arXiv:2003.06164v4 [hep-th] 8 May 2020 neligUdaiywr ugse n[]ad[]a Leibni as [3] and [2] in suggested e.g.[1]) were see U-duality review (for underlying U-dualities called so are theories non-perturbativ to T-dualities Poisson–Lie of Extensions Introduction 1 banda xesosof extensions as obtained product identity ∗ 1 for hlavaty@fjfi.cvut.cz lsicto fsxdmninlLeibniz six-dimensional of Classification eeaie rm ed r given. explic algebras are the fields of frame exte all generalized unique For algebras. and ob Bianchi double are non-Abelian Drinfel’d algebras extens of four-dimensional considered be types Abelian can Two that algebras here. Lie Six-dimensional classified are algebras Lie yn -ult.Algebras U-duality. lying aut fNcerSine n hsclEngineering, Physical and Sciences Nuclear of Faculty n ≤ ◦ ebi algebras Leibniz 4 in [ n + zc ehia nvriyi Prague, in University Technical Czech X n ( ◦ n ( − Y 1) ◦ E / Z n n 2] dmninlLeagbadfiignon-symmetric defining algebra Lie -dimensional )=( = )) eeitoue sagbacsrcueunder- structure algebraic as introduced were ailvHlavatý Ladislav dmninlvco space vector -dimensional algebras zc Republic Czech a 1 2020 11, May E 3 X Abstract eie rmBacithree-dimensional Bianchi from derived ◦ 1 Y ) ◦ Z E + 3 Y ∗ ◦ ( X 1 leri structures Algebraic . ◦ htstse Leibniz satisfies that ymtiso string of symmetries e Z ) . o fsemi- of ion tfrsof forms it sosof nsions algebras z tained: (1) E n 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 alge- bras 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. 1 0 0 M = sin x tan x cos x − sin x sec x , 1 1 2 1 1 2 − cos x1 tan x2 sin x1 cos x1 sec x2
cos x1 − sin x1 cos x2 − sin x2 cos x1 M = sin x cos x cos x − sin x sin x 2 1 1 2 1 2 0 0 cos x2 for Bianchi B9.
3 Malek-Thompson modification of frame fields
In the paper [3] another algebra underlying U-duality was presented starting from a more general form of the frame field
i I ea 0 E = a1a2b ei (19) A − Π b αr [a1 r a2] √2! [i1 i2] !
7 a that for α = 1 coincides with (16). This ansatz for α = eZax leads to modification of the algebra (2) to the form [2]
c Ta ◦ Tb = fab Tc , b1b2 b1b2c [b1 b2]c b1b2 Ta ◦ T = fa Tc +2 fac T − Za T , (20) a1a2 a1a2c [a1 a2] c1c2 a1a2 c1c2 T ◦ Tb = −fb Tc +3 f[c1c2 δb] T +3 Z[b δc1c2] T , a1a2 b1b2 a1a2[b1 b2]d T ◦ T = −2 fd T
This modification enables to define the algebras E3 also for the non-unimodular Bianchi algebras. Leibniz identities for this generalized algebra in case n =3 require Za = b −fab so that they admit also the non-unimodular Bianchi algebras B3, B4, B5, B6a, B7a beside those given in the preceding Section. In these cases bcd b n1 = 0 and fa = 0, Z1 = −f1b = 2a, Z2 = Z3 = 0. In other words, non-vanishing products of these algebras are
[T1, T2]= −aT2 + n3T3, [T2, T3]=0, [T3, T1]= n2T2 + aT3 , 12 12 13 T1 ◦ T = −a T + n2 T , 13 13 12 T1 ◦ T = −a T − n3 T , (21) 23 13 12 T2 ◦ T = −a T − n3 T , 23 12 13 T3 ◦ T = a T − n2 T , where the values of parameters a, n2, n3 are given in the Table 1.
3.1 Twisted generalized frame fields
C It is understandable that for structure constants F ′AB of the modified al- I gebra (21) the generalized frame fields EA of the form (16) do not satisfy relations (13). For example, the formula (16) for Bianchi algebra B5 gives
I x1 x1 1 x1 1 x1 1 2x1 E = diag(1, e , e , , e− , , e− , , e− ) A 2 2 2 and £ˆ I 2x1 E1 E6 = (0, 0, 0, 0, 0, e− ). But, as follows from (21),
C I F ′1,6 EC = (0, 0, 0, 0, 0, 0).
8 To satisfy relations
I C I £ˆ ′ E A E′B = −F ′AB E′C , (22)
C where F ′AB are structure coeficients of the algebra (21) the generalized I J I frame fields must be modified by a twist matrix T as E′A = EA TJ [2, 3]. It concerns the non-unimodular Bianchi algebras B3, B4, B5, B6a, B7a and the twist matrix is 1 0 T I = 3 . (23) J 0 e2ax1 1 3 where a is given in the Table 1. Corresponding twisted generalized frame fields are
x1 e− 00 0 00 0 cosh x sinh x 0 0 0 1 1 I x1 0 sinh x1 cosh x1 0 0 0 E′A = e 1 1 0 0 0 2 cosh x1 − 2 sinh x1 0 1 1 0 0 0 − 2 sinh x1 2 cosh x1 0 1 x1 000 0 0 e− 2 for B3,
10 0 0 00 x1 x1 0 e −e x1 0 00 x1 I 0 0 e 0 00 E′A = ex1 00 0 2 0 0 x1 00 0 1 ex1 x e 0 2 1 2 00 0 0 0 1 2 for B4,
10 0 0 00 0 ex1 0 0 00 x1 I 0 0 e 0 0 0 E′A = ex1 0 0 0 2 0 0 x1 00 0 0 e 0 2 00 0 0 0 1 2 for B5,
9 ax1 e− 00 0 0 0 0 cosh x sinh x 0 0 0 1 1 I ax1 0 sinh x1 cosh x1 0 0 0 E′A = e 1 1 0 0 0 2 cosh x1 − 2 sinh x1 0 1 1 0 0 0 − 2 sinh x1 2 cosh x1 0 1 ax1 000 0 0 e− 2 for B6a,
ax1 e− 00 0 0 0 0 cos x − sin x 0 0 0 1 1 I ax1 0 sin x1 cos x1 0 0 0 E′A = e 1 1 0 0 0 2 cos x1 − 2 sin x1 0 1 1 0 0 0 2 sin x1 2 cos x1 0 1 ax1 00 0 0 0 e− 2 for B7a. These twisted generalized frame fields then satisfy relations (22).
4 Conclusions
We have classified six-dimensional Leibniz algebras (2) and (20) starting from Bianchi classification of three-dimensional Lie algebras. Up to linear transformations from E3 = SL(3) × SL(2) we have obtained seven inequivalent algebras (2). Two of them, obtained from Abelian Lie algebra B1, are six-dimensional Lie algebras (7) that can be considered ex- tensions of semi-Abelian four-dimensional Drinfel’d double. The other five are unique Leibniz extensions (9) of unimodular Bianchi algebras B2, B60, B70, B8, B9 (see Table 1). For all of these algebras we have calculated ex- plicit forms of generalized frame fields and checked that they satisfy relations (13). Beside that we have obtained five inequivalent generalized algebras (20) corresponding to the Bianchi algebras B3, B4, B5, B6a, B7a. Their products are given by relations (21). In this case generalized frame fields given by (16) must be twisted by matrix (23) to satisfy relations (22) for the generalized algebra.
10 As we have complete classification of algebras E3 they represent analogs of Manin triples obtained for n = 3 in [6]. In case we want to study non- Abelian U-dualities or pluralities we have to look for their mutual relation given by linear transformations, more precisely, representations of the group E3 = SL(3) × SL(2). Unfortunately, it seems that no such relations exist. Certainly there is no linear transformation between algebras (7) and (9) or (21) as the former ones are Lie and the latter not. Attempts to find a linear transformations inside these two classes by brute force failed as well. It seems that the only possibility to find non-Abelian U-dualities or pluralities for six-dimensional Leibniz algebras is a further generalization of the form (20).
References
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[2] Yuho Sakatani, U-duality extension of Drinfel’d double, Prog. Theor. Exp. Phys. Volume 2020, Issue 2, February 2020, 023B08, [arXiv:1911.06320]
[3] E. Malek and D. C. Thompson, Poisson-Lie U-duality in Exceptional Field Theory, JHEP 0420 (2020) 058, [arXiv:1911.07833].
[4] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Perga- mon Press, 1987
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