Classification of six-dimensional Leibniz algebras ${\mathcal E}_3

Leibniz algebras ${\mathcal E}_n$ were introduced as algebraic structure underlying U-duality. Algebras ${\mathcal E}_3$ derived from Bianchi three-dimensional Lie algebras are classified here. Two types of algebras are obtained: Six-dimensional Lie algebras that can be considered extension of semi-Abelian four-dimensional Drinfeld double and unique extensions of non-Abelian Bianchi algebras.


Introduction
Algebraic structures underlying U-duality were suggested in [1] and [2] as Leibniz algebras E n obtained by extension of n-dimensional Lie algebra defining non-symmetric product • on [n + n(n − 1)/2]-dimensional vector space in such way that it satisfies Leibniz identity In those papers examples of these Leibniz algebras derived from two-dimensional and four-dimensional Lie algebras are given. Goal of the present note is to write down all algebras that can be derived from three dimensional Lie algebras whose classification given by Bianchi is well known. Namely, let (T a , T a 1 a 2 ), a, a 1 , a 2 ∈ 1, . . . , n, T a 2 a 1 = −T a 1 a 2 is a basis of [n + n(n − 1)/2]-dimensional vector space. The algebra product given in [1] is where f ab c are structure coefficients of n-dimensional Lie algebra and

Bianchi-Leibniz algebras
We are going to classify Leibniz algebras E 3 derived by extension (2) of three dimensional Lie algebras. In this case where ε is totally antisymmetric Levi-Civita tensor. Non-vanishing bilinear forms are T 1 , T 12 2 = T 1 , T 13 3 = T 2 , T 23 3 = 1, First of all we shall show that for dimension three the Leibniz identities are satisfied only for unimodular Lie Algebras 1 , i.e. f ab b = 0. Indeed, Leibniz identity and definitions (2) give and similarly for cyclic permutation of (1, 2, 3).
Inserting (4) and (2) into Leibniz identities (1) we get This can be shown inspecting e.g. identities (1) for We get n 2 f 1 T 13 + n 2 f 2 T 23 = 0, so that By cyclic permutation of (1, 2, 3) we get (5) and it is easy to check that these conditions are sufficient for satisfaction of all Leibniz identities (1). Solution of conditions (5) is either n j = 0, j = 1, 2, 3 or f k = 0, k = 1, 2, 3. It means that we get two types of Bianchi-Leibniz algebras. The first type are algebras depending only on f k with products It is rather easy to check that this product is antisymmetric so that it is a sixdimensional Lie algebra. By linear transformation we can achieve f a ∈ {0, 1} and we get four distinct cases. Only one of them is Drinfel'd double, namely The Bianchi-Leibniz algebras of the second type depend only on n j whose values are given in the Table 1. It means that they are in one to one correspondence with the unimodular Bianchi algebras. Their products are Maximal isotropic algebras in both types of algebras are generated by {T 1 , T 2 , T 3 }, {T 12 , T 13 , T 23 } and {T 1 , T 23 }, {T 2 , T 13 }, {T 3 , T 12 }.
As mentioned in [1], under some conditions we can choose a subalgebra of dimension 2(n − 1) of the Leibnitz algebra E n that is Lie algebra of Drinfel'd double. Leibniz algebra then can be considered as an extension of Drinfel'd double of dimension 2(n − 1). Namely, if we can decompose the generators {T a } as {Tȧ, T z } and {T ab } as {T˙a˙b, T˙a z } (ȧ = 1, . . . , n − 1) so that then the subalgebra spanned by becomes Lie algebra of Drinfel'd double with the bilinear form Tȧ, T˙b := Tȧ, T˙b z = δ˙b˙a .

Malek-Thompson modification of E n
In the paper [2] another algebra undelying U-duality was presented. Its product (for n = 4) written for dual elements where L a = τ a5 − I a . This generalizes product of Sakatani's construction to where Z a = L a /3 (see Note added in [1]). Leibniz identities for this generalized algebra in case n = 3 admit also the non-unimodular Bianchi algebras B3, B4, B5, B6 a , B7 a beside those given in the preceding Section. In this case n 1 = 0 and f a bcd = 0, Non-vanishing products of these algebras are where the values of parameters a, n 2 , n 3 are given in the Table 1.

Conclusions
We have classified six-dimensional Leibniz algebras (2) and (13) starting from classification of three-dimensional Lie algebras. We have obtained nine inequivalent algebras (2) up to linear transformations. Four of them, obtained from Bianchi algebra B1, are six-dimensional Lie algebras (7) that can be considered extension of semi-Abelian four-dimensional Drinfel'd double. The other five are unique Leibniz extensions (8) of unimodular Bianchi algebras B2, B6 0 , B7 0 , B8, B9.
Beside that we have obtained five inequivalent generalized algebras (13) corresponding to the Bianchi algebras B3, B4, B5, B6 a , B7 a . Their products are given by relations (14).