Quasitriangular structure and twisting of the 3D bicrossproduct model

Transcription

1 Quasitriangular structure and twisting of the 3D bicrossproduct model MAJID, SH; Osei, P The Author(s) 2018 This article is distributed under the terms of the Creative Commons Attribution License (CC- BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. For additional information about this publication click this link. Information about this research object was correct at the time of download; we occasionally make corrections to records, please therefore check the published record when citing. For more information contact scholarlycommunications@mul.ac.uk

2 Quasitriangular structure and twisting of the 3D bicrossproduct model S. Majid and P. K. Osei December 14, 2017 Abstract We show that the bicrossproduct model C[SU 2 ] U(su 2) uantum Poincaré group in 2+1 dimensions acting on the uantum spacetime [x i, t] = ıλx i is related by a Drinfeld and module-algebra twist to the uantum double U(su 2 ) <C[SU 2 ] acting on the uantum spacetime [x µ, x ν ] = ıλɛ µνρ x ρ. We obtain this twist by taking a scaling limit as 1 of the -deformed version of the above, where it corresponds to a previous theory of -deformed Wick rotation from -Euclidean to -Minkowski space. We also recover the twist result at the Lie bialgebra level. 1 Introduction and Motivation It is now widely accepted that uantum gravity effects may plausibly lead to spacetime, even flat spacetime, being better modelled by noncommuting coordinates than classical ones. One such model that is is clearly related to 3D uantum gravity (without cosmological constant and with point sources) is the angular momentum algebra [x µ, x ν ] = ıλɛ µνρ x ρ as spacetime, as first proposed by t Hooft in the mid 1990s[1, 2, 3, 4]. Another well-known model from the mid 1990s is the Majid-Ruegg bicrossproduct model[5, 6] with spacetime [x i, t] = ıλx i, which we take in the 3D case. Both models are notably for having Poincaré uantum group symmetries of interest in their own right[2, 7]. In the present paper we show that these well-known models are in fact related and in some sense euivalent via a module-algebra (or Drinfeld-type) twist. Previously it was explained at the -deformed level in [8] that these two models are uantum Born reciprocal or semidual aspects of 3d uantum gravity and that at this -deformed level they are also related by twisting and hence in some sense self-dual up to twisting euivalence. However, the isomorphisms used were highly singular as 1, so only apply strictly with cosmological constant. The interpretation of -deformation here as introducing a cosmological constant is clear from the close link between the uantisation of the relevant Chern-Simons theory (giving the Turaev-Viro invariant), the relevant -deformation uantum group and the relevant WZNW model of conformal field theory[9, 10]. In this context, uantum Born reciprocity interchanges the cosmological and Planck scales for a fixed value of -deformation parameter and the uantum double D(U (su 2 )) = U (so 1,3 ) with the uantum group U (su 2 ) cop U (su 2 ) = U (so 4 ) at the 1

3 level of isometry uantum group, so these are twisting euivalent, a result first introduced in [11] as uantum Wick rotation, see [12]. Our surprising new result is that by working out the structures in great detail and carefully taking the 1 limit while at the same time scaling the generators, i.e in a contraction limit, a remnant of the result survives in the form a module algebra twist between the above two uantum spacetimes and their Poincaré uantum groups D(U(su 2 )) and C[SU 2 ] U(su 2 ) respectively. The role of D(U(su 2 )) in particular for constructing the states of 3d uantum gravity with point sources is well established and we refer to [4, 8] for an introduction. That a scaling contraction limit of U (so 4 ) gives a uantum Euclidean group was first pointed out in [13] and this is presumably isomorphic to C[SU 2 ] U(su 2 ) in the same way as the 4d uantum Poincaré uantum group proposed in [7] by contraction of U (so 2,3 ) was shown in [5] to be a bicrossproduct C[R R 3 ] U(so 1,3 ). Our scaling limit result is striking because the two uantum spacetimes models appear very different and have always been treated as such; one uantum spacetime is the enveloping algebra of a simple Lie algebra and the other of a solvable one. One Poincaré uantum group is uasitriangular while bicrossproducts are not usually uasitriangular, although the 3d one in [13] is, a result which in our version is now explained by twisting as this preserves uasitriangularity. Moreover, whereas the uasitriangular structure of the double exists formally, it does not take a usual algebraic form as the exponential of generators, whereas our universal R-matrix on the bicrossproduct does and this implies such a form also for the double by twisting. Similarly, when uantum spacetimes are related by a module algebra twist then their covariant noncommutative differential geometry is related by twisting[14, 15] and hence that must also be the case here: For the spin model the smallest covariant calculus is known to be 4D [2] and for the standard bicrossproduct models it is known to be one dimension higher than classical [16], so again 4D but now this is explained by our twisting result. Similarly, the construction of particle state representations of D(U(su 2 )) by the Wigner little group method in [8] should have a parallel on the bicrossproduct model side via twisting. Such possible applications will be considered elsewhere. The paper starts in Section 2 with some general Hopf algebra constructions which underly the uantum Wick rotation[11] and semidualisation in [17, 8] but which were not given so explicitly before. We carefully specialise these to U (su 2 ), again giving all constructions in explicit detail in Sections These exact formulae then allow us in Section 3.6 to take the 1 limit with suitably scaled generators. This is a rather tricky process due to 1/(1 2 ) singularities but we remarkably do obtain finite results, which we then verify explicitly, see Corollary 3.1. Section 4 rounds off the paper with the Poisson-Lie or semiclassical level version of our results in line with [19] and mainly as a further check of our calculations (notably, we show that we recover the expected Lie bialgebra double r-matrices). Our results relate to a different Lie bialgebra contraction than [20] but the latter may emerge as a different limit of our results. Another direction for future work is that U (su 2 ) as uantum spacetime is a unit hyperboloid in -Minkowski space and as such its constant-time slices give the 2-parameter Podles spheres [21], all of which may have a parallel on the bicrossproduct model side of the twisting. Another topic for further study is to look carefully at the different signatures (so the spin model 2

4 is the version for Euclideanised 3D uantum gravity) and the relationship between the different real forms as reflected in the applicable -structures. 2 Explicit Hopf algebra isomorphisms This section brings together two different contexts in the book [12]. The first, about semidualisation, was explained in [8] in the present context of 3d uantum gravity while the second about twisting was explained in [11] in the context of uantum Wick rotation. It was also outlined in [8] how to bring these together but now we need to work out the underlying isomorphisms rather explicitly, which is not easy from the literature. We use the conventions for Hopf algebras in [12] namely a Hopf algebra or uantum group H is both an algebra and a coalgebra, with coproduct : H H H which is an algebra homomorphism. There is also a counit ɛ : H k if we work over k and an antipode S : H H defined by (Sh (1) )h (2) = h (1) Sh (2) = ɛ(h) for all h H and notation h = h (1) h (2). We shall refer to a covariant system (H, A) meaning a Hopf algebra H acting on an algebra A as a module algebra, i.e., in the left handed case, h (ab) = (h (1) a)(h (2) b), h 1 = ɛ(h). where is a left action. There is then a left cross-product algebra A> H. We refer to [12] for details. We denote by H a suitable dual Hopf algebra with dual pairing given by a non degenerate bilinear map, and H cop, H op denote taking the flipped coproduct or flipped product. As an easy exercise, if H acts covariantly on A from the right then is a left action of H on A op as another covariant system. 2.1 Semidualisation and the uantum double h a = a S 1 h (2.1) (i) A double crossproduct Hopf algebra H 1 H 2 can be thought of as a Hopf algebra H which factorises into two sub-hopf algebras built on H 1 H 2 as a vector space. By factorisation, we mean a map H 1 H 2 H as an isomorphism of linear spaces. One can then naturally extract the actions : H 2 H 1 H 1 and : H 2 H 1 H 2 of each Hopf algebra on the vector space of the other defined by (1 a).(h 1) = a (1) h (1) a (2) h (2) for the product viewed on H 1 H 2 obeying some further compatibility properties (one says that one has a matched pair of interacting Hopf algebras). Conversely given such data one can reconstruct the algebra of H 1 H 2 from these actions as a double (both left and right) cross product. The coproduct of H 1 H 2 is the tensor one given by the coproduct of each factor and there is a canonical right action of this Hopf algebra on the vector space of H 2 which respects the coalgebra structure of H 2 and thus provides in a canonical way a covariant left action of H 1 H 2 on H2 as an algebra. Here H 1 acts on H2 by dualising the above right action on H 2, and H 2 acts on H2 by the coregular action a φ = φ (1) a, φ (2). Hence we have a covariant system (H 1 H 2, H2) and an associated cross product H2> (H 1 H 2 ). Further details are in [12] and earlier works by the first author. 3

5 (ii) The semidual of this picture associated to the same matched pair data was introduced by the first author, see [12] for details, and is constructed by dualising the data involving H 2 to give a bicrossproduct Hopf algebra H 2 H 1 which then acts covariantly on H 2 from the right as an algebra as the semidual covariant system (H 2 H 1, H 2 ). The remarkable fact is that (H 2 H 1 ) <H 2 = H 2> (H 1 H 2 ) as algebras, i.e. the combined system is the same actual algebra but its interpretation is different in that the role of spacetime coordinates H 2 and momentum H 2 coordinates in the first case is reversed in the other, with rotations H 1 the same. This is the B-model semidualisation referred to in [8]. There is eually well an A-model semidualisation where we dualise H 1 to obtain H 2 H 1 acting on the left on H 1 while H 1 H 2 acts on the right on H 1 and the two covariant systems again have the same cross products, H 1 > (H 2 H 1) = (H 1 H 2 ) <H 1. These ideas go back to the first author as a new foundation ( uantum born reciprocity ) proposed for uantum gravity namely that one can swap position and momentum generators in the algebraic structure [17]. We will particularly need details of the B-model which were not provided explicitly in [8]. Starting with a matched pair H 1, H 2 acting on each other the left action : H 1 H 2 H 2 of H 1 on H 2 and a right coaction R : H 1 H 1 H 2 of H 2 on H 1 are define by (h φ)(a) := φ(a h), φ H 2, a H 2 h H 1 h 0 h 1, a = a h, h H 1, a H 2, R h = h 0 h 1 H 1 H 2. These define the bicrossproduct H 2 H 1 by a left handed cross product H 2> H 1 as an algebra and a right handed cross coproduct H 2 <H 1 as coalgebra: (φ h)(ψ g) =φ(h (1) ψ) h (2) g, h H 1, φ, ψ H 2 (2.2) (φ h) =(φ (1) h 0 (1)) (φ (2) h 1 (1) h (2) ) (2.3) The canonical right action of H 2 H 1 on H 2 is a (φ h) = a (2) h φ, a (1), h H 1, a H 2, φ H 2. (2.4) Note that H 2 1 and 1 H 1 appear as subalgebras with cross relations hψ =(1 h)(ψ 1) = h (1) ψ h (2) = (h (1) ψ 1)(1 h (2) ) = (h (1) ψ)h (2) where we identify h = 1 h and ψ = ψ 1. (iii) We now apply the above construction to the specific case of the Drinfeld uantum double D(H) = H H op due to [18] and viewed as an example of a double crossproduct from work of the first author, see [12] for details. Here the right action of H on H op and the left action of H op on H are given respectively by a h = a (2) h (1), a (1) Sh (2), a (3), a h = h (2) h (1), a (1) Sh (3), a (2), h H, a H op. (2.5) The double cross product H H op then comes out as (h a).(g b) = hg (2) ba (2) g (1), a (1) Sg (3), a (3), h, g H, a, b H, (2.6) 4

6 with the tensor product coproduct. This Hopf algebra canonically acts on (H op ) = H cop from the left as an algebra. The action is (h a) φ = φ (1), a h φ (2), φ H cop (2.7) in terms of the coproduct of H and the action in (2.8). Semidualising, the left action of H on H cop already referred to and the right coaction of H cop on H are respectively h φ = h (1) φsh (2) = Ad h (φ), R h = h (2) h (1) Sh (3) (2.8) so that the product of H cop H is therefore (φ h)(ψ g) = φh (1) ψ h (2) g, h H, φ, ψ H cop (2.9) as the standard cross product H Ad > H and the coproduct is (φ h) = φ (2) h (2) φ (1) h (1) Sh (3) h (4) (2.10) in terms of coproducts of H. This Hopf algebra acts covariantly on H op from the right according to a (φ h) = φh (1), a (1) a (2) Sh (2), a (3), h H, φ H cop, a H. (2.11) Using (2.1), and correctly using the inverse antipode of the bicrossproduct determined by the coproduct (2.10) gives the covariant left action of the bicrossproduct uantum group on H as (φ h) a = Sh (1) Sφ, a (1) a (2) h (2), a (3). (2.12) In summary, the semidual of the left covariant system (D(H), H cop ) is the right covariant system (H cop H, H op ), which is (H cop H, H ) as a left covariant system with action (2.12). This is essentially as in [12] where we denoted H cop H = M(H) the mirror product but now in our current conventions and, critically, keeping track of algebras on which our Hopf algebras act. (iv) Finally, we observe as a right-left flipped version of [12, Prop ] that there is a Hopf algebra isomorphism θ 1 : H cop H H cop H, θ 1 (φ h) = φsh (1) h (2), θ 1 1 (φ h) = φh (1) h (2) (2.13) under which the right action of H cop H on H op by (2.11) is isomorphic to a right action of H cop H on H op by a (φ h) = a H cop Hθ 1 (φ h) = a (2) φ, a (1) Sh, a (3). and by observation (2.1) this is euivalent to H cop H acting on the left on H by (φ h) a = a S 1 (φ h) = a (Sφ S 1 h) = a (2) Sφ, a (1) h, a (3) (2.14) and this is also θ 1 (φ h) a acting by (2.12). In summary, the semidual of the left covariant system (D(H), H) acting by (2.7) is isomorphic to the left covariant system (H cop H, H ) acting by (2.14). This action is euivalent to a left action of H and a right action of another copy of H it is H with a natural H-bimodule structure afforded by the coproduct (the Hopf algebra version of left and right derivatives on H ). 5

7 2.2 Twisting of module algebras and uantum Wick rotation (i) We recall following Drinfeld that a uasitriangular Hopf algebra is a pair (H, R), where H is a Hopf algebra and R is an invertible element of H H satisfying In this case R obeys ( id) (R) =R 13 R 23, (id ) (R) = R 13 R 12 cop (h) =R( h)r 1, h H. (ɛ id)(r) = (id ɛ)(r) = 1, (S id)r = R 1, (id S)R 1 = R, where we write R = R [1] R [2] with the notation that R 12 R 13 R 23 = R 23 R 13 R 12, (2.15) R ij = 1... R [1] 1... R [2]... 1 is the element of H H... H which is R in the i th and j th factors. The identity (2.15) is known as the uantum Yang-Baxter Euation (QYBE) and on account of this R is also called a Universal R-matrix. Next, an element χ H H for any Hopf algebra H is called a twisting 2-cocycle [12] if χ 12 ( id)χ = χ 23 (id )χ, (ɛ id)χ = 1 and in this case there is a new Hopf algebra H χ with the same algebra and [12], χ h = χ( h)χ 1 R χ = χ 21 Rχ 1, S χ h = U(Sh)U 1 h H χ, (2.16) where U = (id S)χ is invertible. Moreover, if H acts covariantly on A from the left then H χ acts covariantly on a new algebra A χ with product a χ b = (χ 1 (a b)). (2.17) This cocycle twisting theory was introduced by the first author in [22, 11] and other works from that era (Drinfeld did not consider 2-cocycles or module algebra twists but rather conjugation by general elements χ in the category of uasi-hopf algebras). Clearly, if H is uasitriangular and we take 2-cocycle χ = R, then H χ = H cop. (ii) Following [11], we similarly see that H cop H acting on H by (2.14) twists via χ 1 = R 1 13 to H H acting on a new algebra, which we will denote H, with product a b = R (a b) = ((R [1] 1) a)((r [2] 1) b) = R(a (1) b (1) )a (2) b (2) Thus the covariant system (H cop H, H ) at the end of Section 2.1 twists to (H H, H ) by χ 1. Moreover, the further twist of H H by the 2-cocycle χ 2 = R 1 23 gives a Hopf algebra which we will denote H R H (it is technically a double cross coproduct) acting 6

8 covariantly on an algebra H with a b = (R 23 (a b)) =a (1) R [1], a (2) b (2) SR [2], b (1) =a (2) b (3) R(a (1) b (2) )R(a (2) Sb (1) ) =a (2) b (3) R(Sa (1) Sb (2) )R(a (3) Sb (1) ) =a (2) b (2) R(Sa (1) a (3) Sb (1) ) for all a, b H, where we view R by evaluation as a map on H 2 and use the axioms of R in dual form. This product makes H with its unchanged coproduct into a braided- Hopf algebra as part of the theory of transmutation so the result in [11] was that this can be seen as a twist (namely by χ = χ 2 χ 1 = R 1 23 R 1 13 = ( id)r 1 ). (iii) Moreover it is known [12, Thm 7.3.5] that there is a Hopf algebra map θ 2 : D(H) H R H, θ 2 (h a) = h (1) R [2] h (2) R [1] a, R [1] R [2] (2.18) according to inclusions i = and j(a) = (id id a)(r 1 31 R 23 ) of H, H op in H R H. Note also that the latter has at least a couple of interesting uasitriangular structures built from R namely, R D = R 1 41 R 1 31 R 24 R 23 = (χ 2 ) 21 R 1 13 R 24 χ 1 2, R L = R 1 41 R 13 R 24 R 23 = (χ 2 ) 21 R 31 R 24 χ 1 2 (2.19) with R D the image under θ 2 of the canonical uasitriangular structure of D(H) (at least if H is finite dimensional so that the latter is defined). In the factorisable case the map θ 2 is an isomorphism of Hopf algebras, where factorisable means Q = R 21 R viewed by evaluation as a map Q : H H by Q(a) = a, Q [1] Q [2] is an isomorphism. This holds formally for the standard uantum groups associated to semisimple Lie algebras. Pulling back under θ 2 we compute using the Hopf algebra and uasitriangularity axioms that D(H) = H H op acts covariantly on H by (h a) b = θ 2 (h a) H cop Hb = b (3) h, (Sb (2) )b (4) R(a (1), b (1) )R(b (5), a (2) ) (2.20) This is also the action on of θ 1 θ 2 (h a) H cop H on b for the action (2.12). Lemma 2.1 Q : H H is a map of covariant algebras, intertwining the action of D(H) in (2.20) with its action on H in (2.7). Proof It is known [12, Prop 7.4.3] that Q is a homomorphism of braided-hopf algebras where H is as above with unchanged coproduct and H has an unchanged product and modified coproduct. In particular, it maps the algebras and Q(b (2) ) h, (Sb (1) )b (3) = h Q(b) as it intertwines the left action given by evaluating with the right adjoint coaction, with the left adjoint action of H. Hence Q((h a) b) = R(a (1), b (1) )R(b (3), a (2) )h Q(b (2) ) = Q(b (1) ), a (1) R((Sb (2) )b (4), a (2) )h Q(b (2) ) = Q(b (1) )R [2], a (hr [1] ) Q(b (2) ) = = Q(b) (1) R [2], a (hr [1] ) Q(b) (2) = Q(b) (1), a h Q(b) (2) = (h a) Q(b) 7

9 where we used that Q(b (1) ) Q(b (2) ) = Q(b) = Q(b) (1) SR [2] R [1] Q(b) (2) and indicated the braided coproduct by the underlining the numerical suffices. The 2nd euality is easily proven by breaking down the 3rd expressions in terms of parings of H with H and using the uasitriangular and Hopf algebra pairing axioms. Putting all the above together, we arrive at our main result: Theorem 2.2 If H is factorisable then the covariant system (D(H), H) in Section 2.1 viewed via Q as a covariant system (D(H), H ) is isomorphic to a twisting of its semidual (H cop H, H ). Here we twist by (θ 1 θ 1 )(χ) = R 1 23 and the isomorphism is given by θ = θ 1 θ 2 : D(H) H cop H, where θ(h a) = h (1) Q [2] Sh (2) h (3) R [1] a, Q [1] R [2] Moreover, H cop H has two uasitriangular structures given by θ 1 θ 1 of R 1 13 R 24 and R 31 R 24. Proof We combine the results above together with an straightforward computation for θ. We recognise χ = χ 2 χ 1 = R 1 23 R 1 13 = ( id)r = (θ1 1 θ1 1 )R 1 23 so under θ 1 this maps over to R 1 23 (H cop H) 2. Because the action of the double on the vector space H in Lemma 2.1 agrees via θ with the action (2.12), it means that Q at the algebra level with the transmuted product and θ at the uantum symmetry level together form an isomorphism of the covariant systems as stated. We can also compute the uasitriangular structures in H cop H explicitly in terms of R s using the axioms of a uasitriangular structure as R BD = (θ 1 θ 1 )(R 1 13 R 24 ) = R 1 13 (S id S id)( )R = R 1 13 (S id S id)r 14 R 24 R 13 R 23 = (id S 1 id id)(r 1 13 R 23 R 13 R 1 14 R 1 24 ) R BL = (θ 1 θ 1 )(R 31 R 24 ) = R 31 (S id S id)( )R = R 31 (S id S id)r 14 R 24 R 13 R 23 = (id S 1 id id)(r 31 R 23 R 13 R 1 14 R 1 24 ) where all expressions are reduced to tensor products of H. 3 Computations for H = U (su 2 ) and 1 scaling limit Here we obtain the main result, starting with explicit formulae in the -deformed case. 3.1 The Hopf algebra U (su 2 ) = C [SU 2 ] We recall that the Hopf algebra U (su 2 ) is defined over formal power series C[[t]] with generators H, X ±, where = e t 2, say. The relations are defined by [H, H] = 0, [H, X ± ] = ±2X ±, [X +, X ] = H H 8. (3.1) 1

10 The coproduct, counit and antipode are given by H = H H (X ± ) = H 2 X± + X ± H 2, ɛ(h) = 0 ɛ(x ± ) = 0, (3.2) S(H) = H, S(X ± ) = ±1 X ±, For real, the -structure takes the form H = H X± = X. The Hopf algebra U (su 2 ) is called the -deformation of the universal enveloping algebra U(su 2 ). It is uasitriangular with real-type universal R-matrix where e z 2 is the -exponential e z 2 = k=0 [k; 2 ][k 1; 2 ]...[1; 2 ]. This means that e z 2 e z 2 R = H H 2 e (1 2 ) H 2 X + H 2 X 2, (3.3) z k [k; 2 ]!, with [k; 2 ] = 1 2k and [k; 2 ]! = 1 2 = 1 and that if AB = 2 BA, then e A+B = e A 2 e B 2. 2 Next, unusually, we write U (su 2 ) as C [SU2 ] where latter has α, β, γ, δ generators of B [SU 2 ] related via the map Q. The coordinate algebra B [M 2 ] is the space of 2 ( 2 braided ) Hermitian matrices [29, 12], or -Minkowski space, with generators u = α β satisfying the relations γ δ βα = 2 αβ, γα = 2 αγ, δα = αδ, [β, γ] = (1 2 )α(δ α), [δ, β] = (1 2 )αβ, [γ, δ] = (1 2 )γα, (3.4) ( ) ( ) α β α γ and real form =. Its uotient by the braided-determinant relation γ δ β δ det(u) = αδ 2 γβ = 1 gives the braided group B [SU 2 ] or -hyperboloid. When 1 this algebra with α 1 adjoined provides a version U (su 2 ) via the map Q the uantum Killing form [12] as ( ) ( ) α β Q = H 1 2 ( 1 ) H 2 X (3.5) γ δ 1 2 ( 1 )X + H 2 H + 1 ( 1 ) 2 X + X which we regard as an identification. If we assume α is invertible then the element δ is determined by the braided-determinant relation and not regarded as a generator. This map can also be viewed as essentially an isomorphism between the braided enveloping algebra BU (su 2 ) (which has the same algebra as U (su 2 )) and its dual which is the braided function algebra B [SU 2 ]. Here, the unbraided coproduct of B [SU 2 ] as inherited from that of U (su 2 ) is α = α α β = 1 β + β α γ = 1 γ + γ α, Sα = α 1 Sβ = 2 α 1 β Sγ = γα 1, ɛ(α ±1 ) = 1, ɛ(β) = ɛ(γ) = 0. (3.6) The R-matrix becomes R = H H 2 e (1 2 ) 1 γ α 1 β 2 ; α = H. (3.7) We denote U (su 2 ) in the form of the algebra of B [SU 2 ] with α invertible and the coproduct in (3.6) as the Hopf algebra C [SU 2 ]. 9

11 3.2 The Hopf algebra C [SU 2 ] = U (su 2) The well-known Hopf algebra C [SU 2 ] is the dual of U (su 2 ) and can be viewed as the uantum deformation of the algebra of functions of SU(2). A set of generators for C [SU 2 ] is given by the matrix elements t i j : U (su 2 ) C in the defining representation of U (su 2 ) where h, t i j = ρ(h) i j, h U (su 2 ), t i j C [SU 2 ] (3.8) ( ) a b and ρ is in the spin- 1 representation. As usual we write 2 ti j = which we recall c d have the relations ba =ab, bc = cb, bd = 1 db, ca =ac, cd = 1 dc, da = ad + ( 1 )bc. (3.9) The coproduct, counit and antipode are given by a =a a + b c, b = a b + b d, c = c a + d c, d = c b + d d, ɛa =ɛd = 1, ɛb = ɛc = 0, Sa = d, Sd = a, Sb = b, Sc = 1 c, (3.10) and the real form by a = d, b = 1 c for real. The duality pairing takes the form ± H 2, a = ± 1 2, ± H 2, d = 1 2, X+, b = 1, X, c = 1. (3.11) Applying a representation (3.8) to one half of the R-matrix leads to the definition of the well-known L-matrices (L + ) i j = R [1] ρ(r [2] ), (L ) i j = ρ(r [1] )R [2], (3.12) where L + = ( H µx + H 2 ) L = ( H µx 0 H 2 ), µ = 1 2. We also, unusually, write C [SU 2 ] with new generators z, x ± defined by ( ) ( ) a b z 1 2 µx =. (3.13) c d 3 2 µx + z (1 + µ 2 x + x ) If we assume a is invertible, the element d is not regarded as a generator as it is fixed by the -determinant relation det (t) = ad 1 bc = 1. The algebra then takes the form [x ±, z] = x ±, [x +, x ] = 0. (3.14) The coproduct, counit and antipode can then be translated as ( z ) = z z + 2 µ 2 x x +, (x ) = z x + x z + µ 2 x z x + x, (x + ) = x + z + z x + + µ 2 x + x z x +, ɛ(z) = 0, ɛ(x ± ) = 0, S( z ) = z (1 + µ 2 x + x ), S(x ± ) = 1 x ±. (3.15) We denote C [SU 2 ] with a invertible as the Hopf algebra U (su 2). The corresponding -structure on U (su 2) is given by x = x +, ( z ) = z (1 + µ 2 x + x ). 10

12 3.3 The uantum double covariant system (D(U (su 2 )), U (su 2 )) The uantum double of U (su 2 ) is the double cross product Hopf algebra D(U (su 2 )) = U (su 2 ) C [SU 2 ] op, with algebra structure given by (3.1), the opposite algebra to (3.9) together with cross relations obtained from (2.6) as [ H 2, a] = 0, H 2 b = 1 b H 2, H 2 c = c H 2, [ H 2, d] = 0, X a = 1 ax + b H 2, [X, b] = 0, [X, c] = ( H 2 d H 2 a), dx = 1 X d + H 2 b, ax + = X + a + H 2 c, [X+, c] = 0, [X +, b] = 1 ( H 2 a H 2 d), X+ d = dx + + c H 2. (3.16) The coproduct, counit and antipode are given by (3.2) for the Hopf subalgebra U (su 2 ) and the coproduct, the counit and inverse of the antipode in (3.10) for the Hopf subalgebra C [SU 2 ] op. The uantum double D(U (su 2 )) canonically acts on U (su 2 ) from the left with (3.1) as algebra, resulting in the covariant system (D(U (su 2 )), U (su 2 )). The left covariance action is given by (2.7) as H H = 0, H X ± = ±2X ±, X ± H = 2 ±1 H 2 X ±, X ± X ± = ( ±2 ±1 ) H 2 X 2 ±, X ± X = H 2 (X ± X ±1 X X ± ), a H = 1 + H, a H 2 = 1 2 H 2, a X ± = 1 2 X ±, b H = 0, b H 2 = 0, b X + = H 2, b X = 0, c H = 0, c H 2 = 0, c X + = 0, c X = H 2, d H = 1 + H, d H 2 = 1 2 H 2, d X ± = 1 2 X ±. (3.17) This is the standard -deformed uantum double system. This 1 corresponds to a cosmological constant in 3d uantum gravity. 3.4 The bicrossproduct covariant system (C [SU 2 ] cop U (su 2 ), U (su 2)) Here we use the alternative description of one of the U (su 2 ) as C [SU2 ] and of C [SU 2 ] as U (su 2) as explained above. From (2.8), we obtain the left action of U (su 2 ) on C [SU2 ] as ( ) ( ) ( ) ( ) α β 0 2β H =, ± H 2 α β α = 1 β γ δ 2γ 0 γ δ ±1, γ δ ( ) ( 3 ) ( ) ( ) α β 2 γ 1 2 (δ α) α β 1 2 β 0 X + =, X γ δ 0 1 =. 2 γ γ δ 1 2 (δ α) 3 2 β Writing α α 1, β β 1, γ γ 1, in the subalgebra C [SU 2 ] cop 1 and H 1 H, X ± 1 X ±, in 1 U [su 2 ] of the mirror product M(U (su 2 )) = C [SU 2 ] cop U (su 2 ), we obtain its cross relations from (2.9) as [H, α] = [H, δ] = 0, [H, β] = 2β, [H, γ] = 2γ, [X +, α] = 3 2 γ H 2, X + β = βx (δ α) H 2, X + γ = 1 γx +, [X +, δ] = 1 2 γ H 2, [X, α] = 1 2 β H 2, X β = βx, X γ = 1 γx (δ α) H 2, [X, δ] = 3 2 β H (3.18)

13 The coproduct is given by (2.10) as α = α α β = β 1 + α β γ = γ 1 + α γ, H = 1 H + H 1, X + = H 2 X + + X + α 1 H µ 1 ( H 2 H 2 ) γα 1 H 2, X = H 2 X + X α 1 H µ 1 ( H 2 H 2 ) α 1 β H 2. (3.19) This Hopf algebra covariantly acts from the left on U (su 2) with (3.14) as algebra giving, the covariant system (C [SU2 ] cop U (su 2 ), U (su 2)). From (2.12), we obtain this left action on C [SU2 ] as ( ) ( ) ( ) ( ) a b 0 2b H =, ± H 2 a b a = 1 b c d 2c 0 c d ±1, c d ( ) ( 3 ) ( ) ( ) a b 2 c 1 2 (a d) a b 1 2 b 0 X + =, X c d 0 1 =, 2 c c d 1 2 (d a) 3 2 b ( ) ( ) ( ) ( ) a b α = 1 a 1 b a b 0 0, β = µ, c d c d c d a b ( ) ( ) a b c d γ = 1 µ. (3.20) c d 0 0 This then translates to the left action on U (su 2) given by z 0 z H x + = 2x +, H 2 x + = x 2x x X + X α γ z x + x z x + x z x + x z x + x = = = = 3 µx + 0 µ 1 ( z z (1 + µ 2 x + x )), 2 µx 2 µ 1 ( z z (1 + µ 2 x + x )) 0 1 z x + 1 x, β z x + x 1 2 µ 2 x z (1 + µ 2 x + x ) = z x + 1 x, z 0,,. (3.21) This is the mirror product covariant system semidual to the uantum double one. The mirror product Hopf algebra[12] for generic 1 here is isomorphic to a tensor product so it not usually considered of interest, though it is for us. 3.5 Twisting euivalence of the -deformed covariant systems In this section we work out the algebra isomorphism and the twisting of the preceding two covariant systems as established in Theorem 2.2. Here the algebra isomorphism θ : D(U (su 2 )) C [SU 2 ] cop U (su 2 ) 12

14 is defined in Theorem 2.2 by θ(h t) = h (1) Q [2] Sh (2) h (3) R [1] t, Q [1] R [2], h U (su 2 ), t C [SU 2 ]. If R = R [1] R [2], then Q = R 21 R = R [2] R [1] R [1] R [2], so that Q 1 = R [1] R [2] R [2] R [1]. Therefore Q [2] t i k, Q [1] = R [2] R [1] t i k, R [1] R [2] = R [2] R [1] t i k, R [1] R [2] = R [2] R [1] t i m, R [1] t m k, R [2] = R [2] SR [1] t i m, SR [1] t m k, R [2] = L i msl +m k. This combination is a conjugate map Q to the one we used before. Hence θ(1 t i j) = Q [2] R [1] t i j, Q [1] R [2] = Q [2] R [1] t i k, Q [1] t k j, R [2] = L i msl +m k L +k j. Also, for h 1 U (su 2 ) 1, we have θ(h 1) = h (1) Q [2] Sh (2) h (3) R [1] 1, Q [1] R [2] = h (1) Sh (2) h (3) = 1 h. (3.22) In terms of our generators this means that θ identifies the H 2, X ± generators of the two uantum groups and ( ) a b θ c d = = ( H + 3 µ 2 X X µx H µ H 2 X + H ( 2 (δ µα) 2 β 2 γ α ) ( H µx + H 2 ) ( ) H µx + H 2 ) (3.23) where δ = α 1 (1+ 2 βγ) and in the 2nd expression we replace by α, β, γ for the generators of C [SU 2 ] cop. One can check that this is indeed an algebra isomorphism as dictated by the theorem. The Drinfeld twist of the bicrossproduct C [SU 2 ] cop U (su 2 ), defined in Theorem 2.2 as χ B = (θ 1 θ 1 )χ = R 1 23 C [SU 2 ] cop U (su 2 ) C [SU 2 ] cop U (su 2 ) is given by 1 H χ B = e X+ α 1 β H H 1 1 = e 2 KX+ α 1 β H H. (3.24) where H = α when viewed in C [SU2 ] cop and K = H 2 in U (su 2 ) and the second euality uses the identifications C [SU2 ] cop C [SU2 ] cop 1 and U (su 2 ) 1 U (su 2 ) in C [SU2 ] cop U (su 2 ). One can check that χ B ( )χ 1 B where is the coproduct of C [SU2 ] cop U (su 2 ), gives us a coalgebra isomorphic by θ to the coalgebra of the uantum double (so θ is not just an algebra isomorphism if we take this twisted coproduct) as per the theorem. For example, (θ θ) d = (θ θ)(d d + c b) = αk 1 αk 1 + ( 2 γk + α 1 2 µx+ ) ( 2 βk 1 ) 13

15 χ B ( θ(d))χ 1 B 1 = 2 KX+ α 1 β e (αk 1 αk 1 1 )e 2 KX+ α 1 β 2 2 which gives the same answer on writing A = 1 2 KX + α 1 β and B = αk 1 αk 1 and C = 3 γk K 1 β so that AB = 2 BA + C, AC = CA using the commutation relations in C [SU 2 ] cop U (su 2 ). Operators with these relations formally obey which we use. e A 2Be A 2 = B + ( 2 1)BA + C, The map of covariant algebras provided by Lemma 2.1 is computed from Q(t) = t, Q [1] Q [2] = L + SL and was already given in (3.5) when one notes that B [SU 2 ] and C [SU 2 ] have the same coalgebra and the same generators (but different algebra relations). From this point of view Q : U (su 2) U (su 2 ) is not an algebra map (we would have to use the transmuted or twisted product on the first algebra) and obeys ( z 1 2 µx Q 3 2 µx + z (1 + µ 2 x + x ) ) ( = H 1 2 µx + H µ H 2 X H + µ 2 X + X Here a n = a a a (n times) when one looks carefully at the transmuted product on the generator a = z, which implies at the level of U (su 2) generators. Q(z) = H, Q(x ) = H 2 X, Q(x + ) = 1 X + H 2. (3.25) We also obtain two uasitriangular structures for the bicrossproduct C [SU 2 ] cop U (su 2 ) defined in Theorem 2.2. Then we find expressions for R BD, R BL C [SU 2 ] cop U (su 2 ) C [SU 2 ] cop U (su 2 ) as follows: From Theorem 2.2, we have R BD = (id S 1 id id)(r 1 13 R 23 R 13 R 1 14 R 1 and similarly for R L with R 31 in place of R Writing R for U (su 2 ), this is R BD = (id S 1 id id) ( H H 1 e µkx + H K 1 X 1 e µh KX + K 1 X e µkx + 1 K 1 X 1 2 e µkx K 1 X 2 ). 24 ) e µh KX + 1 K 1 X 1 (H 1 1 H+1 H 1 H)) 2 2 = (id S 1 id id) ( H H 1 1 e 2 α KX+ α 1 β 1 e 1 µ γ H α 1 β e 1 µ γ 1 α 1 β 1 2 e 1 2 γ 1 1 K 1 X 2 e µα KX + 1 K 1 X ( H 1 1 H+1 H 1 H) ) (3.26) where K = H 2 and α = H viewed in C [SU2 ] cop. Here, we have written R BD as an element of U (su 2 ) U (su 2 ) U (su 2 ) U (su 2 ) in the first euality. In the second euality, we view the first and the third legs in C [SU2 ] cop using the map Q in (3.5) 1 and used the fact that e 2 α KX+ α 1 β 1 commutes with e 1 µ γ H α 1 β 1 2. Note that 2 14

16 S 1 reverses order, resulting in more complicated expressions if we apply this. Similarly, Theorem 2.2 gives R BL = (id S 1 id id) ( 1 2 H 1 H 1 e µk 1 X 1 KX H H 1 e µ1 KX + K 1 X H 1 H 1 e µkx + 1 K 1 X 1 2 e µ1 KX + 1 K 1 X e µkx K 1 X H 1 1 H (1 H 1 H)) = (id S 1 id id) ( 1 H 1 H 2 1 e 1 µ α 1 β 1 γ H H 1 1 e 2 1 KX+ α 1 β H 1 H 1 e 1 µ γ 1 α 1 β 1 2 e 1 2 γ 1 1 K 1 X 1 H 1 1 H 2 2 e µ1 KX + 1 K 1 X (1 H 1 H)). (3.27) 3.6 Limiting twist between the spin model and the bicrossproduct model We are now in position to consider the degenerations of the two covariant systems by scaling the various generators appropriately to recover the results of Theorem 2.2 in the limit 1. (i) For the uantum double D(U (su 2 )) = U (su 2 ) C [SU 2 ] op, the U (su 2 ) part has no problem with the limit 1 and so U (su 2 ) U(su 2 ), with the standard Lie brackets and cocommutative coalgebra [H, H] = 0, [H, X ± ] = ±2X ±, [X +, X ] = H, (3.28) H = H H (X ± ) = 1 X ± + X ± 1, ɛ(h) = 0 ɛ(x ± ) = 0, S(H) = H, S(X ± ) = X ±, (3.29) Similarly, C [SU 2 ] op C(SU 2 ), the commutative algebra of functions on SU 2 as 1. The cross relations in the limit can easily be extracted from (3.16) as [H, a] = 0, [H, b] = b, [H, c] = 2c, [H, d] = 0, [X, a] = b, [X, b] = 0, [X, c] = d a, [X, d] = b, [X +, a] = c, [X +, c] = 0, [X +, b] = a d), [X +, d] = c, (3.30) and the coproduct is the tensor product one. The uantum group U (su 2 ) is also the uantum spacetime algebra for the covariant system (D(U (su 2 )), U (su 2 )). In the limit the covariant action of D(U(su 2 )) = U(su 2 ) <C(SU 2 ) on U(su 2 ) is given by the 1 limit of (3.31) as H H = 0, H X ± = ±2X ±, X ± H = 2X ±, X ± X ± = 0, X ± X = H, a H = 1 + H, a X ± = X ±, b H = 0, b X + = 1, b X = 0, c H = 0, c X + = 0, c X = 1, d H = 1 + H, d X ± = X ±. (3.31) 15

17 This limit action was first computed in [2] and to match standard conventions, we choose generators ( e H = 2J 0, X ± = J 1 ± ıj 2, t i λp 3 + ı λ j = P 2 0 ı λ(p ) 2 1 ıp 2 ) ı λ(p ıp 2 ) e λp 3 ı λp, (3.32) 2 0 where λ R is a deformation parameter and P 3 not regarded as a generator but is determined by the det(t) = 1 condition. This gives the algebra as and the coproducts turns out to be [J a, J b ] = ıɛ abc J c, [P a, J b ] = ıɛ abc P c, [P a, P b ] = 0, (3.33) J a = J a J a, P a = P a P a λ 2 ɛ abcp b P c. (3.34) For the uantum spacetime algebra for the covariant system, we write λh = 2x 0, λx ± = x 1 ± ıx 2 (3.35) and then the limit of relations (3.28) gives the spin model spacetime algebra [x µ, x ν ] = ıλɛ µνρ x ρ. (3.36) This is the 1 limit (D(U(su 2 )), U(su 2 )) as a deformation of U(iso(3)) on R 3 [2]. (ii) In the covariant system (C [SU2 ] cop U (su 2 ), U (su 2)), we have C [SU2 ] cop C[SU2 ] cop with commutative algebra and U (su 2 ) U ( su 2 ) with the standard algebra (3.28). From (3.18), the cross relations becomes [H, α] = [H, δ] = 0, [H, β] = 2β, [H, γ] = 2γ, [X +, α] = γ, [X +, β] = (δ α), [X +, γ] = 0, [X +, δ] = γ, [X, α] = β, [X, β] = 0, [X, γ] = (δ α), [X, δ] = β. (3.37) The coproduct is obtained from (3.19) as α = α α β = β 1 + α β γ = γ 1 + α γ, H = 1 H + H 1 X + = 1 X + + X + α 1 + H 2 γα 1, X = 1 X + X α 1 + H 2 α 1 β. (3.38) In this limit the covariant action of C[SU 2 ] cop U(su 2 ) on U(su 2) from (3.21) is H z = 0, H x ± = ±2x ±, X ± x ± = 0, X ± z = 2x ±, X ± x = ±z α ±1 z = z 1, α x ± = x ±, β z = 0, β x + = 1, β x = 0, γ z = 0 γ x + = 0, γ x = 1. (3.39) To match standard conventions for the bicrossproduct uantum group C[SU 2 ] cop U(su 2 ), we now identify new generators α = e λp 0, β = λp +, γ = λp, P ± = p 2 ± ıp 1, H = 2M, X + = N 2 ın 1, X = N 2 + ın 1. (3.40) 16

18 Then the relations (3.37) become [p a, p b ] = 0, [M, N 1 ] = N 2, [M, N 2 ] = N 1, [N 1, N 2 ] = M, [M, p 0 ] = 0 [M, p i ] = ıɛ ij p j, [N i, p 0, ] = ıɛ ij p j e λp 0 ( ) [N i, p j ] = ı ɛ 2 ij e λp 0 e 2λp 0 1 λ p 2, i, j = 1, 2, λ (3.41) where p 2 = p p 2 2 and coproduct (3.38) gives p 0 = p p 0, M = 1 M + M 1, p i = p i 1 + e λp 0 p i, N i = 1 N i + N i e λp 0 + λm p i e λp 0, i = 1, 2. (3.42) For the model spacetime U (su 2), we set x 0 = ıλz, x 1 = ıλ(x + + x ), x 2 = λ(x + x ) (3.43) and then take the limit 1 to get from (3.14), the bicrossproduct model spacetime [x i, x 0 ] = ıλx i. (3.44) In terms of these standard generators (3.40), the covariant actions (3.39) can be translated as M x 0 = 0, M x i = ɛ ij x j, N i x 0 = ıx i, N i x j = ıδ ij x 0, p 0 x 0 = ı, p 0 x i = 0, p i x 0 = 0, p i x j = iɛ ij, i, j = 1, 2. (3.45) This is the bicrossproduct model covariant system (C[SU 2 ] cop U(su 2 ), U(su 2)) as a uantum Poincare group in three dimensions acting on the Majid-Ruegg uantum spacetime as a 3d version of [5]. (iii) We now look at the 1 limit of the twist between the two covariant systems. We remind the reader that for handling of the cocycle and R-matrices we reduced expressions to the tensor product of the underling Hopf algebras. In effect in what follows we euip the vector space of C[SU 2 ] cop U(su 2 ) with two products, one is the cross product algebra as part of the bicrossproduct construction and the other is the tensor product algebra. In the following, we use the convention that all exponentials are multiplied in the tensor product algebra, which does not impact (3.46) but is important for the correct reading of (3.47). Corollary 3.1 From the above analysis, we arrive at the degeneration limit 1 of our result that the covariant system (D(U(su 2 )) = U(su 2 ) C[SU 2 ] op, U(su 2 )) is isomorphic to a twisting of the covariant system (C[SU 2 ] cop U(su 2 ), U(su 2)). The twist is derived from (3.24) in the limit 1 as χ B0 = e X + α 1β e 1 2 H (α 1) (C[SU 2 ] cop U(su 2 )) 2. (3.46) 17

19 The degeneration limit 1 of (3.26) also provides an R-matrix for the bicrossproduct C[SU 2 ] cop U(su 2 ) given by R B0 = e 1 2 γh α 1β : e αx + α 1β : e 1 2 H (α 1) e γ X e 1 2 (α 1) H, (3.47) where : e αx + α 1β ( 1) n := α n X+ n (α 1 β) n. n! n=0 Proof We write α α 1, β β 1, γ γ 1, in the subalgebra C [SU2 ] cop 1 and H 1 H, X ± 1 X ±, in 1 U (su 2 ) of C [SU2 ] cop U (su 2 ), with = e t. Then in the limit t 0, the algebra isomorphism becomes θ : D(U(su 2 )) C[SU2 ] cop U(su 2 ) given by ( ) ( ) a b δ β θ =, θ(h) = h, h U(su c d γ α 2 ) (3.48) It is easy to check that θ is indeed an algebra isomorphism. For example [θ(x + ), θ(b)] = θ(a) θ(d), etc. Now we write α = e t 2 H = 1 + t 2 H + O(t 2 ), so that 1 2 H H = ( t) n n=0 n! H n H n 4 n = n=0 ( 1 H ) n (α 1) n +O(t 2 ) = e 1 2 H (α 1) +O(t 2 ), n! 2 and therefore from (3.24), we see that χ B χ B0 = e X + α 1β e 1 2 H (α 1) as t 0. To obtain the limit for the R-matrix, we first note that for X commuting with H, ( lim t 0 e X µ e 2 t H e X µ 2 2 ) = n=0 ( 1 X ) n n 1 (H 2i). n! 2 Then form (3.26), the limit of R BD as 1 becomes ( R B0 =(id S id id) e H (α 1) 1 e α X + α 1 β 1 ( 1) n ( γ 1 α 1 β 1 ) n 1 n (1 (H 2i) 1 1) n!2 n n=0 e γ 1 1 X e 1 2 (α 1) 1 1 H ), (3.49) where we used that X = γ 1 α 1 β 1 commutes with 1 H 1 1 and also that S 2 = id on U(su 2 ). Next, we also note that for any elements (id S id id) ( (a b c d) (A B C D) ) = (id S id id)(aa bb cc dd) = aa S(bB) cc dd = Aa (SB)(Sb) Cc Dd = (id S id id)(a B C D) (id S id id)(a b c d) provided dd = Dd, since the first and third legs are in C[SU 2 ] cop which is already commutative. Here, indicates that the product is in the tensor product one of the Hopf 18 i=1 i=1

20 algebra. Using this observations, (3.49) becomes R B0 =(id S id id) ( n=0 n 1 1 n!2 n γn (H + 2i) (α 1 β) n 1) ) i=1 (id S id id) ( e H (α 1) 1 e α X + α 1 β 1 ) e γ 1 1 X e 1 2 ((α 1) 1 1 H) (3.50) because the first two lines of (3.49) are of the form A B C 1 and the last line is unchanged under the action of (id S id id). We evaluate the first two lines of the above euation as follows: In the first line, we have (id S id id) = n=0 n=0 ( 1) n n 1 n!2 n γn (H 2i) (α 1 β) n 1 i=1 n 1 1 n!2 n γn (H + 2i) (α 1 β) n 1 = e 1 2 γh α 1β. i=1 The second line of (3.50) is evaluated as (id S id id) ( e H (α 1) 1 e α X + α 1 β 1 ) = e α X + α 1 β 1 e H (α 1) 1 = =: e αx + α 1β : e 1 2 H (α 1). ( 1) n α n X+ n (α 1 β) n e 1 2 H (α 1) n! n=0 Finally, we check that U(su 2 ) is a module algebra twist by χ B0 of U(su 2 ), i.e. U(su 2) χb0 = U(su 2 ), where the twisted product a χb0 b is given by (2.17). With χ 1 B 0 = e H 2 (α 1) e X + α 1β, we have that for example z χb0 x + = (χ 1 B 0 (z x + )) = (( ) m ) 1 1 H ((α ) X n m! n! 2 + z 1) m (α 1 β) n x + = (( ) m 1 H z) ((α 1) m x + ) + (( ) m 1 H ( 2x + )) ((α 1) m ( 1)) m! 2 m! 2 = zx + + 2x +, and x + χb0 z = (( ) m ) 1 1 H ((α X n m! n! 2 + x + 1) m (α 1 β) n z ) = (( ) m ) 1 H x + ((α 1) m z) m! 2 = x + z x +. 19

21 Computing all possible combination of products, we obtain the following twisted algebra [x +, x ] χb0 = [x +, x ] + z, [x ±, x ± ] χb0 = [x ±, x ± ], [z, z] χb0 = [z, z] [x +, z] χb0 = [x +, z] 3x +, [x, z] χb0 = [x, z] + x = 2x which on evaluating the product in U(su 2) gives We see that [x +, x ] χb0 = z, [x ±, x ± ] χb0 = 0, [z, z] χb0 = 0, [x ±, z] χb0 = 2x ±. Φ(H) = z, Φ(X ± ) = x ± (3.51) defines an isomorphism of U(su 2 ) with U(su 2) after twisting. This is manifestly the inverse of Q : U(su 2) U(su 2 ) of covariant algebras in Lemma 2.1 which in our case by (3.25) is just Q(z) = H, Q(x ± ) = X ± in the 1 limit. By construction, the identification must be covariant but it is a useful check to see this directly. For example, θ(x + ) Φ(H) = X + z = 2x + = Φ(X + H) θ(a) Φ(H) = δ z = z + 1 = Φ(1 + H) = Φ(a H) θ(d) Φ(X ) = α x = Φ(X ) = Φ(d X ). 4 Semiclassical limit of results In this section describe our twisting result at the infinitesimal Lie bialgebra level, i.e. the classical double as a Lie bialgebra twist of the bicross sum, both in the case when is switched on and in the scaling limit 1 with a parameter λ. We begin with a brief review of Lie bialgebras and classical r-matrices and refer the reader to [12, 23] and references therein for details. 4.1 Double and Semidual Lie Bialgebras and classical r-matrices A Lie bialgebra in the sense of Drinfeld provides a semiclassical or infinitesimal notion of a Hopf algebra. A Lie bialgebra (g, [, ], δ) is a Lie algebra (g, [, ]) over a field k euipped with a cocommutator δ : g g g is a skew-symmetric linear map, i.e. δ : g 2 g satisfying the cojacobi identity and that for all ξ, η g, (δ id) δ(ξ) + cyclic = 0, ξ g δ([ξ, η]) = (ad ξ ad ξ )δ(η) (ad η ad η )δ(ξ). There exist a Lie bialgebra version of the uasitriangular Hopf algebra which arise naturally in the following way: Since δ is a 1-cocycle, an element r = r [1] r [2] g g provides a coboundary structure for the Lie bialgebra (g, [, ], δ) if δ = r, i.e. δ(ξ) = 20

22 ad ξ (r) = [ξ ξ, r]. This reuires that ad ξ (r + r 21 ) = 0 for all ξ g to have δ antisymmetric. For any Lie algebra g, we define the map g 2 g 3, r [[r, r]] = [r 12, r 13 ] + [r 12, r 23 ] + [r 13, r 23 ]. (4.1) This map restricts to the map 2 g 3 g. The euation [[r, r]] = 0 (4.2) is called the classical Yang-Baxter euation(cybe) and any solution of the CYBE in g g is called a classical r-matrix. The classical r-matrix provides a uasitriangular structure for the Lie bialgebra. It is triangular if it satisfies the CYBE and r 21 = r and called factorisable if it satisfies the CYBE and r + r 21 : g g is a linear surjection. The classical r-matrix therefore provides a natural infinitesimal version of the universal R-matrix while the factorisable case correspond to R factorisable. If (g, [, ], r) is a uasitriangular Lie bialgebra and χ c g g obeys [[r, χ c ]] + [[χ c, r]] + [[χ c, χ c ]] = 0, ad ξ (χ c + χ c 21) = 0, ξ g, (4.3) then (g, [, ], r + χ c ) is also a uasitriangular Lie bialgebra. The element χ is called a Lie bialgebra twist and δ χ c = δ + χ c is also a Lie bialgebra. A uantised enveloping algebra roughly speaking means a Hopf algebra over C[[t]] generated by a vector space g, with relations and coproduct of the form ξη ηξ = [ξ, η] + O(t), ( cop )ξ = tδξ + O(t 2 ), (4.4) where t is a formal deformation parameter. Further, if the Hopf algebra has a uasitriangular structure of the form R = 1 + tr + O(t 2 ), (4.5) then δ = r, and our Lie bialgebra is uasitriangular. This interpretation is also compatible with twisting. More explicitly, if χ = 1 + tf + O(t 2 ), then from (2.16), we have χ c = f 21 f (4.6) Classical double and bicross sum Lie bialgebras provide a semiclassical version of the uantum doubles and bicrossproduct uantum groups respectively, described in Section 2.1. For any finite dimensional Lie bialgebra g with dual g, there is a uasitriangular Lie bialgebra, D(g), the classical double of g built on g g as a vector space, with [ξ φ, η ψ] = ([ξ, η] + ξ[1] ψ, ξ[2] η[1] φ, η[2] ) ([ψ, φ] + ψ[1] ψ[2], ξ φ[1] φ[2], η ), δ(ξ φ) = (0 φ[1]) (0 φ[2]) + (ξ[1] 0) (ξ[2] 0), r = a (0 f a ) (e a 0). 21

23 Here g op, g, appear as sub-lie bialgebras, where () op denotes the opposite (negated) Lie bracket. The set {e a } is a basis of g and {f a }, a dual basis. Moreover, if (g, r) is factorisable then D(g) =(g g) χ c 2 where we mean twisting by a certain cocycle χ c 2, which as we saw amounts to adding χ c 2 to the ( r 21 ) r if we want r D and to r r for another r-matrix r L. We can further view this as a twisting of g cop g by a certain other cocycle χ c 1 built from r. Semidualisation can also be defined for Lie bialgebras which are double cross sums. Given a matched pair of Lie algebras (g, m), one can define the double cross sum g m as the vector space g m. The semidual gives the the bicross sum Lie bialgebra m g built on m g with [f ξ, h η] = (ξ h η f) [ξ, η], δ(f ξ) = a (0 e a ξ) (f a 0) (f a 0) (0 e a ξ) + (f[1] 0) (f[2] 0), for all f ξ, h η m g, where δ(f) = f[1] f[2] is the Lie coalgebra given by the dualisation of the Lie bracket of m. For a detailed account of these constructions, we refer to [12]. In particular, the splitting from the double semidualises to a bicross-sum g cop g =g cop g as Lie bialgebras. Putting these facts together gives an isomorphism of uasitriangular Lie bialgebras θ c : D(g) (g cop g) χ c B, with θ c (ξ) = ξ, θ c (φ) = 2 r + (φ) + (id φ)r, χ c B = r 23 r 41, (4.7) for all ξ g and φ g, where r + = (r + r 21 )/2 is viewed as a map g g and tilde indicates that the result is viewed in the g cop copy. As a check, one has (θ c θ c )(f a e a ) = r 24 r 14 r 41 = r BD + χ c B, r BD = r 24 r 23 r 14 so the canonical r-matrix for the double when mapped over under the isomorphism is the twist of r BD. There is also r BL = r 31 r 23 + r 13 r 14 + r 24 which twists to the image under θ c θ c of the other r-matrix r L on D(g). This is the Lie bialgebra version of the general theory in Section 2. We now verify everything on our examples as a check. 4.2 Infinitesimal limit of results in the limit 1 In the infinitesimal Lie bialgebra limit, the uantum double D(U(su 2 )) becomes the Lie bialgebra double D(su 2 ) = su 2 <su op 2 = su 2 <R 3. Here, the su 2 parts has its standard Lie bracket and su op 2 = R 3 has a commutative algebra. The relations are given by (3.33) [J a, J b ] = ıɛ abc J c, [P a, J b ] = ıɛ abc P c, [P a, P b ] = 0. (4.8) 22

24 The bicrossproduct uantum group C[SU2 ] cop U(su 2 ) described in Section 3.6 in terms of the basis (3.40) can be viewed as a deformation of U(su cop 2 su 2 ), where λ is the deformation parameter. In the semiclassical limit, C[SU2 ] cop U(su 2 ) becomes the bicross sum su cop 2 su 2, where su 2 has its standard Lie bracket and su cop 2 = R 3 has a commutative Lie bracket. From (3.41) the relations are given by [p a, p b ] = 0, [M, N 1 ] = N 2, [M, N 2 ] = N 1, [N 1, N 2 ] = M, [M, p 0 ] = 0 [M, p i ] = ıɛ ij p j, [N i, p 0, ] = ıɛ ij p j, [N i, p j ] = ıɛ ij p 0 i, j = 1, 2, In terms of the basis (3.40), the R-matrix becomes (4.9) R B0 = e λp M λp + e λp 0 : e eλp 0 X + λp + e λp 0 : e M (eλp 0 1) e λp X e (eλp 0 1) M, (4.10) where we have kept the X ± for simplicity. Then the semiclassical limit of the bicrossproduct R-matrix (4.10) gives a classical r-matrix for the bicross sum as r B0 = M p 0 p 0 M P X X + P + = M p 0 p 0 M (p 2 N 2 + p 1 N 1 + N 2 p 2 + N 1 p 1 ) ı(p 2 N 1 p 1 N 2 + N 2 p 1 N 1 p 2 ) (4.11) It is interesting to see that if we set M = J 0, J i = N i and P a = p a, to match standard notation, we get an r-matrix for the bicross sum su cop 2 su 2 as r B0 = P a J a + J a P a ı(p 1 J 2 P 2 J 1 ) where m is a unit time-like vector. = P a J a + J a P a + ım a ɛ abc P b J c, m 2 = 1, (4.12) We observe that the symmetric part of the r-matrix (4.12) is eual to the Casimir associated to the invariant, non-degenerate symmetric bilinear form used in the Chern- Simons action [24, 25] and therefore suitable for constructing the Poisson structure on the classical phase space via the Fock-Rosly construction [26]. This shows that the bicrossproduct with r-matrix depending on a time-like deformation vector and with complex antisymmetric part is compatible with 3d gravity via the Fock-Rosly compatibility condition. This is different from the family of classical bicross sum r-matrices associated to 3d gravity with vanishing cosmological constant obtained in [27, 19]. In the later, the r-matrices are real and depend on space-like deformations vectors. See also [20] where a complete classification of all r-matrices compatible to 3d gravity with vanishing cosmological is constructed via semidualisation of Lie bialgebras which are double cross sums. Now, rewriting the twist in Corollary 3.1 in terms of the basis (3.40), we get χ B0 = e X + λp + e λp 0 e M (eλp 0 1), (4.13) and the semiclassical limit for the twist gives the Lie bialgebra twist by (4.6) as χ c B 0 = X + P + + M p 0 P + X + p 0 M = M p 0 p 0 M + N 2 p 1 + N 1 p 1 p 2 N 2 p 1 N 1 + ı(n 2 p 1 N 1 p 2 p 2 N 1 p 1 N 2 ) (4.14) 23