Classification of Six-Dimensional Leibniz Algebras ${\Mathcal E} 3

$Classification of Six-Dimensional Leibniz Algebras ${\Mathcal E} 3$ Classification of six-dimensional Leibniz algebras E3 Ladislav Hlavatý∗ Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Czech Republic May 11, 2020 Abstract Leibniz algebras En were introduced as algebraic structure underlying U-duality. Algebras E3 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 Drinfel’d double and unique extensions of non-Abelian Bianchi algebras. For all of the algebras explicit forms of generalized frame fields are given. 1 Introduction arXiv:2003.06164v4 [hep-th] 8 May 2020 Extensions of Poisson–Lie T-dualities to non-perturbative symmetries of string theories are so called U-dualities (for review see e.g.[1]). Algebraic structures underlying U-duality were suggested in [2] and [3] as Leibniz algebras En obtained as extensions of n-dimensional Lie algebra defining non-symmetric product ◦ in [n + n(n − 1)/2]-dimensional vector space1 that satisfies Leibniz identity X ◦ (Y ◦ Z))=(X ◦ Y ) ◦ Z + Y ◦ (X ◦ Z). (1) ∗hlavaty@fjfi.cvut.cz 1for n ≤ 4 1 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. a1a2 a2a1 a1a2 Namely, let (Ta, T ), a, a1, a2 ∈ 1,...,n, T = −T is a basis of [n + n(n − 1)/2]-dimensional vector space. The algebra product given in [2] is c Ta ◦ Tb = fab Tc , b1b2 b1b2c [b1 b2]c Ta ◦ T = fa Tc +2 fac T , (2) a1a2 a1a2c [a1 a2] c1c2 T ◦ Tb = −fb Tc +3 f[c1c2 δb] T , a1a2 b1b2 a1a2[b1 b2]d T ◦ T = −2 fd T , c b1b2b3 where fab are structure coefficients of n-dimensional Lie algebra and fa = [b1b2b3] fa . Moreover, bilinear forms on En are defined b1b2 [b1 b2] a1a2 b1b2 a1 a2 b1 b2 hTa, T ic = 2! δa δc , hT , T ic1 c4 = 4! δ c1 δc2 δc3 δc4 . (3) ··· [ ] 2 Bianchi-Leibniz algebras We are going to classify Leibniz algebras E3 derived from three-dimensional b1b2b3 b1b2b3 Lie algebras. In this case fa = fa ε where ε is totally antisymmetric Levi-Civita symbol. Non-vanishing bilinear forms are 12 13 23 hT1, T i2 = hT1, T i3 = hT2, T i3 =1, 12 13 23 hT2, T i1 = hT3, T i1 = hT3, T i2 = −1. First of all we shall show that for dimension three the Leibniz identities 2 b are satisfied only for unimodular Lie Algebras , i.e. fab =0. Indeed, Leibniz identity 23 23 23 T ◦ (T1 ◦ T1)=(T ◦ T1) ◦ T1 + T1 ◦ (T ◦ T1) and definitions (2) give 2 3 2 23 b 2 23 0=2(f12 + f13 ) T =2(f1b ) T 2This is not true in general as can be shown explicitely for dimension four or by two- dimensional example in [2]. 2 Class a n1 n2 n3 B1 0 0 0 0 B2 0 1 0 0 B3 1 0 1 -1 B4 1 0 0 1 B5 1 0 0 0 B60 0 1 -1 0 B6a (a> 0, a =6 1) a 0 1 -1 B70 0 1 1 0 B7a (a> 0) a 0 1 1 B8 0 1 1 -1 B9 0 1 1 1 Table 1: Bianchi algebras and similarly for cyclic permutation of (1, 2, 3). Next point in our computations is the well known classification of 3– dimensional real Lie algebras. Non–isomorphic Lie algebras can be divided into eleven classes, traditionally known as Bianchi algebras. Their Lie algebra products are (see e.g. [4]) [X1,X2]= −aX2 + n3X3, [X2,X3]= n1X1, [X3,X1]= n2X2 + aX3, (4) where the parameters a, n1, n2, n3 have the values given in the Table 1. Uni- modular Bianchi algebras are those with a = 0, i.e. B1 (Abelian), B2 (Heisenberg), B60 (Euclidean), B70 (Poincare), B8 (so(2,1)), and B9 (so(3)). Inserting (4) and (2) into Leibniz identities (1) we get njfk =0, j,k =1, 2, 3. (5) This can be shown inspecting e.g. identities (1) for 23 12 X = T1, Y = T , Z = T , and 23 13 X = T1, Y = T , Z = T . We get 13 23 n2f1 T + n2f2 T =0, 3 12 23 n3f1 T + n3f3 T =0, so that n2f1 =0, n2f2 =0, n3f1 =0, n3f3 =0. (6) 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 nj =0, j =1, 2, 3 or fk =0, k =1, 2, 3. It means that we get two types of Bianchi-Leibniz algebras. The first type are algebras depending only on fk with products Ta ◦ Tb =0 , b1b2 b1b2c Ta ◦ T = fa ε Tc , a1a2 a1a2c (7) T ◦ Tb = −fb ε Tc , a1a2 b1b2 a1a2[b1 b2]d T ◦ T = −2 fd ε T . It is rather easy to check that this product is antisymmetric so that they are six-dimensional Lie algebras. The simplest one is Abelian where all nj = 0 and fk = 0. If at least one of fk is not zero then by linear transformation from E3 = SL(3) × SL(2) we can achieve f1 =1, f2 = f3 =0 so that [Ta, Tb]=0 , b1b2 b1b2c [T1 ◦ T ]=2 ε Tc , (8) [T 23, T 12]=2 T 12, [T 23, T 13]=2 T 13 . The Bianchi-Leibniz algebras of the second type depend only on nj whose values are given in the Table 1. It means that they are in one to one corre- spondence with the unimodular Bianchi algebras. Their products are Ta ◦ Tb = [Ta, Tb] , b1b2 b1 b1c b2 b,c Ta ◦ T = δa εab2c nb2 T − δa εab1c nb1 T , a1a2 (9) T ◦ Tb =0 , T a1a2 ◦ T b1b2 =0 . b1b2 Explicit forms of products Ta ◦ T are 12 23 13 T1 ◦ T = −T3 ◦ T = n2 T , 13 23 12 T1 ◦ T = T2 ◦ T = −n3 T , 4 12 13 23 T2 ◦ T = T3 ◦ T = n1 T . Maximal isotropic algebras in both types of algebras are generated by 12 13 23 23 13 12 {T1, T2, T3}, {T , T , T } and {T1, T }, {T2, T }, {T3, T }. As mentioned in [2], under some conditions we can choose a subalgebra of dimension 2(n − 1) of the Leibnitz algebra En 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 ab a˙b˙ az˙ {Ta} as {Ta˙ , Tz} and {T } as {T , T } (a˙ =1,...,n − 1) so that z b b1b2b3 b˙1b˙2b˙3 fab =0 , faz =0 , fz =0 , fa˙ =0 , (10) then the subalgebra spanned by a˙ a˙ az˙ (TA˙ ) ≡ (Ta˙ , T ) (T ≡ T ) (11) becomes Lie algebra of Drinfel’d double with the bilinear form b˙ b˙ b˙ hTa˙ , T i := hTa˙ , T iz = δa˙ . (12) However for n = 3, the conditions (10) and unimodularity are satisfied only for Abelian Bianchi algebra B1. It means that Leibniz algebras (7) can be considered extensions of Lie algebras of four-dimensional Drinfel’d doubles 13 23 [5] generated by T1, T2, T , T and 13 23 13 23 [T1, T2]=0, [T , T ]= −f1 T − f2 T , 13 23 hT1, T i = hT2, T i =1. 2.1 Generalized frame fields Now we will present explicit forms of the so called generalized frame fields I EA required to satisfy £ˆ I C I EA EB = −FAB EC , (13) C where FAB are structure constants of the Bianchi-Leibnitz algebras, A,B,C,I ∈ (1, 2, 3, {1, 2}, {1, 3}, {2, 3}). 5 The generalized Lie derivative can be expressed by £ wi £ˆ I v V W = (£v w2 ιwdv2)i i , (14) − 1 2 √2! ! generalized vectors V and W are parameterized as i I v V = vi1i2 , (15) √2! 1 i j v2 = 2 vijdx ∧ dx and similarly for W and w. I Generalized frame fields EA have block triangular form e i i I ea 0 ea 0 a1a2b i EA = a1a2i a1a2 = Π eb [a1 a2] , (16) E E i1,i2 − r r √2! [i1 i2] ! i where ea are components of right-invariant vector fields with respect to a a1a2b Bianchi groups, ri are components of right-invariant 1-forms and Π is the a1a2b so called Nambu-Poisson tensor that in the dimension three is π(x1, x2, x3) ε where ε is the Levi-Civita symbol. I From the formula (16) follows explicit form of matrices EA for the first type algebras (7), namely 1 0 E I = 3 (17) A Π 1 1 3 2 3 where 13 is three-dimensional unit matrix, 0 0 −π(x1, x2, x3) Π = 0 π(x , x , x ) 0 , (18) 3 1 2 3 −π(x1, x2, x3)0 0 and π(x1, x2, x3)= f1x1+f2x2+f3x3+const where fa are constants appearing in (7). I Explicit form of matrices EA for the second type algebras (9) is M 0 E I = 1 A 0 1 M 2 2 6 where 1 00 M = 0 10 , 1 −n1x2 x1 1 1 −x1 −n1x2 M = 01 0 2 00 1 for Bianchi algebras B2, B60, B70. 1 0 0 M = − sinh x tanh x cosh x sinh x sech x , 1 1 2 1 1 2 − cosh x1 tanh x2 sinh x1 cosh x1 sech x2 cosh x1 − sinh x1 cosh x2 − sinh x2 cosh x1 M = − sinh x cosh x cosh x sinh x sinh x 2 1 1 2 1 2 0 0 cosh x2 for Bianchi B8.

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