Poisson-Lie T-duality and Bianchi type algebras

15 July 1999

Physics Letters B 458 Ž1999. 477–490

Poisson-Lie T-duality and Bianchi type algebras M.A. Jafarizadeh

a,b,1

, A. Rezaei-Aghdam

a,2

a

b

Department of Theoretical Physics, Tabriz UniÕersity, Tabriz 51664, Iran Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Iran Received 21 March 1999 Editor: L. Alvarez-Gaume´

Abstract All Bianchi bialgebras have been obtained. By introducing a non-degenerate adjoint invariant inner product over these bialgebras the associated Drinfeld doubles have been constructed, then by calculating the coupling matrices for these bialgebras several s-models with Poisson-Lie symmetry have been obtained. Two simple examples as prototypes of Poisson-Lie dual models have been given. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction It is a well known fact that one of the most important symmetries of string theories, or in general s-models, is target-space, T-duality. There was a common belief among the experts in this field that the existence of T-duality mainly should depend on the isometry symmetry of the original target manifold such that, depending on the kind of isometry it was called abelian or non-abelian T-duality Žfor a review see w1x.. Of course there were some difficulties in the non-abelian case, for example it was not possible to obtain the original model from the dual one, because the latter might not have isometry w2x. Indeed Klimcik and Severa, by introducing the Poisson-Lie T-duality in their pioneering work w3x, could remove the requirement of isometry in obtaining the dual model. Hence they could remove the above mentioned difficulties in T-duality investigation. Indeed, Poisson-Lie duality does not require the isometry in the original target manifold; the integrability of the Noether’s current associated with the action of group G on the target manifold is good enough to have this symmetry. In other words, the components of the Noether’s current play the role of flat connection, that is, they satisfy Maurer-Cartan equations with group structure of G˜ Žwith the same dimension as G . w3x; such that G and G˜ have Poisson-Lie structure and their Lie algebras form a bialgebra w4,5x. Classically the dual models are equivalent, since one can obtain one from the other through a canonical transformation in their respective phase space. This canonical equivalence has already been shown, both in Poisson-Lie T-duality w3,6,7x and abelian or non-abelian T-duality w8x. Actually, the quantum equivalence of both abelian and non-abelian dual models have

1 2

E-mail: [email protected] E-mail: [email protected]

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 5 7 1 - 7

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

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already been investigated. This equivalence can be shown by obtaining a general relation between the Weyl-anomaly coefficients Žand b-functions. of the original model and those of its dual w9,10x. But the quantum equivalence of the Poisson-Lie dual models is still a challenging problem; the quantum equivalence has been shown only in some special examples w10,11x. Therefore, in order to understand the quantum features of the Poisson-Lie T-duality we have a long way ahead. Hence, we need to investigate too many s-models with Poisson-Lie symmetry, since so far few examples with Poisson-Lie symmetry have been obtained w3,10–12x. This suggests us trying to find further examples with Poisson-Lie symmetry. In this paper, we find explicitly all possible dual algebras of all three dimensional real Lie algebras ŽBianchi algebras.. Then by introducing a non-degenerate adjoint invariant inner product over the Lie algebras of Drinfeld doubles, we find many bialgebras, where we have associated a pair of Poisson-Lie dual s-models to each of these bialgebras which contain all examples mentioned above and many other examples. This paper is organized as follows. In order the paper to be self-contained and also to fix the notations, we give a brief review of the Poisson-Lie T-duality in Section 2. In Section 3 we obtain all possible dual algebras of each of Bianchi algebras, where the bialgebras thus obtained have been listed in Table 2. Then, we introduce a non-degenerate adjoint invariant inner product over Žsix dimensional. Lie algebras of Drinfeld doubles, such that each of bialgebras are isotropic with respect to this inner product. In Section 4 some of the general formulas related to the calculation of coupling matrices of s-models, associated with the bialgebras of Table 2, have been constructed. Actually these informations help us to write a pair of Poisson-Lie dual s-models associated with each bialgebra of Table 2. At the end of this section we give two simple examples. Finally, the paper ends with a brief conclusion and two appendices.

2. Poisson-Lie T-duality In this section we review briefly the Poisson-Lie T-duality. According to w3x the Poisson-Lie duality based on the concepts of Drinfeld double and Manin triple. Drinfeld double D is a Lie group where its Lie algebra D can be decomposed Žas a vector space. into direct sum of two Lie subalgebras g and g, ˜ such that this is maximal isotropic with respect to a non-degenerate invariant bilinear form ² ,: over D. The doublet Žg,g˜ . and triplet Ž D,g,g˜ . are called bialgebra and Manin triple respectively w4,5x. Actually taking the sets  X i 4 and  X˜ i 4 as the bases of the Lie algebras g and g, ˜ respectively, we have: Xi , X j s fi j k Xk ,

X˜ i , X˜ j s f˜i j k X˜ k ,

X i , X˜ j s f˜jk i X k q f k i j X˜ k .

Ž 1.

The isotropy of the subalgebras with respect to bilinear form means that: ² X i , X j : s ² X˜ i , X˜ j : s 0,

² X i , X˜ j : s d i j .

Ž 2.

Also, the Jacobi identity of Lie algebra D imposes the following relations over the structure constants of Lie algebras g and g˜ w3,5x: f m k i f˜jm l y f m l i f˜jm k y f m k j f˜i m l q f m l j f˜i m k s f k l m f˜i j m .

Ž 3.

As we will see in the next section the above relations put rather strong constraint on the structure constants of subalgebras. Hence, obtaining of the bialgebra and Manin triple becomes rather a tough job. In order to define s models with Poisson-Lie duality symmetry, we need to consider the following relations w3x: gy1 X i g s a Ž g . i j X j ,

ij

i

gy1 X˜ i g s b Ž g . X j q d Ž g . j X˜ j ,

P Ž g . s b Ž g . ay1 Ž g . ,

Ž 4.

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with g and g˜ as elements of Lie groups G and G˜ associated with Lie algebras g and g, ˜ respectively. The invariance of inner product with respect to adjoint action of group together with Ž2. and Ž4. requires the above matrices to possess the following properties: d Ž g . s ayt Ž g . ,

ay1 Ž g . s a Ž gy1 . ,

b t Ž g . s b Ž gy1 . ,

P t Ž g . s yP Ž g . .

Ž 5. ˜ ˜ ˜ The matrices a˜Ž g˜ ., bŽ g˜ . and P Ž g˜ . associated with group G can be defined in a similar way. Now, we can define below the s-model with d-dimensional target manifold M, where the group G acts freely on it w3,6,7x: i

j

i

S s y 12 d jq d jy Ei j Ž Eq ggy1 . Ž Ey ggy1 . q F Ž1. i a Ž Eq ggy1 . Ey y a q F Ž2. a i Eq y a Ž Ey ggy1 .

H

i

qFab Eq y aEy y b ,

Ž 6.

where the coupling matrices are: E s Ž E0 y1 q P

y1

.

,

F Ž1. s EE0 y1 F Ž1. , F Ž2. s F Ž2. E0 y1 E, F s F y F Ž2.P EE0 y1 F Ž1. .

a

y1

Ž 7.

The y with Ž a s 1, . . . ,d y dim G . are coordinates of the MrG manifold. The matrices E0 , F , F and F are arbitrary functions of y a only. Similarly, the coupling matrices of the dual s-model can be written as w6,7x: E˜ s Ž E0 q P˜

y1

.

,

Ž1.

˜ Ž1. , F˜ Ž2. s yF Ž2. E, ˜ F˜ s F y F Ž2. EF ˜ Ž1. . F˜ Ž1. s EF

Ž2.

Ž 8.

The target space of dual model is d-dimensional manifold M˜ with the group G˜ acting freely on it. The corresponding dual action can be written as: S˜s y 12 d jq d jy w E˜ i j Ž Eq gg ˜˜y1 . i Ž Ey gg ˜˜y1 . j q F˜ Ž1.i a Ž Eq gg ˜˜y1 . i Ey y a q F˜ Ž2. a Ei q y a Ž Ey gg ˜˜y1 . i

H

q F˜ab Eq y a Ey y b x .

Ž 9.

The actions Ž6. and Ž9. correspond to Poisson-Lie dual s models w3x 3. Notice that if the group GŽ G˜ . besides having free action on M Ž M˜ ., acts transitively over it, then the corresponding manifolds M Ž M˜ . will be the same as the groups GŽ G˜ .. In this case only the first term appears in the actions Ž6. and Ž9.. Also if the group G ˜ then we get the standard nonabelian becomes the isometry group of the manifold M with dual abelian group G, duality w2x.

3. Bianchi bialgebras In this section, we use Ž3. to obtain the dual Lie algebras g˜ of a given 3-dimensional real Lie algebras g. Then, by introducing a nondegenerate adjoint invariant bilinear form we get the corresponding Manin triples. According to Behr’s classification w13,14x of 3-dimensional Bianchi Lie algebras w15x, the commutation relation of these algebras can be generally written as:

w X1 , X 2 x s yaX 2 q n 3 X 3 , w X 2 , X 3 x s n1 X1 , w X 3 , X1 x s n 2 X 2 q aX 3 ,

Ž 10 .

4

where the structure constants are given in Table 1 .

3

Actually the Noether’s current associated with the action of GŽ G˜ . on M Ž M˜ . satisfy the Maurer-Cartan equation with structure constants of dual Lie group G˜Ž G .. 4 For three algebras III, VIa and VIIa , with nonzero parameters a, n 2 and n 3 , rescaling of the basis yields the ratio a 2 r n 2 n 3 constant. For other algebras one can choose a basis with structure constants "1, as in Table 1.

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Table 1 Bianchi classification of three dimensional Lie algebras Type

a

n1

n2

n3

I II VII0 VI0 IX VIII V IV VIIa III VIa

0 0 0 0 0 0 1 1 a

0 1 1 1 1 1 0 0 0

0 0 1 y1 1 1 0 0 1

0 0 0 0 1 y1 0 1 1

a

0

1

y1

Ž as1 . Ž a/1 .

5

For all Lie algebras of Table 1 we have an1 s 0. The algebras with a s 0 are called class A while those with nonvanishing a are called class B. Now we can determine the dual Lie algebra g, ˜ by using the structure constant of Lie algebra g given in Ž10.. Due to the antisymmetricity of Ž3. with respect to the set of indices Ž i, j . and Ž k,l ., these relations, after some algebraic calculations and with the help of Ž10., lead to the following nine equations n1 f˜23 1 y n 2 f˜13 2 s n 3 f˜12 3 ,

n 2 f˜13 3 s n 2 f˜12 2 ,

n 3 f˜13 3 s n 3 f˜12 2 ,

n1 f˜23 3 s yn1 f˜12 1 ,

n1 f˜23 2 s n1 f˜13 1 , yn1 f˜32 1 q n 3 f˜12 3 s yn 2 f˜13 2 , a f˜32 2 q f˜31 1 s n 3 f˜12 1 q f˜23 3 ,

ž

/

ž

/

a f˜32 3 q f˜12 1 s n 2 f˜13 1 y f˜23 2 ,

ž

/

ž

/

yaf˜31 3 q n 2 f˜31 2 y n 3 f˜21 3 y af˜21 2 s n1 f˜23 1 .

Ž 11 .

In the above equations some coeffecients can be eliminated, but the reason for their presence is due to their vanishing for some algebras. Now, in order to determine the structure constants of dual Lie algebras g˜ associated with those listed in Table 1, we could write Ž11. for each of them. Then, by using the Jacobi identity for g, ˜ we can determine f˜i j k 5. The algebras g˜ reduces to Lie algebras of Bianchi type if Ža. we use the following general form of structure constant of a given 3-dimensional Lie algebra: f˜i j k s e i jk n˜ l k q d i k a˜ k y d j k a˜ i , with a˜ i as components of a 3-vector and n˜ i j as elements of symmetric matrix with the constraint n˜ i j a˜ j s 0; and Žb. if we work in a basis that diagonalizes the matrix n˜ i j with diagonal elements Ž n˜ 1 ,n˜ 2 ,n˜ 3 . and vector a˜ k with components Ž a,0,0 ˜ .. But, instead of performing the above steps, we can choose another method with rather less camputations and obtain the same results, as follows. Similar to f i j k we assume that the f˜i j k have the following form: X˜1 , X˜2 s yaX ˜ ˜2 q n˜ 3 X˜3 ,

5

X˜2 , X˜3 s n˜ 1 X˜1 ,

X˜3 , X˜1 s n˜ 2 X˜2 q aX ˜ ˜3 .

After using Ž11. for each Lie algebra, the number of equations becomes less than nine.

Ž 12 .

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Table 2 Bianchi bialgebras g Žclass A.

g Žclass B .

I II VII0 VI0 IX VIII

g˜ all Bianchi algebras all Bianchi algebras except types of IX,VIII I, II,V, IV I, II,V, IV I,V I,V

V IV VIIa III VIa

class A I, II,VII0 ,VI0 Žclass A except types of IX,VIII . I, II,VIIa˜ Ž as ˜ 1a . I, II, III I, II,VIa˜ Ž as ˜ 1a .

Then Ž11. reduces to n1 n˜ 1 s 0,

aa˜ s n 2 n˜ 2 s n 3 n˜ 3 .

Ž 13 .

Hence the structure constants Ž a,n ˜ ˜ 1 ,n˜ 2 ,n˜ 3 . of the dual Lie algebra g˜ can be determined in terms of the structure constants Ž a,n1 ,n 2 ,n 3 . of the Lie algebra g. As an example, for g s IX, from Ž13. and Table 1 we have: n˜ 1 s n˜ 2 s n˜ 3 s 0 and a˜ is an arbitrary constant. Now, for a˜ s 0 we obtain Bianchi of type I and for a˜ / 0 we get Bianchi of type V if we rescale the basis such that a˜ s 1. Therefore, the Lie algebra IX has two different dual Lie algebras I, V. These bialgebras have already been used in Refs. w10,16x. As another example we consider g s VII0 , then we have n˜ 1 s n˜ 2 s 0 and a˜ together with n˜ 3 become arbitrary. In this case the dual Lie algebra g˜ can be one of the following four different Bianchi Lie algebras: 1. For n˜ 3 s a˜ s 0, we get Bianchi of type I. 2. For n˜ 3 / 0 and a˜ s 0 via rescaling and change of basis, we get Bianchi of type II. 3. For n˜ 3 s 0 and a˜ / 0 via rescaling of basis, we get Bianchi of type V. 4. Finally, for n˜ 3 / 0 and a˜ / 0 via rescaling of basis, we get Bianchi of type IV. Therefore the Lie algebra VII0 has four different dual Lie algebras I, II, V, IV. We have also determined the possible dual Lie algebras of other Bianchi Lie algebras, where the results are given in Table 2. As in Table 2, all Bianchi Lie algebras, except types of IX and VIII, have more than three different dual Lie algebras. Also Bianchi algebras II, III,VIa and VIIa can be self dual. The number of different bialgebras listed in Table 2, is 28 6 . So far we have been able to obtain all possible bialgebras associated with Bianchi algebras. As it is mentioned in the previous section, in order to have a Manin triple associated with these bialgebras, we need to have a nondegenerate ad-invariant inner product over the algebra D s g [ g, ˜ such that the algebras g and g˜ become isotropic with respect to it. In order to obtain this inner product we choose the set TA 4 s  X 1 , X 2 , X 3 , X˜1 , X˜2 , X˜34 as the basis of D. Notice that we have more than one D Lie algebra, but considering the algebras g and g˜ with commutation relation Ž10. and Ž12., respectively, we can give the general form of D.

6

It is clear that if the pair Žg,g˜ . form a bialgebra, then, also, the pair Žg,g ˜ . will lead to another bialgebra. Therefore the total number of bialgebras of Table 2 is 56.

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

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Now, writing the commutation relation of the algebras D as:

w TA ,TB x s CA B C TC , with A, B s 1, . . . ,6, the structure constants CA B C can be obtained by using Ž1., Ž10. and Ž12.. Then the basis TA will have the following matrix form in the adjoint representation of D :

Ž14. where Xi , the adjoint representation of basis of Bianchi Lie algebra g, and the antisymmetric matrices Yi are: 0 X1 s 0 0

0 a n2

0 yn 3 , a

X2 s

0 Y1 s 0 0

0 0 n1

0 yn1 , 0

0 Y2 s ya yn 2

 

0  0 

0 0 yn1

ya 0 0 a 0 0

n3 0 , 0

0 X 3 s n1 0

0  0 

n2 0 , 0

Y3 s

0 n3 ya

yn 2 0 0 yn 3 0 0

ya 0 , 0

0

a 0 . 0

0

Ž 15 .

Similarly, X˜ i , adjoint representation of the basis of dual Bianchi Lie algebras g, ˜ and Y˜ i have the same form as Xi and Yi with the difference that we must replace the set Ž a,n1 ,n 2 ,n 3 . with Ž a,n ˜ ˜ 1 ,n˜ 2 ,n˜ 3 .. Since, in general, Lie algebra D is non-semisimple, the nondegenerate ad-invariant inner product on it can not be obtained via the trace of bilinear product of those algebras in adjoint representation. In other words Killing form is degenerate for these algebras. Therefore, in order to obtain the nondegenerate ad-invariant metric V A B in the above basis, we have to solve the following equation w17x: CA B DV C D q CAC DV B D s 0.

Ž 16 .

Eq. Ž16. means that the matrices TA . V must be antisymmetric matrices. Hence we must find symmetric matrix V such that the above relations holds. Now, in order to determine V for each bialgebra of Table 2, we must write TA explicitly and then, using the antisymmetricity of TA . V , we find the explicit form of V . As an example, for the pair of Ž V, IX ., the form of V is: yb 0 0 Vs k 0 0



0 0 0 0 k yb

0 0 0 0 b k

k 0 0 b 0 0

0 k b 0 b 0

0 yb k , 0 0 b

0

3

det V s y Ž k 2 q b 2 . .

Now by choosing k s 1 and b s 0 we get: Ž17. Actually, by using Ž14. and Ž15. one can easily show that the symmetric metric Ž17. can be ad-invariant metric of all 28 different bialgebras of Table 2, that is

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is antisymmetric for all bialgebras of Table 2. Hence, we choose V as the required inner product. It is also interesting to see that the Lie subalgebras of all bialgebras of Table 2 are isotropic with respect to this inner product, that is we have: ²TA ,TB : s V A B ,

² X i , X˜ j : s ²Ti ,Tjq3 : s V i , jq3 s d i j ,

² X˜ i , X j : s ²Tiq3 ,Tj : s V iq3, j s d i j .

Ž 18 .

Therefore, all bialgebras of Table 2, together with the ad-invariant metric Ž17., form Manin triples. Now we can construct s-models with Poisson-Lie symmetries associated with the Manin triples obtained above.

4. s-models and Bianchi Lie groups As mentioned in Section 2, in order to construct s-models with Poisson-Lie symmetry associated with bialgerba Žg,g˜ ., we need to know aŽ g . and bŽ g .. Indeed for every bialgebras of Table 2, we can evaluate aŽ g . and bŽ g .. Hence we will have a s-model as well as its dual, which possess the Poisson-Lie symmetry. In this section we give general formulas for aŽ g . and bŽ g . in terms of the sets of parameters Ž a,n1 ,n 2 ,n 3 . and Ž a,n ˜ ˜ 1 ,n˜ 2 ,n˜ 3 .. Therefore, by appropriate choices of these parameters, aŽ g . and bŽ g . of all bialgebras of Table 2 can be evaluated. Hence we can construct all s-models associated with them. To achieve this goal we parametrize the group G in the following form: g s e x1 X 1 e x 2 X 2 e x 3 X 3 ,

Ž 19 .

where x i are coordinates of the group manifold G. In order to calculate aŽ g . and bŽ g ., we need to calculate expressions such as eyx i X i X j e x i X i and eyx i X i X˜ j e x i X i . To do this, we must use the algebraic relations Ž1., Ž10. and Ž12.. To make the calculation simple, we need to work with basis of D, where we have 7: eyx i X i TA e x i X i s eyx i T i TA e x i T i s Ž e x i T i . A B TB . Therefore, we must calculate matrices e general, they have the following form:

xi Ti

Ž 20 .

, where due to the particular block diagonal form of Ti Ž14., in

Ž21. To calculate Ž21. we need to know the block diagonal form Di of matrices Ti together with matrices Si , then we have: Si Ti Sy1 i s Di ,

xi Di e x i T i s Sy1 Si . i e

Ž 22 .

Now we put each Ti into its block diagonal form separately. For T1 we have: Ž23. where 1 Ns 0 0

ž

7

0 0 y1

0 1 . 0

/

Here, repeated indices do not imply summation.

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

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On the other hand, by using the following matrices: 1 Ss



0 n3

( (

0

( (

n2 q n3

0

n2

yi

0 n3

i

n2 q n3

0

n2 q n3 n2 n2 q n3

0 Ds 0 0



,

0 a q i n2 n3

(

0 0

0

a y i n2 n3

(

0

,

we write the matrix X 1 in digonal form, then using Ž22. for these matrices we calculate e x 1 X 1 :

e x1 X 1 s e a x 1

eya x 1

0

0

cosh1



(

0

n2 n3

(

y

sinh1

0 n3

sinh1

n2

cosh1

(

0 0

,

Ž 24 .

where h1 s n 2 n 3 x 1. With the help of Ž22., Ž23. and Ž24. and after some algebraic calculations we have: 0 B1 s y

n˜ 1 a

sinh a x 1



(

0

0 n2 n3

0 sinh1

cosh1

(

ycosh1

0

n3 n2

sinh1

.

Ž 25 .

Similarly, for the matrices T2 and T3 , we have calculated the matrices S2 and S3 , which transform them into their Jordan forms D 2 and D 3 Žthese matrices are given in appendix A.. Finally, using these Jordan forms together with Ž22. and considering Ž21., we obtain the following expressions for matrices e x 2 T 2 and e x 3 T 3 : a

cosh 2 e x2 X2 s



(n n 1

(

0 n1

y

n3

sinh 2

y 3

(

n3

1 a

sinh 2

y n3

sinh 2

n1

0

0

Ž cosh2 y 1 .

cosh 2

,

Ž 26 .

(

where h 2 s n1 n 3 x 2 , and

(

y



B2 s y

n1 n3

j x 2 n˜ 2 sinh 2

j

'

n 3 n1 n 3

sinh 2 q

an˜ 2 x 2 n3

'

n 3 n1 n 3 cosh 2

2 aj y n1 n 3

2

sinh 2 y

an˜ 2 x 2

Ž cosh 2 y1. y

j y n˜ 2 x 2 cosh 2

n1 n 3

n3

Ž cosh 2 y1. q

cosh 2

a2 n˜ 2 x 2

'

n 3 n1 n 3 an˜ 2 x 2

'n n

1 3

n˜ 2 x 2 cosh 2

j sinh 2

sinh 2

n1 n 3

Ž cosh 2 y1. q

an˜ 2 x 2

'n n

1 3

(

y

n3 n1

x 2 n˜ 2 sinh 2

0

sinh 2 .

Ž 27 . where j s an ˜ 3 q n˜ 2 a.

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

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Also

e x3X3 s



(

cosh 3

(

n1

y

sinh 3

n2

n2 n1

a

sinh 3

(n n 1

a

cosh 3

0

sinh 3

y

n2

2

Ž cosh3 y 1 .

0

1

0

,

Ž 28 .

(

where h 3 s n1 n 2 x 3 , and

(

y

B3 s



n1 n2

j˜ x 3 n˜ 3 sinh 3

y n˜ 3 x 3 cosh 3

(

n˜ 3 x 3 cosh 3

j˜ y

'

n 2 n1 n 2

sinh 3 q

y

an˜ 3 x 3 n2

n2 n1

y n1 n 2

an˜ 3 x 3

j˜ x 3 n˜ 3 sinh 3

j˜ cosh 3

'

n 2 n1 n 2

sinh 3 y

Ž cosh 3 y1. y

an˜ 3 x 3

'n n

y n1 n 2 sinh 3

1 2

2 a j˜ y n1 n 2

2

Ž cosh 3 y1. y

n2

cosh 3

an˜ 3 x 3

'n n

sinh 3

1 2

Ž cosh 3 y1. y

a2 n˜ 3 x 3

'

n 2 n1 n 2

sinh 3

0

.

Ž 29 . where j˜s an ˜ 2 q n˜ 3 a. Now, considering the defenitions of bŽ g . and aŽ g . given in Ž4., in general we have w3x:

Ž30. Hence due to Ž19. and Ž20. we have: Ž31. Finally, by using Ž21. we get: aŽ g . s e x1 X 1 e x 2 X 2 e x 3 X 3 ,

t

t

t

b Ž g . s B1 e x 2 X 2 e x 3 X 3 q eyx 1 X 1 B2 e x 3 X 3 q eyx 1 X 1 eyx 2 X 2 B3 .

Ž 32 .

Indeed by the above mentioned prescription, we can evaluate a˜Ž g˜ . and b˜ Ž g˜ .. Since the set of aŽ g . and bŽ g . associated with bialgebra Žg,g ˜ . is the same as the set a˜Ž g˜ . and b˜ Ž g˜ . of bialgebra Žg,g˜ ., therefore in order to write a˜Ž g˜ . and b˜ Ž g˜ ., it is enough to replace the parameters x i and Ž a,n1 ,n 2 ,n 3 . with x˜ i and Ž a,n ˜ ˜ 1 ,n˜ 2 ,n˜ 3 . in Ž31. and Ž32., respectively. As an example, in order to determine a˜Ž g˜ . and b˜ Ž g˜ . of the pair Ž IX,V ., we just need to calculate aŽ g . and bŽ g . of the pair Ž V, IX ., provided that we replace the coordinates x i with x˜ i. So far all calculations have been done on the basis of the parametrization Ž19. for Bianchi Lie groups G. In the case of usual parametrization of each of Bianchi groups 8 , one only needs to consider the order of multiplication of matrices e x i T i in Ž31., based on the chosen group parametrization, then the required relations Ž32. can be obtained for this new representation.

8

For the group IX this parametrization is of Eulerian type, and for other groups the form of parametrization is g s e x 1 X 3 e x 3 X 2 e x 2 X 1 w15x.

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

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Now, knowing the matrices P Ž g . and P˜ Ž g˜ . we can write the related s-models and their Poisson-Lie dual models. As it has been already mentioned, we can associate a pair of dual s models to each bialgebra of Table 2. Of course some of these models have already been studied. For example models related to the bialgebras Ž I,g. Žstandard nonabelian duality. with M as four dimensional space-time manifold ŽBianchi cosmological models. have already been studied in details w18x. Dual s-models associated with bialgebra Ž IX,V . have been investigated in w10,11,16x. Also in w12x the dual models of bialgebra Ž VIII,V . have been considered. Here in this article we just express two pair of dual s-models associated with bialgebras of Table 2. In the following examples, we assume that M is a four dimensional manifold with Ž y a 4 s  t 4.. Also, we assume that F Ž1. s F Ž2. s 0 and F s 1, while the group parametrization is the same as Ž19.. Hence the action Ž6. reads: S s y 12 d jq d jy

H

Ž E0 y1 q P .

i

y1 ij

Ž Eq ggy1 . Ž Ey ggy1 .

j

q Eq tEy t ,

Ž 33 .

and from Ž9. its dual action is: S˜s y 12 d jq d jy

H

Ž E0 q P˜ .

y1 ij

˜˜y1 . i Ž Ey gg ˜˜y1 . j q Eq tEy t Ž Eq gg

.

Ž 34 .

As an example we first start with bialgebra Ž II, II . which is a typical example of self-dual bialgebras. Using the informations of Table 1 together with the relations Ž24. – Ž29. we can evaluate all matrices needed to determine P Ž g . 9 , where these have been given in appendix B. Finally the matrix P Ž g . of this model can be written as: 0 P Ž g . s B1 s 0 0

0 0 x1



0 yx 1 . 0

0

Ž 35 .

Also we have dg P gy1 s Ž d x 1 q x 2 d x 3 . X 1 q d x 2 X 2 q d x 3 X 3 .

Ž 36 .

Now by considering E0 s I, from Ž33. we have Žin world sheet coordinate.: S s y 12 d s dt Em x 1 E m x 1 q 2 x 2 Em x 1 E mx 3 q

H

ž

q x2 2 q

1 1 q x1

2

/

1 1 q x1

Em x 3 E mx 3 q Em tE m t .

E 2 m

x 2 E mx 2

Ž 37 .

Actually this is a self-dual model with respect to Poisson-Lie dual transformation. Indeed if we choose E0 as an arbitrary invertible constant matrix then the dual s-model again will be the same as the original one but we must replace elements of E0 with those of E0 y1 . Actually this particular example is similar to two dimensional Borelian dual models of Ref. w3x.

9

In caculating these matrices we must remove a series of ambiguities.

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

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Another example is related to the bialgebra Ž II,V .. Again, after calculating the matrices Ž24. – Ž29. given in appendix B, P Ž g . can be written as:

x2 0 0

0 P Ž g . s yx 2 yx 3



x3 0 . 0

0

Ž 38 .

Here dg P gy1 has the same form as Ž36.. Hence for E0 s I, from Ž33. the action of the original model is: S s y 12 d s dt

½

q Ž1 q 2 x 2

2

H

1 1 q x2 2 q x3

. Em x 3 E mx 3

Em x 1 E m x 1 q 2 x 2 Em x 1 E mx 3 q Ž 1 q x 3

2

2

. Em x 2 E mx 2

5

q Em tE m t .

Ž 39 .

Finally, by calculating matrices Ž24. – Ž29., as given in appendix B, we determine P˜ Ž g˜ .: 0

0

P˜ Ž g˜ . s eyx˜ 1 0

0



0

0 &

&

0

ysinhx 1 .

sinhx 1

0

Ž 40 .

On the other hand, we have: dg˜ P g˜y1 s d x˜ 1 X˜ 1 q ey x˜ 1 d x˜ 2 X˜ 2 q ey x˜ 1 d x˜ 3 X˜ 3 .

Ž 41 .

Then from Ž34. the dual model can be written as: S˜s y 12 Hd s dt Em x˜ 1 E m x˜ 1 q

ey2 x˜ 1 1 q ey2 x˜ 1 sinh2x˜ 1

žE

m

x˜ 2 E mx˜ 2 q Em x˜ 3 E mx˜ 3 q Em tE m t .

/

Ž 42 .

5. Concluding remarks We have obtained dual algebras of all Bianchi type algebras. Then by introducing a non-degenerate adjoint invariant inner product over Žsix dimensional. Lie algebra of Drinfeld double, we have obtained many real bialgebras listed in Table 2. As it has been shown above, we can associate a pair of Poisson-Lie dual s-models to every pair of bialgebras of Table 2. This point is explained by two examples; the rest will apear in furture works. Beside introducing associated s-models with Poisson-Lie dualities, there are other important problems to be studied, and we mention some of them in the following. Actually determination of the Drinfeld doubles of Table 2 can be very important for the following reasons. Due to the existence of non-degenerate adjoint invariant metric on these groups, we can construct some important physical models such as WZNW models or gauge theories over them Žmostly non-compact.. Also one can determine the modular space of each of these Drinfeld doubles. Below we give the known doubles. Two of these doubles have already been known, namely the double Ž IX,V . is SO Ž3,1. and Ž I, I . is ŽUŽ1.. 6 . Our investigations show that the bialgebras of Table 2 are the following types. The Drinfeld doubles of Ž III, I .,Ž III, II .,Ž VI0 , I .,Ž VII0 , I . and Ž V, I . are real forms of the complexification of H4 m UŽ1. m UŽ1.. Also Ž III, III . is UŽ2. m UŽ1. m UŽ1.. On the other hand Ž V, II . and Ž VIIa , I . are real forms of the complexification of six dimensional Heisenberg algebra. Also the doubles Ž VII0 ,V .,Ž VI0 ,V .,Ž IX, I . and Ž VIII, I . correspond to the

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

488

real forms of ISO Ž3,c .. The doubles Ž II, II . and Ž VIII,V . are real forms of SO Ž4,c .. Finally doubles Ž IV, I .,Ž IV, II .,Ž VIIa , II . and Ž VIa , II . are solvable. The exact and thorough characterization of these doubles and those mentioned in Table 2 are under further investigation. On the other hand, by using the information of Table 2 together with the prescription of Refs. w20x and w21x one can study the Poisson-Lie T-duality in N s 2 superconformal WZNW models related to the bialgebras of Table 2, for both classical and quantum cases, similar to w20x and w21x. Furthermore for some bialgebras of Table 2 one can extend these studies to N s 4 superconformal WZNW models. Also, because of canonical transformation nature of Poisson-Lie T-duality, it would be interesting to find the generating function of these canonical transformations for each bialgebra of Table 2 w7x; similar to the work done in w11x for Ž IX,V .. Notice that here the algebra D can have more than one decomposition such as D s Žg [ g˜ . and D s Žk [ ˜k.. Hence it is possible to find canonical transformation which relates the s-model with group G to the one with group K w6x. On the other hand, one can research the commutativity of the renormalisation flow with Poisson-Lie T-duality, by explicit computing of the beta functions and Weyl anomaly coefficients at the 1-loop level for these s-models and their duals. Of course this work has already been done in w10x for Ž IX,V .. Finally, there is a possibility that some of these models have application in string cosmology w19x; then the Poisson-Lie T-duality will play a key role in interelating different cosmological models and their solutions.

Acknowledgements We wish to thank S.K.A. Seyed Yagoobi for carefully reading the article and for his constructive comments. Also we are very grateful to S.E. Parkhomenko for his useful comments about Poisson-Lie T-duality in SCWZNW models.

Appendix A Here in this appendix we give the required similarity transformations for obtaining Jordan forms of the matrices T2 and T3 . The similarity transformation related to the matrix T2 and its Jordan form are:

°

¶

j 0

1

0

1

y n1 0

0

n3

a

0

0

0

0

yin1

0

0

0

in1

0

0

j 0 S2 s

(n n 1

0

yi n3 ia

3

yin 3

(n n 1

3

0

,

j 0 yin˜ 2 n1

¢

i

0

n3

yan˜ 2

(

n1 n3

(

n˜ 2 n1 n 3

(n n 1

0

3

ß

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

489

where:

A1 s



(

i n1 n 3

in˜ 2

(

n1 n3

(

0

i n1 n 3

0

,



A2 s

(

yi n1 n 3

1

0

yi n1 n 3

(

0

.

Similarly, the similarity transformation related to the matrix T3 and its Jordan form are:

°

¶

j˜ 0

0

y n1

0

0

1

0

a

yn 2

0

0

0

in1

0

0

0

yin1

0

0

0

j˜ 0 S3 s

0

(n n 1

1

n2

in 2

2

(n n

yi ia

2

0

,

j˜ 0

0

¢yin˜ n

3 1

i

(

yn˜ 3 n1 n 2

(n n 1

n2

yan˜ 3

(

n1

2

0

n2

ß

where A 3 and A 4 are similar to A1 and A 2 except that we must replace Ž n 3 , n˜ 2 . by Ž n 2 , n˜ 3 ., respectively.

Appendix B In this appendix we give coupling matrices of bialgebras of the examples quoted at the end of Section 4. For bialgebra Ž II, II . we have: e

x1 X 1

s I,

e x2 X 2 s



0 B1 s 0 0 1 0 yx 2



0 0 x1

0 1 0

0 0 , 1

0

0 yx 1 , 0

0

B2 s B3 s 0,

1 e x3X3 s x3 0



0 1 0

x2 0 0

0 0 , 0

0 0 . 1

0

Similarly, for bialgebra Ž II,V . we have: e x 1 X 1 s I,

B1 s 0,

0 B2 s yx 2 0



0  B3 s

0 0 yx 3

0 0 0

x3 0 , 0

0

M.A. Jafarizadeh, A. Rezaei-Aghdamr Physics Letters B 458 (1999) 477–490

490

where the matrices e x 2 X 2 and e x 3 X 3 have the same form as the matrices for bialgerba Ž II, II ., as mentioned above. Finally the matrices related to the dual model of bialgebra Ž II,V . are: ˜1

e x˜ 1 X s e x˜ 1 I, 1 ˜2 e x˜ 2 X s 0 0



0 B˜1 s 0 0



yx˜ 2 1 0

0 0 , 1

0

0 0 sinh x˜ 1

0 ysinh x˜ 1 , 0

0

1 ˜3 e x˜ 3 X s 0 0



0 1 0

B˜2 s B˜3 s 0,

yx˜ 3 . 0 1

0

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