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April 17, 1998math.QA/9804084

On the bicrossproduct structures for the

U λ(isoω2...ωN (N )) family of algebras

J. C. Perez Bueno1

Departamento de Fısica Teorica and IFIC Centro Mixto Universidad de Valencia–CSIC

E–46100 Burjassot (Valencia), Spain

Abstract

It is shown that the family of deformed algebras U λ(isoω2...ωN (N )) has a different

bicrossproduct structure for each ωa = 0 in analogy to the undeformed case.

1 Introduction

Deformed algebras (usually called ‘quantum groups’) have received great attention sincethe original works of Drinfel’d, Jimbo and Faddeev, Reshetikhin and Takhtajan [ 1, 2, 3, 4]which gave a (unique) deformation procedure for simple Lie algebras. However, the defor-mation of non-simple Lie algebras has been characterized by the lack of a definite prescrip-

tion and this explains why inhomogeneous algebras do not have a unique deformation.A possible approach to deforming non-simple algebras is by extending the contraction

of Lie algebras to the framework of deformed Hopf algebras, an idea originally introducedby Celeghini et al. [5, 6]. As is well known, the standard Inonu-Wigner [7] contraction of (simple) Lie algebras leads to non-simple algebras which have a semidirect structure, wherethe ideal is the abelianized part of the original algebra. By introducing higher powers in thecontraction parameter or, equivalently, by performing two (or more) successive contractionsit is also possible to arrive to algebras with a central extension structure.

This simple mechanism becomes difficult to implement for deformed algebras for whichit is usually necessary to redefine the deformation parameter in terms of the contraction

one to have a well-defined contraction limit [5, 6]. This is the case, for instance, of theκ-Poincare algebra [8, 9] in which the deformation parameter κ appears as a redefinitionof the original (adimensional) parameter q of soq(3, 2) in terms of the De Sitter radius R.

A way to skip some of the problems of the standard contraction procedure for de-formed algebras is to use the method of ‘graded’ contractions. This mechanism was putforward by Moody, Montigny and Patera [10, 11] for Lie algebras and has been appliedrecently to describe a large set of deformed Hopf algebras [12, 13]. The scheme provides

1E-mail: [email protected]

1

http://arxiv.org/abs/math/9804084v1






































http://arxiv.org/abs/math/9804084

http://arxiv.org/abs/math/9804084




the deformation of all motion algebras of flat affine spaces in N dimensions (the deformedCayley-Klein (CK) algebras U λ(isoω2...ωN

(N )) 2) including, the κ-Poincare algebra in arbi-trary dimensions, other deformations of the Poincare N -dimensional algebra, the Galileialgebra, etc.

A different point of view to study inhomogeneous deformed algebras is provided byMajid’s bicrossproduct structure [16, 17, 18]. In this construction we find the analogue of the Lie algebra semidirect structure (and of the central extension structure in the moregeneral case) for Hopf algebras and provides, for this reason, an appropriate setting forthe study of deformations of inhomogeneous algebras. This structure covers most of thedeformed algebras obtained by contraction but not all (see [19]). Thus, in the case of deformed algebras, the correspondence between contraction and semidirect structure thatexists in the Lie algebra setting is not straightforward.

Nevertheless, the study of the particular algebras for which the structure of bicrossprod-

uct is present, turns out to be useful to understand its properties because the deformationis mainly encoded in the action and coaction mappings that characterize the bicrossprod-uct, whereas the (two) Hopf algebras from which the bicrossproduct deformed algebra isconstructed are usually undeformed. In the appropriate limit of the deformation parame-ter we obtain the undeformed algebra, the coaction mapping is trivialized and the actionmapping is given by the Lie algebra commutators so that we recover the semidirect productstructure. A particular example is the κ-Poincare algebra [8] the bicrossproduct structureof which was found by Majid and Ruegg [20].

Recently [21] (see also [22]) has been shown that the whole family of deformed inhomo-geneous CK algebras U λ(isoω2...ωN

(N )) has a bicrossproduct structure, in analogy to thesemidirect one that appears after the contraction which goes from so( p,q) to iso( p,q) 3

and that it remains under all the possible graded contractions. However, the question thatnaturally arises is whether these contractions carry new bicrossproduct structures relatedto the semidirect ones of the undeformed algebra which are the result of each contraction(see (2.2) below). We prove in this paper that this is indeed the case so that, for everygraded contraction in the inhomogeneous deformed CK family U λ(isoω2...ωN

(N )), we havean associated bicrossproduct structure. The (a priori non-obvious) fact that all the pos-sible semidirect product structures of the undeformed inhomogeneous CK algebras have adirect counterpart in the deformed case is the main result of this paper.

The paper is organized as follows. In sec. 2 we provide an account of the (undeformed)CK algebras and their graded contractions. In sec. 3 some of the results in [21] are

summarized. They will permit us in sec. 4 to show that for each possible graded contractionin the CK family a new bicrossproduct structure arises. Our results are illustrated at theend with some examples.

2 The orthogonal Cayley-Klein family of algebras are the Lie algebras of the motion groups of realspaces with a projective metric [14, 15].

3 Note that associated to each contraction there exists, in the undeformed level, a semidirect productstructure and, in the deformed level, a possible bicrossproduct structure associated to it. In this particularcase the contraction so( p, q) → iso( p, q) gives rise to a semidirect structure in which we have p+ q abelian(momentum) generators and a (pseudo-)orthogonal group acting on them.

2



2 Cayley-Klein algebras

Let us start by recalling the definition of the orthogonal Cayley-Klein family of algebras.The (orthogonal) real Lie algebra so(N +1) can be endowed with a Z⊗N

2 grading group andcorresponding to its graded contractions we may introduce a set of Lie algebras dependingon 2N − 1 real parameters [23]. This set includes the original so(N + 1) algebra, all thepossible pseudo-orthogonal ones and many contracted algebras, as well as the N (N + 1)/2dimensional abelian one. The simplicity of the original so(N +1) algebra is lost for arbitrary

contractions and different algebras in this set may have different properties (as the numberof independent Casimir operators).

However there exists a subfamily, the members of which share many properties with the(parent) simple Lie algebra and hence may be called ‘quasi-simple’. This family, denotedby soω1...ωN

(N +1), is a set of algebras characterized by N real parameters (ω1, . . . , ωN ) and

corresponds to a natural subset of all possible graded contractions that may be obtainedfrom so(N +1) (within this family we find, for instance, the original so(N +1) algebra, theN -dimensional Poincare algebra, the Euclidean algebra, etc.). These algebras correspondexactly to the motion algebras of the geometries of a real space with a projective metricin the Cayley–Klein sense [14, 15] and are therefore called CK orthogonal algebras. Theirnon-zero brackets are

[Jab, Jac] = ωabJbc , [Jab, Jbc] = −Jac , [Jac, Jbc] = ωbcJab , (2.1)

where ωab =b

s=a+1 ωs and a < b < c. By simple rescaling of the generators the values ωi

may be brought to one of the values 1, 0 or –1.

The structure of these algebras may be defined by two main statements:• When all ωi are non-zero the algebra is isomorphic to a certain (pseudo-)orthogonal

algebra.• When a constant ωa = 0 the resulting algebra soω1,...,ωa=0,...,ωN

(N + 1) has the semidi-rect structure

soω1,...,ωa=0,...,ωN (N + 1) ≡ t⊙ (soω1,...,ωa−1(a)⊕ soωa+1,...,ωN

(N + 1 − a)) , (2.2)

where t is an abelian subalgebra of dimension dim t = a(N + 1 − a) and the remainingsubalgebra is a direct sum. In particular, when a = 1 we obtain the usual (pseudo)orthogonal inhomogeneous algebras soω1=0,ω2,...,ωN

(N + 1) with semidirect structure

soω1=0,ω2,...,ωN (N + 1) ≡ isoω2,...,ωN

(N ) = tN ⊙ soω2,...,ωN (N ) . (2.3)

The structure behind the decomposition (2.2) can be described visually by setting thegenerators in a triangular array (see Fig. 2.1). The generators spanning the subspace t arethose inside the rectangle, while the subalgebras soω1,...,ωa−1(a) and soωa+1,...,ωN

(N + 1− a)correspond to the two triangles to the left and below the rectangle respectively. In theω1 = 0 (ωN = 0) case the box is reduced to a single row (column) in the large triangle.

3



J01 J02 . . . J0a−1 J0a J0 a+1 . . . J0N

J12 . . . J1a−1 J1a J1a+1 . . . J1N

. ..

..

.

..

.

..

.

..

.Ja−2a−1 Ja−2 a Ja−2a+1 . . . Ja−2N

Ja−1 a Ja−1a+1 . . . Ja−1N

Jaa+1 . . . JaN

. . ....

JN −1N

Figure 2.1: Generators of the CK soω1,...,ωN (N + 1) algebra

H P 1 P 2 P 3

V 1 V 2 V 3

J 3 −J 2

J 1

,

T −1 T −1T L−1 T −1T L−1 T −1T L−1

T L−1 T L−1 T L−1

1 1

1

Figure 2.2: Generators of the Galilei algebra and its dimensional assignment

To distinguish between the generators we shall denote by X those inside the box (abelianalgebra) and by J those in the two triangles. Namely,

Xij ⇒ i < a and j ≥ a ,Jij ⇒ i ≥ a or j < a .

(2.4)

When two constants are set equal to zero (ωa and ωb say) we have two different semidirectdecompositions (2.2) corresponding to the constant ωa = 0 or to the constant ωb = 0.For instance, the (3,1)–Galilei algebra appears in this context for ω1 = 0, ω2 = 0, ω3 =1, ω4 = 1 and accordingly has two different semidirect structures which correspond to theconstants ω1 and ω2. In the triangular array this may be seen in Fig. 2.2 (for a discussion

on the dimensional analysis of the different contractions see [21, Section 2.2]).

3 The deformed family of inhomogeneous CK alge-bras

Let us start with the set of inhomogeneous CK algebras isoω2...ωN (N ) (see (2.3)). There ex-

ists [12, 13] a family of Hopf algebras, denoted by U λ(isoω2...ωN (N )), that are a deformation

of these CK algebras and, therefore, may be called ‘quantum’ inhomogeneous CK algebras.

4



In [21] it was shown that all these deformed algebras are endowed with a bicrossproductstructure that corresponds to the undeformed semidirect one (2.3) in which the abelianalgebra is given by the single row with generators J

0i(see Fig. 2.1 for a = 1).

Explicitly the deformed Hopf algebra U λ(isoω2...ωN (N )) is given (in the basis in which

its bicrossproduct structure is displayed) by• Commutators

[J0i, J0 j] = 0 , [J0i, J0N ] = 0 ,[Jij, Jik] = ωijJ jk , [Jij, J jk ] = −Jik , [Jik, J jk ] = ω jkJij ,[Jij, JiN ] = ωijJ jN , [Jij, J jN ] = −JiN , [JiN , J jN ] = ω jN Jij ,[Jij, J0k] = δikJ0 j − δ jkωijJ0i , [Jij , J0N ] = 0 ,

[JiN , J0 j] = δij

1− e−2λJ0N

2λ−

λ

2

N −1

s=1ωsN J

20s

+ λωiN J0iJ0 j ,

[JiN , J0N ] = −ωiN J0i .

(3.1)

• Coproduct

∆(J0i) = e−λJ0N ⊗ J0i + J0i ⊗ 1 , ∆(J0N ) = 1⊗ J0N + J0N ⊗ 1 ,

∆(Jij) = 1⊗ Jij + Jij ⊗ 1 ,

∆(JiN ) = e−λJ0N ⊗ JiN + JiN ⊗ 1 + λi−1s=1

ωiN J0s ⊗ Jsi − λN −1s=i+1

ωsN J0s ⊗ Jis .

(3.2)

• Counit ε(J0i) = ε(J0N ) = ε(Jij) = ε(JiN ) = 0 . (3.3)

• Antipode

γ (J0i) = −eλJ0N J0i , γ (J0N ) = −J0N , γ (Jij) = −Jij ,

γ (JiN ) = −eλJ0N JiN + λeλJ0N

i−1s=1

ωiN J0sJsi − λeλJ0N

N −1s=i+1

ωsN J0sJis .(3.4)

In this basis it is easy to check the following

Theorem 3.1 ([21])The deformed Hopf CK family of algebras U λ(isoω2...ωN

(N )) has a bicrossproduct structure

U λ(isoω2...ωN (N )) = U (soω2...ωN

(N ))β ⊲◭α U λ(T N ) (3.5)

relative to the right action

α(J0i, J jk) ≡ J0i ⊳ J jk := [J0i, J jk ] (3.6)

5



and left coaction β

β (Jij) = 1⊗ Jij ,

β (JiN ) := e−λJ0N ⊗ JiN + λi−1s=1

ωiN J0s ⊗ Jsi − λN −1s=i+1

ωsN J0s ⊗ Jis ,(3.7)

where U λ(T N ) is the abelian Hopf algebra generated by J0i and U (soω2...ωN (N )) is the

undeformed cocommutative Hopf algebra (with primitive coproduct) generated by Jij withthe commutation relations given in the second and third line of (3.1).

Let us now set ωa = 0; then the algebra U λ(isoω2...ωa=0...ωN (N )) is given (with the

notation in (2.4)) by• Commutators

X− sector

[Xij ,Xkl] = 0 (3.8)

J− sector

[J0i, J0k] = 0[Jij, J0k] = δikJ0 j − δ jkωijJ0i

[JiN , J0 j] = λωiN X0iJ0 j

[Jij, Jik] = ωijJ jk , [Jij , J jk ] = −Jik , [Jik, J jk ] = ω jkJij

[Jij, JiN ] = ωijJ jN , [Jij, J jN ] = −JiN , [JiN , J jN ] = ω jN Jij

(3.9)

JX− sector

[J0i,X0 j] = [J0i,X0N ] = 0[Jij,Xik] = ωijX jk , [Jij ,X jk ] = −Xik

[J jk ,Xij ] = Xik , [J jk ,Xik] = −ω jkXij

[Jij,XiN ] = ωijX jN , [Jij ,X jN ] = −XiN

[J jN ,Xij] = XiN , [J jN ,XiN ] = −ω jN Xij

[J0k,Xij] = −δikX0 j , [Jij ,X0k] = δikX0 j − δ jkωijX0i

[Jij,X0N ] = 0 , [JiN ,X0N ] = −ωiN X0i

[J0i,X jN ] = −δij

1− e−2λX0N

2λ−

λ

2

N −1s=a

ωsN X20s

[JiN ,X0 j] = δij

1− e−2λX0N

2λ−

λ

2

N −1s=a

ωsN X20s

+ λωiN X0iX0 j

(3.10)

• Coproduct

X− sector

∆X0N = 1⊗X0N + X0N ⊗ 1 , ∆Xij = 1⊗ Xij + Xij ⊗ 1∆X0i = e−λX0N ⊗ X0i + X0i ⊗ 1

∆XiN = e−λX0N ⊗XiN + XiN ⊗ 1− λN −1s=a

ωsN X0s ⊗Xis

(3.11)

6



J− sector

∆J0i = e−λX0N ⊗ J0i + J0i ⊗ 1 , ∆Jij = 1⊗ Jij + Jij ⊗ 1

∆JiN = e−λX0N ⊗ JiN + JiN ⊗ 1 + λa−1s=1

ωiN J0s ⊗Xsi

+ λi−1s=a

ωiN X0s ⊗ Jsi − λN −1s=i+1

ωsN X0s ⊗ Jis

(3.12)

• Counitε(Jij) = ε(J0i) = ε(JiN ) = 0ε(Xij) = ε(X0i) = ε(X0N ) = ε(XiN ) = 0

(3.13)

• Antipode

X− sector

γ (X0N ) = −X0N , γ (Xij) = −Xij

γ (X0i) = −eλX0N X0i

γ (XiN ) = −eλX0N XiN − λeλX0N

N −1s=a

ωsN X0sXis

(3.14)

J− sector

γ (Jij) = −Jijγ (J0i) = −eλX0N J0i

γ (JiN ) = −eλX0N JiN + λeλX0N

a−1s=1

ωiN J0sXsi

+λeλX0N

i−1s=a

ωiN X0sJsi − λeλX0N

N −1s=i+1

X0sJis

(3.15)

This algebra, as result of theorem 3.1, has a bicrossproduct structure (3.5). However thetheorem does not give us information about the (possible) bicrossproduct structure for thedecomposition given in (2.2). This is the problem that we address now.

4 Bicrossproduct structure

The algebra given above (3.8)-(3.15) does not present directly a bicrossproduct structure for

the decomposition (2.2). This is due to the term λa−1

s=1 ωiN J0s⊗Xsi in the JiN coproduct(second line in (3.12)) and to the commutator [JiN , J0 j] (third line in (3.9)) that does notclose a J algebra4.

Let us define

J iN = λa−1s=1

ωiN J0sXsi (4.1)

4 Nevertheless in the particular case a = N (ωN = 0) these terms are not present, and the change of basis given in (4.4) is not necessary (notice that for ωN = 0 the change of basis is trivial).

7



that verifies

∆J iN = e−λX0N

⊗ J iN + J iN ⊗ 1 + λ

a−1s=1

ωiN J0s ⊗Xsi + λ

a−1s=1

ωiN e−λX0N

Xsi ⊗ J0s (4.2)

and[J iN , J0 j] = λωiN J0 jX0i . (4.3)

Thus, the change of basis JiN → JiN − J iN solves the two difficulties pointed out before.Specifically, if we introduce the new set of generators

J 0i = J0i , X 0i = X0i , X 0N = X0N ,J ij = Jij , X ij = Xij , X iN = XiN ,

J iN = JiN − J iN

(4.4)

the algebra U λ(isoω2,...,ωa=0,...,ωN (N )) is written as

• Commutators

X − sector

[X 0i, X 0 j ] = [X 0i, X 0N ] = 0[X ij, X 0k] = [X ij, X kl] = [X ij , X kN ] = 0[X iN , X 0 j] = [X iN , X 0N ] = [X iN , X jk] = [X iN , X jN ] = 0

(4.5)

J − sector

[J 0i, J 0k] = 0[J ij , J 0k] = δikJ 0 j − δ jkωijJ 0i[J iN , J 0 j ] = 0

[J ij , J ik] = ωijJ jk , [J ij , J jk ] = −J ik , [J ik, J jk ] = ω jkJ ij[J ij , J iN ] = ωijJ jN , [J ij , J jN ] = −J iN , [J iN , J jN ] = ω jN J ij

(4.6)

JX −sector

[J 0i, X 0 j] = [J 0i, X 0N ] = 0[J ij , X ik] = ωijX jk , [J ij, X jk ] = −X ik[J jk , X ij] = X ik , [J jk , X ik] = −ω jkX ij[J ij , X iN ] = ωijX jN , [J ij, X jN ] = −X iN , [J jN , X ij] = X iN

[J jN , X iN ] = −ω jN X ij + λω jN

1− e−2λX0N

2λ−

λ

2

N −1s=a

ωsN X 20s

X ij

[J 0k, X ij ] = −δikX 0 j , [J ij, X 0k] = δikX 0 j − δ jkωijX 0i[J ij , X 0N ] = 0 , [J iN , X 0N ] = −ωiN X 0i[J kN , X ij] = δ jkX iN + λωkN X 0 jX ik

[J 0i, X jN ] = −δij

1− e−2λX0N

2λ−

λ

2

N −1s=a

ωsN X 20s

[J iN , X 0 j] = δij

1− e−2λX0N

2λ−

λ

2

N −1s=a

ωsN X 20s

+ λωiN X 0iX 0 j

(4.7)

8



• Coproduct

X − sector

∆X 0N = 1⊗X 0N + X 0N ⊗ 1 , ∆X ij = 1⊗X ij + X ij ⊗ 1

∆X 0i = e−λX0N ⊗X 0i + X 0i ⊗ 1

∆X iN = e−λX0N ⊗X iN + X iN ⊗ 1− λN −1s=a

ωsN X 0s ⊗X is

(4.8)

J − sector

∆J 0i = e−λX0N ⊗ J 0i + J 0i ⊗ 1 , ∆J ij = 1⊗ J ij + J ij ⊗ 1

∆J iN = e−λX0N ⊗ J iN + J iN ⊗ 1− λa−1s=1

ωiN e−λX0N X si ⊗ J 0s

+ λi−1

s=a

ωiN X 0s ⊗ J si − λN −1

s=i+1

ωsN X 0s ⊗ J is

(4.9)

• Counitε(J ij) = ε(J 0i) = ε(J iN ) = 0ε(X ij) = ε(X 0i) = ε(X 0N ) = ε(X iN ) = 0

(4.10)

• Antipode

X − sector

γ (X 0N ) = −X 0N , γ (X ij) = −X ijγ (X 0i) = −eλX0N X 0i

γ (X iN ) = −eλX0N X iN − λeλX0N

N −1

s=a

ωsN X 0sX is

(4.11)

J − sector

γ (J ij) = −J ijγ (J 0i) = −eλX0N J 0i

γ (J iN ) = −eλX0N J iN − λeλX0N

a−1s=1

ωiN X siJ 0s

+λeλX0N

i−1s=a

ωiN X 0sJ si − λeλX0N

N −1s=i+1

ωsN X 0sX is

(4.12)

In this basis we now state the following

Theorem 4.1The algebra U λ(isoω2...ωa=0...ωN

(N )) has the bicrossproduct structure

U λ(isoω2...ωa=0...ωN (N )) = U (soω1=0,...,ωa−1(a))⊕U (soωa+1,...,ωN

(N +1−a))β ⊲◭α U λ(T a(N +1−a))(4.13)

where U (soω1=0,...,ωa−1(a)) is the undeformed Hopf algebra generated by {J ij, i < (a −

1), j < a}, U (soωa+1,...,ωN (N +1−a)) is spanned by the generators {J ij , i > (a−1), j > a}

and U λ(T a(N +1−a)) is the deformed abelian algebra generated by X ij (recall that those

9



generators are restricted to the indices i < a and j ≥ a, (2.4)). The right action α : U λ(T a(N +1−a))⊗U (soω1=0,...,ωa−1(a))⊕U (soωa+1,...,ωN

(N + 1−a)) → U λ(T a(N +1−a)) is definedby (4.7) through

α(X ij , J kl) ≡ X ij ⊳ J kl := [X ij, J kl] (4.14)

and the (left) coaction β : U (soω1=0,...,ωa−1(a))⊕U (soωa+1,...,ωN

(N +1−a)) → U λ(T a(N +1−a))⊗ U (soω1=0,...,ωa−1(a))⊕U (soωa+1,...,ωN

(N +1−a)) is designed to reproduce the coproduct (4.9)

β (J ij) = 1⊗ J ijβ (J 0i) = e−λX0N ⊗ J ij

β (J iN ) = e−λX0N ⊗ J iN − λa−1s=1

ωiN e−λX0N X si ⊗ J 0s + λ

i−1s=a

ωiN X 0s ⊗ J si

−λ

N −1

s=i+1 ωsN X 0s ⊗ J is

(4.15)

From theorem 4.1 it is easy to check the following

Corollary 4.1Associated to each graded contraction in the inhomogeneous CK family of deformed alge-bras we have a different bicrossproduct structure related to the corresponding Lie algebrasemidirect structure that appears in the contraction (see (2.2)). These bicrossproductstructures are preserved under further (graded) contraction processes.

5 Examples5.1 N = 3 case

In the N = 3 case we obtain (in the basis of (3.1)-(3.4)) the following equations

[J0i, J0 j] = 0 , i, j = 1, 2, 3 ,[J12, J13] = ω2J23 , [J13, J23] = ω3J12 , [J12, J23] = −J13 ,[J12, J03] = 0 , [J12, J01] = J02 , [J12, J02] = −ω2J01 ,[J13, J03] = −ω2ω3J01 , [J23, J03] = −ω2J02 ,

[J13, J01] =1− e−2λJ03

2λ

+λ

2

ω3(ω2J201 − J

202) ,

[J23, J02] =1− e−2λJ03

2λ−

λ

2ω3(ω2J

201 − J

202) ,

[J13, J02] = λω2ω3J01J02 , [J23, J01] = λω3J01J02 ,

(5.1)

∆J0i = e−λJ03 ⊗ J0i + J0i ⊗ 1 , i = 1, 2 ,∆J03 = 1⊗ J03 + J03 ⊗ 1 , ∆J12 = 1⊗ J12 + J12 ⊗ 1 ,∆J13 = e−λJ03 ⊗ J13 + J13 ⊗ 1− λω3J02 ⊗ J12 ,

∆J23 = e−λJ03 ⊗ J23 + J23 ⊗ 1 + λω3J01 ⊗ J12 .

(5.2)

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We have stressed with a box the terms that might not allow the algebra to be abicrossproduct (see first paragraph in sec. 4). If ω3 = 0 these terms cancel5 but if ω2 = 0we keep them and we need the change of basis given in (4.4). In the new basis we have

∆J 23 = e−λX03 ⊗ J 23 + J 23 ⊗ 1− λω3X 12e−λX03 ⊗ J 01 (5.3)

and[J 23, J 01] = 0 . (5.4)

Thus, the terms marked above have disappeared after the change of basis and we have abicrossproduct structure as given in theorem 4.1.

5.2 A particular case: the Heisenberg-Weyl algebra

Now we are going to study the case a = N (i.e., ωN = 0). First, let us rename the

generators in the basis (4.4) as

J 0i = X i , J ij = J ij , X iN = Y i , X 0N = Ξ (5.5)

(note that for ωN = 0 the X sector is reduced to a single column in the triangular arrayin Fig. 2.1). Now the equations (4.5), (4.6) and (4.7) acquire the form

[X i, Ξ] = [Y i, Ξ] = [J ij, Ξ] = 0 ,[X i, X j] = 0 , [Y i, Y j] = 0 ,[J ij , J ik] = ωijJ jk , [J ij, J jk ] = −J ik , [J ik, J jk] = ω jkJ ij ,[J ij , X k] = δikX j − δ jkωijX i , [J ij , Y k] = δikωijY j − δ jkY i ,

[X i, Y j] = −δij1− e−2λΞ

2λ

.

(5.6)

In this way, we easily recognize the deformed Heisenberg-Weyl (HW) algebra [19] whereΞ is the central generator and the J ij generators act as a rotation group on the X i and Y igenerators. The coproduct (4.8)-(4.9) takes the form

∆Ξ = 1 ⊗ Ξ + Ξ ⊗ 1 , ∆J ij = 1⊗ J ij + J ij ⊗ 1 ,∆X i = e−λΞ ⊗X i + X i ⊗ 1 , ∆Y i = e−λΞ ⊗ Y i + Y i ⊗ 1 .

(5.7)

From the arguments given above we know that this algebra has two different bicrossproduct(semidirect like) structures, one for the abelian algebra generated by {X i, Ξ}, and the otherfor the abelian algebra generated by {Y i, Ξ} [19].

But, in this case, we have an additional cocycle-bicrossproduct structure (analogue tothe undeformed central extension structure of the HW-algebra). To see this let us define thealgebra H as the undeformed algebra generated by {X i, Y i, J ij} with primitive coproductand commutators

[J ij , J ik] = ωijJ jk , [J ij, J jk ] = −J ik , [J ik, J jk] = ω jkJ ij ,[J ij , X k] = δikX j − δ jkωijX i , [J ij , Y k] = δikωijY j − δ jkY i ,[X i, X j] = 0 , [Y i, Y j] = 0 , [X i, Y j] = 0

(5.8)

5Note that for ω3 = 0 (i.e., ωN = 0) the change of basis (4.4) is trivial (see (4.1) and footnote 4).

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(note that all the commutators are identical to those in (5.6) but the [X i, Y j] one that nowis abelian). The algebra A is the undeformed algebra U (Ξ). Now if we define the rightaction ⊳ :

A ⊗ H→AΞ ⊳ J ij = 0 , Ξ ⊳ X i = 0 , Ξ ⊳ Y i = 0 (5.9)

(central extension means trivial action), the left coaction β : H→A ⊗H

β (J ij) = 1⊗ J ij , β (X i) = e−λΞ ⊗X i , β (Y i) = e−λΞ ⊗X i , (5.10)

the antisymmetric two-cocycle ξ : H ⊗H→A 6

ξ(X i, Y j) = −ξ(Y j, X i) = −δij2

1− e−2λΞ

2λ

(5.11)

and a trivial ‘two-cococycle’ the HW algebra is given by the bicrossproduct

U λ(HW ) = H⊲◭ξ A . (5.12)

In this form it is easy to recover the dual algebra Funλ(HW) [19]. Let Rij be the dualgenerators corresponding to the undeformed ‘rotation’ algebra generated by J ij

7 and letxi, y j be the dual coordinates to the generators X i, Y j. Then, the algebra H dual to H isgiven by

∆Rij = Rik ⊗Rkj ,∆xi = 1⊗ xi + xk ⊗Rki , ∆yi = 1⊗ yi + yk ⊗R−1

ik ;(5.13)

[Rij , Rkl] = [Rij , xk] = [Rij , yk] = [xi, y j] = 0 . (5.14)

If we introduce the coordinate χ dual to the central generator Ξ we may complete the dual

algebra by dualizing the left coaction (5.10) and the two-cocycle (5.11). The left action isdefined as the dual to the left coaction

χ ⊲ xi = [χ, xi] = −λxi , χ ⊲ yi = [χ, yi] = −λyi , χ ⊲ Rij = [χ, Rij ] = 0 (5.15)

and the dual to the two-cocycle defines the two-cococycle8

ψ(χ) =1

2(yi ⊗R−1

ji x j − xi ⊗Rijy j) . (5.16)

Thus, the coproduct is given by

∆χ = 1⊗ χ + χ⊗ 1 +1

2(yi ⊗R−1

ji x j − xi ⊗Rijy j) . (5.17)

As we may see the bicrossproduct structure (with cocycle in this case) allows us to recoverFunλ(HW) in an easy way from the enveloping (dual) algebra U λ(HW).

6 The antisymmetric form of the cocycle is a matter of convention; different forms of the cocycle arerelated by a coboundary change (see [24] for an explicit example).

7This algebra is a true rotation algebra for ωi = 1 i = 1, . . . .N − 1; in general it is an inhomoge-neous algebra (if some ω = 0) or a pseudo-orthogonal algebra. The dual algebra is given by the matrixrepresentation Rij.

8 As said in footnote 6 we may choose a different form of the two-cococycle. For instance ψ(χ) =yi ⊗R−1ji xj is also a two-cococycle (related to (5.16) by the cocoboundary 1

2yixi).

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Acknowledgements

The author thanks J. A. de Azcarraga for his comments on the manuscript. This paperhas been partially supported by a research grant (PB096–0756) from the MEC, Spain. Theauthor wishes to acknowledge an FPI grant from the Spanish Ministry of Education andCulture and the CSIC.

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