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About Leibniz cohomology and deformations of Lie algebras

We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding whether it has more Leibniz deformations than just the Lie ones. We also give the complete description of a Leibniz (and Lie) versal deformation of the 4-dimensional diamond Lie algebra, and study the case of its 5-dimensional analogue.


Introduction
Leibniz algebras, along with their Leibniz cohomologies, were introduced in [8] as a non antisymmetric version of Lie algebras. Lie algebras are special Leibniz algebras, and Pirashvili introduced [17] a spectral sequence, that, when applied to Lie algebras, measures the difference between the Lie algebra cohomology and the Leibniz cohomology. Now, Lie algebras have deformations as Leibniz algebras and those are piloted by the adjoint Leibniz 2-cocycles. In the present paper, we focus on the second Leibniz cohomology groups HL 2 (g, g), HL 2 (g, C) for adjoint and trivial representations of a complex Lie algebra g. We adopt a very elementary approach, not resorting to the Pirashvili sequence, to compare HL 2 (g, g) and HL 2 (g, C) to H 2 (g, g) and H 2 (g, C) respectively. In both cases, HL 2 appears to be the direct sum of 3 spaces: H 2 ⊕ ZL 2 0 ⊕ C where H 2 is the Lie algebra cohomology group, ZL 2 0 is the space of symmetric Leibniz-2-cocycles and C is a space of coupled Leibniz-2-cocycles the nonzero elements of which have the property that their symmetric and antisymmetric parts are not Leibniz cocycles. Our comparison gives some useful practical information about the structure of Lie and Leibniz cocycles. We analyse the case of Heisenberg algebras, the 4-dimensional diamond algebra and its 5-dimensional analogue. We completely describe a versal Leibniz and Lie deformation of the diamond algebra.

Leibniz cohomology and deformations
Recall that a (right) Leibniz algebra is an algebra g with a (non necessarily antisymmetric) bracket, such that the right adjoint operations [·, Z] are required to be derivations for any Z ∈ g. In the presence of antisymmetry, that is equivalent to the Jacobi identity, hence any Lie algebra is a Leibniz algebra.
Let K be a field of zero characteristic. We recall the notion of deformation of a Lie (Leibniz) algebra g (L) over a commutative algebra base A with identity, with a fixed augmentation ε : A → K and maximal ideal M. Assume dim(M k /M k+1 ) < ∞ for every k (see [2,4]). Definition 1. A deformation λ of a Lie algebra g (or a Leibniz algebra L) with base (A, M), or simply with base A is an A-Lie algebra (or an A-Leibniz algebra) structure on the tensor product A ⊗ g (or A ⊗ L) with the bracket [, ] λ such that is an A-Lie algebra (A-Leibniz algebra) homomorphism.
A deformation of the Lie (Leibniz) algebra g (L) with base A is called infinitesimal, or first order, if in addition to this M 2 = 0. We call a deformation of order k, if M k+1 = 0. A deformation with base is called local if A is a local algebra over K, which means A has a unique maximal ideal.
Suppose A is a complete local algebra ( M is the maximal ideal in A. Then a deformation of g (L) with base A which is obtained as the projective limit of deformations of g (L) with base A/M n is called a formal deformation of g (L).
Definition 2. (see [2]) Let C be a complete local algebra. A formal deformation η of a Lie algebra g (Leibniz algebra L) with base C is called versal, if (i) for any formal deformation λ of g (L) with base A there exists a homomorphism f : C → A such that the deformation λ is equivalent to f * η; (ii) if A satisfies the condition M 2 = 0, then f is unique.
is finite dimensional, then there exists a versal deformation of g (similarly for L).
In [1] a construction for a versal deformation of a Lie algebra was given and it was generalized to Leibniz algebras in [4]. The computation for a specific Leibniz algebra example was given in [3].
3. Comparison of the cohomology spaces HL 2 and H 2 for a Lie algebra In [17] the relation between Chevalley-Eilenberg and Leibniz homology with coefficients in a right module is considered via spectral sequence. The statements are valid in cohomological version as well. As a corollary, one deduces Proposition 1. [17] Let g be a Lie algebra over a field K and M be a right g-module. If H * (g, M) = 0, then HL * (g, M) = 0.
As the similar statement is true for cohomologies, it implies that rigid Lie algebras are Leibniz rigid as well. Now we describe the Leibniz 2-cohomology spaces with the help of Lie 2-cohomology space of a Lie algebra g.

Recall that a symmetric bilinear form
Since the projection π : g → g/C 2 g induces an isomorphism ̟ : ker I → S 2 g/C 2 g * , . Note also that the restriction of δ C to (S 2 g * ) g is −I.
For more details on I-null Lie algebras, see [13].
If c denotes the center of g, c ⊗ (S 2 g * ) g is the space of invariant c-valued symmetric bilinear map and we denote F = Id ⊗ I : c ⊗ (S 2 g * ) g → C 3 (g, g) = g ⊗ 3 g * . Then Im F = c ⊗ Im I.
consists of coupled Leibniz 2-cocycles, i.e. the nonzero elements have the property that their symmetric and antisymmetric parts are not cocycles. To get such a supplementary subspace, pick any supplementary subspace W to ker I in (S 2 g * ) g and take C = The class of adjoint ZL 2 -uncoupling Lie algebras is rather extensive since it contains all zero-center Lie algebras and all I-null Lie algebras. For non zero-center, adjoint ZL 2 -uncoupling implies trivial . The reciprocal holds obviously true for I-exact Lie algebras. However we do not know if it holds true in general (e.g. we do not know of a nilpotent Lie algebra which is not I-exact).
Remark 3. This fact also follows from the cohomological version of Theorem A in [17].
Proof. Let g be a Lie algebra and M be a right g-module. Consider the product map m : g ⊗ Λ n g −→ Λ n+1 in the exterior algebra. This map yields an epimorphism of chain complexes is the reduced chain complex: Define the chain complex CR * (g) such that CR * (g [1] is the kernel of the epimorphism C * (g, g) −→ C * (g, K)[−1]. Denote the cohomology of CR * (g) by HR * (g).
Let us recall Theorem A in [17]. There exists a spectral sequence .
As the center of our Lie algebra is 0, it follows that E 2 00 = 0, and so we get H rel 0 (g, g) = 0.
But then from the exact sequence in [17] 0 Corollary 3. For any reductive algebra Lie g with center c, HL 2 (g, g) = Proof. g = s ⊕ c with s = C 2 g semisimple. We first prove that g is adjoint . Suppose first s simple. Then any bilinear symmetric invariant form on s is some multiple of the Killing form K.
For any ψ 0 ∈ c ⊗ (S 2 g * ) g , F (ψ 0 ) is then some linear combination of copies of I K . As is wellknown, I K is no coboundary. Hence if we suppose that F (ψ 0 ) is a coboundary, necessarily F (ψ 0 ) = 0. g is adjoint ZL 2 -uncoupling when s is simple. Now, if s is not simple, s can be decomposed as a direct sum s 1 ⊕ · · · ⊕ s m of simple ideals of s.

Examples
For ω, π ∈ g * , ⊙ stands for the symmetric product ω ⊙ π = ω ⊗ π + π ⊗ ω. Example 2. For g = gl(n), , where (x i ) 1 i n 2 is a basis of g such that (x i ) 1 i n 2 −1 is a basis of sl(n) and x n 2 is the identity matrix, and (ω i ) 1 i n 2 the dual basis to (x i ) 1 i n 2 . Hence there is a unique Leibniz deformation of gl(n). the dual basis to (x i ) 1 i 2N +1 (⊙ stands for the symmetric product Proof. (i) Follows from ker I = S 2 (g/C 2 g) * .  (4N 2 − 1). Example 3. The case N = 1 has been studied in [3]. In that case, dim ZL 2 0 (H 1 , H 1 ) = 3 and the 3 Leibniz deformations are nilpotent, in contradistinction with the 5 Lie deformations. The authors completely describe a Leibniz versal deformation of the 3-dimensional Heisenberg algebra.
Example 4. The 4-dimensional solvable "diamond" Lie algebra d has basis (x 1 , x 2 , x 3 , x 4 ) and nonzero commutation relations (with anticommutativity) The relations show that d is an extension of the one-dimensional abelian Lie algebra Cx 1 by the Heisenberg algebra n 3 with basis x 2 , x 3 , x 4 . It is also known as the Nappi-Witten Lie algebra [15] or the central extension of the Poincaré Lie algebra in two dimensions. It is a solvable quadratic Lie algebra, as admits a nondegenerate bilinear symmetric invariant form. Because of these properties, it plays an important role in conformal field theory. We can use d to construct a Wess-Zumino-Witten model, which describes a homogeneous fourdimensional Lorentz-signature space time [15]. It is easy to check that d is I-exact. In fact, one verifies that all other solvable 4-dimensional Lie algebras are I-null (for a list, see e.g. [16]). We get a basis satisfying the usual commutation relations (4) by letting (5) x 1 = ie 4 , x 2 = e 3 , x 3 = i(−e 2 + e 3 ), x 4 = ie 1 .
One should mention that even though these two forms are equivalent over C, they represent the two nonisomorphic real forms of the complex diamond algebra.
We found that by considering Leibniz algebra deformation of d one gets more structures. Indeed it gives not only extra stucture but also keeps track of Lie structures obtained by considering Lie algebra deformations. To get the precise deformations we need to consider the cohomology groups.
We compute cohomologies necessary for our purpose. First consider the Leibniz cohomology space HL 2 (L; L). Our computation consists of the following steps: (i) To determine a basis of the space of cocycles ZL 2 (L; L), (ii) to find out a basis of the coboundary space BL 2 (L; L), (iii) to determine the quotient space HL 2 (L; L). Therefore, in terms of the ordered basis {e i ⊗ e j } 1≤i,j≤4 of L ⊗ L and {e i } 1≤i≤4 of L,transpose of the matrix corresponding to ψ is of the form are in C . Let φ i ∈ ZL 2 (L; L) for 1 ≤ i ≤ 14, be the cocyle with x i = 1 and x j = 0 for i = j in the above matrix of ψ. It is easy to check that {φ 1 , · · · , φ 14 } forms a basis of ZL 2 (L; L).
Let g(e i ) = a 1 i e 1 + a 2 i e 2 + a 3 i e 3 + a 4 i e 4 for i = 1, 2, 3, 4. The matrix associated to g is given by for 0 ≤ i, j ≤ 4. The transpose matrix of δg can be written as Since ψ 0 = δg is also a cocycle in CL 2 (L; L), comparing matrices δg and M we conclude that the transpose matrix of ψ 0 is of the form 2,4,5,6,8,9,10,12,13 be the coboundary with x i = 1 and x j = 0 for i = j in the above matrix of ψ 0 . It Now we have to compute the Massey brackets [φ i , φ j ] which are responsible for obstructions to extend infinitesimal deformations. We find That means that the two infinitesimal Lie deformations can be extended to real deformations, with the new nonzero brackets (and their anticommutative version) The first of the deformations represents a 2-parameter projective family d(λ, µ), for which each projective parameter (λ, µ) defines a nonisomorphic Lie algebra (in fact, the diamond algebra is a member of this family with (λ, µ) = (1, −1)): is isomorphic to sl(2, C) ⊕ C for every nonzero value of t, see [5]. We note that this Leibniz algebra is not nilpotent.
For the bracket [φ 11 , φ 11 ] we get a nonzero 3-cocycle, so the infinitesimal Leibniz deformation with infinitesimal part being φ 11 can not be extended even to the next order.
The nontrivial mixed brackets [φ i , φ j ] determine relations on the base of versal deformation.
Thus we need to check the Massey 3-brackets which are defined, namely In these five possible Massey 3-brackets, only < φ 3 , φ 3 , φ 7 > is represented by nontrivial cocycle.
So we now proceed to compute the possible Massey 4-brackets. We get that four of them are nontrivial: At the next step, we get that all the 5-order Massey products are either not defined or are trivial. Example 5. The quadratic 5-dimensional nilpotent Lie algebra g 5,4 [11] has commutation relations [x 1 , This is an extension of the trivial Lie algebra Cx 1 by the 4-dimensional Lie algebra Cx 4 × n 3 (n 3 the 3-dimensional Heisenberg Lie algebra [x 2 , x 3 ] = x 5 ). As it is moreover the only 5-dimensional indecomposable nilpotent Lie algebra which is not I-null, it can be considered as a 5-dimensional analogue of the diamond algebra d.