}], b, c] + [a, [b, [x, y_{2}, e], y_{1}], c].

On the other hand we have

[[a, b, c], x, y] =

[[[a, b, c], x, y_{1}], y_{2}, e] − [[[a, b, c], y_{2}, e], x, y_{1}] − [[a, b, c], [x, y_{2}, e], y_{1}].

Similarly

−[[a, x, y], b, c] =
−[[[a, x, y_{1}], y_{2}, e], b, c] + [[[a, y_{2}, e], x, y_{1}], b, c] + [[a, [x, y_{2}, e], y_{1}], b, c]

and

−[a, [b, x, y], c] =

−[a, [[b, x, y_{1}], y_{2}, e], c] + [a, [[b, y_{2}, e], x, y_{1}], c] + [a, [b, [x, y_{2}, e], y_{1}], c].

One checks that after substitution in (3.1) all terms cancel and therefore K[Y] has a well defined structure of Leibniz 3-algebra. If L is any Leibniz 3-algebra and x ∈ L, then by induction one can check that there exists a unique homomorphism

f : K[Y] → L

such that f(e) = x and Proposition 4.2 is proved.

Now we can formulate the following

Theorem 4.3. The vector space spanned by the set Ȳ = Y − {e} has a unique Leibniz algebra structure such that

[x ⋆ y, z ⋆ e] = (x ⋆ z) ⋆ y + x ⋆ (y ⋆ z).

It is a free Leibniz algebra over the set Y′ = {x ⋆ e | x ∈ Y } ⊂ Ȳ. Moreover one has isomorphisms of Leibniz algebras

D_{2}(K[Y]) ≅ K[Ȳ] ≅ T (K[Y′]).

Proof. 1^{st} Step: Uniqueness. Assume that such a Leibniz algebra structure exists. Then ad(u) = [−, u] is uniquely determined when u = x ⋆ e. We will prove by induction on l(q) that ad(u) is uniquely determined when u = x ⋆ q. If l(q) > 1, then q = y ⋆ z and by assumption

u = [x ⋆ y, z ⋆ e] − (x ⋆ z) ⋆ y.

Therefore

ad(u) = −ad((x ⋆ z) ⋆ y) + ad[x ⋆ y, z ⋆ e] =
= −ad((x ⋆ z) ⋆ y) + ad(z ⋆ e) · ad(x ⋆ y) − ad(x ⋆ y) · ad(z ⋆ e)

and by induction assumption ad(u) is uniquely determined.

2^{nd} Step: Bijection. There is a linear isomorphism K[Y]⊗K[Y] ≅ K[Ȳ], which is given by

(x, y) ↦ x ⋆ y, for x, y ∈ Y.

It yields indeed a bijection because

K[Y] ⊗ K[Y] ≅ ⨁_{p,q≥1} K[Y_{p}] ⊗ K[Y_{q}] ≅ ⨁_{p,q≥1} K[Y_{p} × Y_{q}]

≅ ⨁_{m≥2} ⨁_{p+q=m} K[Y_{p} × Y_{q}] ≅ ⨁_{m≥2} K[Y_{m}] = K[Ȳ].

3^{rd} Step: Algebra isomorphism D_{2}(K[Y]) ≅ K[Ȳ]. Let us consider K[Ȳ] as a Leibniz algebra induced by the linear isomorphism from Step 2. Since x ⋆ y and z ⋆ e are the images of x ⊗ y and z ⊗ e ∈ K[Y] ⊗ K[Y] under the isomorphism of Step 2, we have to show that, in this algebra, the following identity

[x ⋆ y, z ⋆ e] = (x ⋆ z) ⋆ y + x ⋆ (y ⋆ z)

holds. By definition of the functor D_{2} one has

[x ⊗ y, z ⊗ e] = [x, z, e] ⊗ y + x ⊗ [y, z, e] = (x ⋆ z) ⊗ y + x ⊗ (y ⋆ z)

and this element goes to (x ⋆ z) ⋆ y + x ⋆ (y ⋆ z) under the isomorphism of Step 2. This proves also the existence part of the Theorem.

4^{th} Step: Y′ generates K[Ȳ]. Indeed let X be the subalgebra of K[Ȳ] generated by Y′. We have to prove that Ȳ ⊂ X. Let x ⋆ y be an element in Ȳ. We will show by induction on l(y) that x ⋆ y ∈ X. When l(y) = 1, then x ⋆ y = x ⋆ e ∈ Y′ ⊂ X. If l(y) > 1 then y = y_{1} ⋆ z and by the assumption

x ⋆ y = −(x ⋆ z) ⋆ y_{1} + [x ⋆ y_{1}, z ⋆ e].

But, by the induction assumption, one has (x ⋆ z) ⋆ y_{1}, x ⋆ y_{1}, z ⋆ e ∈ X. Therefore x ⋆ y ∈ X as well.

5^{th} Step: K[Ȳ] as a graded Leibniz algebra. For x ∈ Ȳ we let d(x) to be l(x) − 1. Then K[Ȳ] can be considered as a graded vector space by declaring that the degree of an element x ∈ Ȳ is d(x). We claim that under this grading K[Ȳ] is a graded Leibniz algebra, that is, if d(x) = k and d(y) = m, then [x, y] is a linear combination of elements of degree k + m. The claim is clear when y = x′ ⋆ y′ and l(y′) = 1 and it can be proved by the same induction arguments as in Step 1.

6^{th} Step: Y′ freely generates K[Ȳ]. Let us recall from [5] that for a vector space U the free Leibniz algebra generated by U is the unique Leibniz algebra structure on

T̄(U) = ⨁_{m≥1} U^{⊗m}

such that for any u ∈ U one has

[x, u] = x ⊗ u, x ∈ T̄(U).

Take U = K[Y′]. Then we obtain the natural epimorphism

φ : T̄(K[Y′]) → K[Ȳ].

We have to show that φ is injective. The vector space K[Y′] is a graded subspace of K[Ȳ]. Therefore T̄(K[Y′]) is also graded and φ is a morphism of graded Leibniz algebras. Since each component is of finite dimension, it is enough to show that the degree p part of T̄(K[Y′]) and K[Ȳ] have the same dimension.

The dimension of the degree p part of K[Ȳ] is equal to C_{p} (Catalan number), while the dimension of the degree p part of T̄(K[Y′]) is equal to the coefficient of t^{p} in the expansion of

g(t) = ∑_{m=1}^{∞}(∑_{k=1}^{∞} C_{k−1}t^{k})^{m},

because the degree m part of K[Y′] is of dimension C_{m−1}. Hence we have to prove that g(t) = f(t) − 1. But

g(t) = ∑_{m=1}^{∞}(tf(t))^{m} = \frac{tf(t)}{1 − tf(t)}

and it is equal to f(t) − 1, thanks to (4.1).

Let us now show the following parametrized version of Theorem 4.3.

Theorem 4.4. Let V be a vector space, and put

F (V) := ⨁_{m≥1} K[Y_{m}] ⊗ V^{⊗2m−1}.

(i) There exists a unique Leibniz 3-algebra structure on F (V) such that

[ω_{1} ⊗ x_{1}, ω_{2} ⊗ x_{2}, e ⊗ x] = (ω_{1} ⋆ ω_{2}) ⊗ x_{1} ⊗ x_{2} ⊗ x,

where ω_{1} ∈ K[Y_{p}], x_{1} ∈ V^{⊗(2p−1)}, ω_{2} ∈ K[Y_{q}], x_{2} ∈ V^{⊗(2q−1)} and x ∈ V.

(ii) Equipped with this structure F (V) is a free Leibniz 3-algebra generated by V.

(iii) The Leibniz algebra D_{2}(F (V)) is isomorphic to the free Leibniz algebra generated by the vector space

E = ⨁_{m≥1} K[Y_{m}] ⊗ V^{⊗2m}.

Proof. The proof is similar to the proof of Theorem 4.2. At the end, in order to show that the vector spaces F (V) and T (E) have the same dimension we use the following identity of formal power series:

(x+x^{3} +2x^{5} +5x^{7} +14x^{9} +⋯)^{2} = ∑_{m=1}^{∞}(x^{2} +x^{4} +2x^{6} +5x^{8} +14x^{10} +⋯)^{m},

which is an immediate consequence of the functional equation (4.1).

Let us now state the results for (n + 1)-Leibniz algebras. Since the proofs follow the same pattern as in the case n + 1 = 2 + 1, we mention only the main modifications. By definition an n-magma is a set M together with a map (n-ary operation)

(−, ⋯, −) : \underbrace{M × ⋯ × M}_{n copies} → M .

Let Z be the free n-magma with one generator e. It can be described as follows. The sequence of sets (Z_{m})_{m≥1} is given by:

Z_{1} = {e}, Z_{m} = ∐_{p_{1}+···+p_{n}=m} Z_{p_{1}} × ⋯ × Z_{p_{n}}, (m ≥ 2; p_{i} ≥ 1).

Observe that Z_{m} = ∅ unless m = (n − 1)k + 1 for some k ≥ 0. We let Z be the disjoint union

Z = ∐_{m≥1} Z_{m}.

One defines (−, ⋯, −) : Z × ⋯ × Z → Z by means of

Z_{(n−1)k_{1}+1} × ⋯ × Z_{(n−1)k_{n}+1} → Z_{(n−1)(k_{1}+···+k_{n}+1)+1} ⊆ Z.

Then Z is a n-magma, which is freely generated by e. Let D_{k} be the number of elements of Z_{(n−1)k+1}. Clearly D_{0} = 1, D_{1} = 1 and

D_{k+1} = ∑_{k_{1}+···+k_{n}=k} D_{k_{1}} ⋯ D_{k_{n}}.

Hence the function f(t) = ∑_{k=0}^{∞} D_{k}t^{k} satisfies the functional equation

(4.5)  f(t) − 1 = tf(t)^{n}.

If ω ∈ Z_{(n−1)k+1}, then we say that ω is of length k and we write l(ω) = k. Clearly if ω ∈ Z, then ω = e or ω = (ω_{1}, ⋯, ω_{n}), for some unique elements ω_{1}, ⋯, ω_{n} in Z. Moreover l(ω) = l(ω_{1}) + ⋯ + l(ω_{n}) + 1. Recall that the elements of Z_{m} can be interpreted as n-ary planar trees, that is each vertex has one root and n leaves. Under this interpretation the operation (−, ⋯, −) is simply the grafting operation. Observe that the number of vertices (resp. edges) of a tree in Z_{(n−1)k+1} is k (resp. kn + 1).

Proposition 4.6 Let K[Z] be the vector space spanned by Z. Then there exists a unique structure of (n + 1)-Leibniz algebra on K[Z] such that

[ω_{1}, ⋯, ω_{n}, e] = (ω_{1}, ⋯, ω_{n}).

Moreover K[Z] with this structure is a free (n+1)-Leibniz algebra generated by e.

Theorem 4.7. The vector space spanned by the set Z̄ = Z − {e} has a unique Leibniz algebra structure such that

[(ω_{1}, ⋯, ω_{n}), (ω′_{1}, ⋯, ω′_{n−1}, e)] = ∑_{i=1}^{n}(ω_{1}, ⋯, (ω_{i}, ω′_{1}, ⋯, ω′_{n−1}), ⋯, ω_{n}).

As a Leibniz algebra it is free over the Z′ = {(ω_{1}, ⋯, ω_{n−1}, e) | ω_{i} ∈ Z}. Moreover one has isomorphisms of Leibniz algebras:

D_{n}(K[Z]) ≅ K[Z̄] ≅ T (K[Z′]).

Proof. Let us just mention the computation of the dimensions of the vector spaces. Let E_{k} be the number of n-ary trees with k vertices which are of the form (ω_{1}, ⋯, ω_{n−1}, e). One has

E_{k} = ∑_{k_{1}+···+k_{n−1}+1=k} D_{k_{1}} × ⋯ × D_{k_{n−1}}.

Hence we get ∑ E_{k}t^{k} = t ∑ D_{k_{1}}t^{k_{1}} ⋯ D_{k_{n−1}}t^{k_{n−1}} = tf(t)^{n−1}. Therefore the generating series for T (K[Z′]) is \frac{tf(t)^{n−1}}{1−tf(t)^{n−1}} . By the functional equation (4.6) this is equal to f(t) − 1, which is the generating series for K[Z̄].

Theorem 4.8. Let V be a vector space, and put

F (V) := ⨁_{k≥0} K[Z_{(n−1)k+1}] ⊗ V^{⊗nk+1}.

(i) There exists a unique (n+1)-Leibniz algebra structure on F (V) such that

[ω_{1} ⊗ x_{1}, ⋯, ω_{n} ⊗ x_{n}, e ⊗ x] = (ω_{1}, ⋯, ω_{n}) ⊗ x_{1} ⊗ ⋯ ⊗ x_{n} ⊗ x,

where ω_{i} ∈ K[Z_{(n−1)k_{i}+1}], x_{i} ∈ V^{⊗nk_{i}+1} and x ∈ V.

(ii) Equipped with this structure F (V) is a free (n + 1)-Leibniz algebra generated by V.

(iii) The Leibniz algebra D_{n}(F (V)) is isomorphic to the free Leibniz algebra generated by the vector space

E = ⨁_{k≥0} K[Z_{(n−1)k+1}] ⊗ V^{⊗n(k+1)}.

Remark 4.9 By the same kind of argument one can show that the functor D_{kn}^{k} :_{kn+1} Lb →_{k+1} Lb sends free objects to free objects.

5. Cohomology of Leibniz n-algebras

An abelian extension of Leibniz n-algebras

(5.1)  0 → M → K → L → 0

is an exact sequence of Leibniz n-algebras such that [a_{1}, ⋯, a_{n}] = 0 as soon as a_{i} ∈ M and a_{j} ∈ M for some 1 ≤ i ≠ j ≤ n. Here a_{1}, ⋯, a_{n} ∈ K. Clearly then M is an abelian Leibniz n-algebra, that is the bracket vanishes on M. Let us observe that the converse is true only for n = 2.

If (5.1) is an abelian extension of Leibniz n-algebras, then M is equipped with n actions

[−, ⋯, −] : L^{⊗i} ⊗ M ⊗ L^{⊗n−1−i} → M, 0 ≤ i ≤ n − 1

satisfying (2n − 1) equations, which are obtained from (3.1) by letting exactly one of the variables x_{1}, ⋯, x_{n}, y_{1}, ⋯ y_{n−1} be in M and all the others in L.

By definition a representation of the Leibniz n-algebra L is a vector space M equipped with n actions of [−, ⋯, −] : L^{⊗i} ⊗ M ⊗ L^{⊗n−1−i} → M satisfying these (2n − 1) axioms. For example L is a representation of L. The notion of representation of a Leibniz n-algebra for n = 2 coincides with the corresponding notion given in [5]. Let L be a Leibniz n-algebra and let M be a representation of L. Let

(K)  0 → M → K → L → 0

be an abelian extension, such that the induced structure of representation of L on M induced by the extension is the prescribed one. If this condition holds, then we say that we have an abelian extension of L by M. Two such extensions (K) and (K′) are isomorphic when there exists a Leibniz n-algebra map from K to K′ which is compatible with the identity on M and on L. One denotes by Ext(L, M) the set of isomorphism classes of extensions of L by M.

Let f : L^{⊗n} → M be a linear map. We define an n-bracket on K = M ⊕ L by

[(m_{1}, x_{1}), (m_{2}, x_{2}) ⋯, (m_{n}, x_{n})] :=

(∑_{i=1}^{n} [x_{1}, ⋯, m_{i}, ⋯, x_{n}] + f(x_{1}, ⋯, x_{n}), [x_{1}, ⋯, x_{n}]).

Then K is a Leibniz n-algebra if and only if

(5.2)  f([x_{1}, ⋯, x_{n}], y_{1}, ⋯, y_{n−1}]) + [f(x_{1}, ⋯, x_{n}), y_{1}, ⋯, y_{n−1}] =

∑_{i=1}^{n} (f(x_{1}, ⋯, [x_{i}, y_{1} ⋯, y_{n−1}], ⋯, x_{n}]+[x_{1}, ⋯, f(x_{i}, y_{1} ⋯, y_{n−1}), ⋯, x_{n}])

for all x_{1}, ⋯, x_{n}, y_{1}, ⋯, y_{n−1} ∈ L. If this condition holds, then we obtain an extension

0 → M → K → L → 0

of Leibniz n-algebras. Moreover this extension is split in the category of Leibniz n-algebras if and only if there exists a linear map g : L → M such that

(5.3)  f(x_{1}, ⋯, x_{n}) = ∑_{i=1}^{n} [x_{1}, ⋯, g(x_{i}), ⋯, x_{n}] − g([x_{1}, ⋯, x_{n}]).

An easy consequence of these facts is the following natural bijection :

(5.4)  Ext(L, M) ≅ Z(L, M)/B(L, M).

Here Z(L, M) is the set of all linear maps f : L^{⊗n} → M satisfying (5.2) and B(L, M) is the set of such f which satisfy (5.3) for some k-linear map g : L → M.

Let L be a Leibniz n-algebra and let M be a representation of L. A map f : L → M is called a derivation if

f([a_{1}, ⋯, a_{n}]) = ∑_{i=1}^{n} [a_{1}, ⋯, f(a_{i}), ⋯, a_{n}].

We let Der(L, M) be the vector space of all derivations from L to M.

The next goal is to construct a cochain complex for Leibniz n-algebras so that the derivations and the elements of Z are cocycles in this complex. It turns out that this problem reduces to the case n = 2, that is for Leibniz algebras, which was the subject of the paper [5]. Let us recall the main construction of [5]. Let g be a Leibniz algebra and let M be a representation of g. We let CL^{∗}(g, M) be a cochain complex given by

CL^{m}(g, M) := Hom(g^{⊗m}, M), m ≥ 0,

where the coboundary operator d^{m} : CL^{m}(g, M) → CL^{m+1}(g, M) is defined by

(d^{m}f)(x_{1}, ⋯, x_{m+1}) := [x_{1}, f(x_{2}, ⋯, x_{m+1})]

+ ∑_{i=2}^{m+1}(−1)^{i}[f(x_{1}, ⋯, x̂_{i}, ⋯, x_{m+1}), x_{i}]

+ ∑_{1≤i<j≤m}(−1)^{j+1}f(x_{1}, ⋯, x_{i−1}, [x_{i}, x_{j}], x_{i+1}, ⋯, x̂_{j}, ⋯, x_{m}).

According to [5] cohomology of the Leibniz algebra g with coefficients in the representation M is defined by

HL^{∗}(g, M) := H^{∗}(CL^{∗}(g, M), d).

In order to generalize this notion to Leibniz n-algebras for n ≥ 3 we need the following Proposition. Let us recall that if L is an (n + 1)-Leibniz algebra, then the Leibniz algebra D_{n}(L) was defined in Section 3. Let L be an (n + 1)-Leibniz algebra and let M be a representation of L. One defines the maps

[−, −] : Hom(L, M) ⊗ D_{n}(L) → Hom(L, M)
[−, −] : D_{n}(L) ⊗ Hom(L, M) → Hom(L, M)

by

[f, x_{1} ⊗ ⋯ ⊗ x_{n}](x) := [f(x), x_{1}, ⋯, x_{n}] − f([x, x_{1}, ⋯, x_{n}]),

[x_{1}⊗⋯⊗x_{n}, f](x) := f([x, x_{1}, ⋯, x_{n}])−[f(x), x_{1}, ⋯, x_{n}]−⋯−[x, x_{1}, ⋯, f(x_{n})].

The proof of the next result is a straightforward (but somehow tedious) calculation.

Proposition 5.5 Let L be an (n + 1)-Leibniz algebra and let M be a representation of L. Then the above homomorphisms define a structure of representation of D_{n}(L) on Hom(L, M).

Let L be a Leibniz n-algebra and let M be a representation of L. One defines the cochain complex_{n}CL^{∗}(L, M) to be CL^{∗}(D_{n−1}(L), Hom(L, M)). We also put

nHL^{∗}(L, M) := H^{∗}(_{n}CL^{∗}(L, M))

Thus, by definition one has _{n}HL^{∗}(L, M) ≅ HL^{∗}(D_{n−1}(L), Hom(L, M)). Let us observe that for n = 2, one has _{2}CL^{m}(L, M) ≅ CL^{m+1}(L, M) for all m ≥ 0. Thus

2HL^{m}(L, M) ≅ HL^{m+1}(L, M), m ≥ 1

and _{2}HL^{0}(L, M) ≅ Der(L, M). Comparison of the definitions shows that

nHL^{0}(L, M) ≅ Der(L, M)

holds for any Leibniz n-algebras L. Similarly one has

Ker (d : _{n}CL^{1}(L, M) → _{n}CL^{2}(L, M)) ≅ Z(L, M)

and therefore

(5.6)  Ext(L, M) ≅ _{n}HL^{1}(L, M).

Proposition 5.7. Let L be a free n-Leibniz algebra and let M be a representation of L. Then

nHL^{m}(L, M) = 0, m ≥ 1.

Proof. The main result of Section 4 shows that D_{n−1}(L) is a free Leibniz algebra. Thanks to Corollary 3.5 of [5] we have HL^{i}(D_{n−1}(L), −) = 0 for i ≥ 2 and thus _{n}HL^{m}(L, −) = 0, m ≥ 1.

Let us recall that in [7] Quillen developped the cohomology theory in a very general framework. This theory can be applied to Leibniz n-algebras. It has the following description. Let L be a Leibniz n-algebra and let M be a representation of L. Then Quillen cohomology of L with coefficients in M is defined by

H_{Quillen}^{∗}(L, M) := H^{∗}(Der(P_{∗}, M)).

Here P_{∗} → L is an augmented simplicial n-Leibniz algebra, such that P_{∗} → L is a weak equivalence and each component of P_{∗} is a free Leibniz nalgebra.

Corollary 5.8. Let L be a Leibniz n-algebra and let M be a representation of L. Then

H_{Quillen}^{∗}(L, M) ≅ _{n}HL^{∗}(L, M).

Proof. Since P_{∗} → L is a weak equivalence, we obtain from the Künneth theorem that _{n}CL^{m}(P_{∗}, M) → _{n}CL^{m}(L, M) is also a weak equivalence. This fact, together with Proposition 5.7, shows that both spectral sequences for the bicomplex _{n}CL^{∗}(P_{∗}, M) degenerate and give the expected isomorphism.

Acknowledgments

The first author was supported by the Government of Galicia, grant XUGA 37101 A 97. The third author is very grateful to the University of Vigo and Université Louis Pasteur for hospitality. He was partially supported by the grant INTAS-93-2618-Ext and by the TMR network K-theory and algebraic groups, ERB FMRX CT-97-0107.

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