Approaches to Quantum Gravity

Algebraic approach to Quantum Gravity II 483and it has been conjectured that the resulting bicrossproducts are all (nontrivially)isomorphic to certain contractions of the q-deformation quantum groupsU q (so m,n ). In the 4D case the contraction of U q (so 3,2 ) was found first [8] withthe bicrossproduct of form found later in [23]. Note that the physical interpretationof the generators coming from contractions is completely different from thebicrossproduct one.24.3.3 Bicrossproduct C λ [Poinc] quantum groupWe now apply the same matched pair factorisation data (24.20)–(24.22) but nowto construct the dual Hopf algebra. We start with C[SO 1,1 ] naturally described bygenerators s = sinh(θ) and c = cosh(θ) with relations c 2 − s 2 = 1 (which formthe matrix μ ν) and matrix coproduct( ) ( ) ( ) ( ) ( )c s c s c s c s c −s = ⊗ , S =.s c s c s c s c −s cTo see how this arises in our theory, recall that we worked with S = sinh( θ 2 ) andC = cosh( θ 2 ) which (similarly) describe the double cover of SO 1,1 in coordinateform. We differentiate (24.22) written in terms of S by ∂ |∂p μ p μ =0 to obtain the vectorfields β and infinitesimal left action of the Lie algebra [a 0 , a 1 ]=ıλa 1 on functionsof θ:β ıa0 =∂∂p | 0 0 =−2λCS ∂∂θ , β ıa 1= ∂∂p | 1 0 =−2λS 2 ∂∂θ∂a 0 ⊲S =−ı∂p | 0 0 sinh( θ ′2 ) = ∂ıλSC2 , a 1 ⊲S =−ı∂p | 1 0 sinh( θ ′2 ) = ıλCS2 .Note also that sinh(θ) = 2CS and cosh(θ) = C 2 + S 2 . Hence from (24.15) we findthe relations( ( ( ( c s c s[a 0 , ]=ıλs , [as)c)1 , ]=ıλ(c − 1)s)c)of the bicrossproduct C λ [Poinc 1,1 ]≡C[SO 1,1 ]◮⊳U(R>⊳R). Finally, differentiate(24.20)–(24.21) to have the coaction R of C[SO 1,1 ] on the a μ :⇒∂p 0′∂p 0 | 0 = C 2 + S 2 = ∂p1′∂p 1 | 0,( C S R (a 0 , a 1 ) = (a 0 , a 1 ) ⊗S∂p 0′∂p | 1 0 = 2CS = ∂p1′∂p | 0 0) 2 ( c s= (aC 0 , a 1 ) ⊗s cwhich along with the antipode completes the Hopf algebra structure constructedfrom (24.15)–(24.17). One can similarly describe C λ [Poinc 3,1 ] =)