Approaches to Quantum Gravity

474 S. MajidPositionMomentumGravity Curved Noncommutative∑μ x2 μ = 1γ 2 [p i , p j ]=2ıγɛ ijk p kCogravity Noncommutative Curved∑[x i , x j ]=2ıλɛ ijk x k μ p2 μ = 1λ 2Quantum Mech.[x i , p j ]=ıδ ijFig. 24.1. Noncommutative spacetime means curvature in momentum space. Theequations are for illustration.independent physical effect and comes therefore with its own length scale whichwe denote λ. These ideas were introduced in this precise form by the author in themid 1990s on the basis of the quantum group Fourier transform [9]. Other workson the quantum group Fourier transform in its various forms include [5; 6; 7]24.3 Bicrossproduct quantum groups and matched pairsWe will give an explicit construction of the bicrossproduct quantum groups of interest,but let us start with a general theorem from the theory of Hopf algebras. Thestarting point is a theory of factorisation of a group X into subgroups M, G suchthat X = MG. It means every element of X can be uniquely expressed as a normalordered product of elements in M, G. In this situation, define a left action ⊲ of Gon M and a right action ⊳ of M on G by the equationThese actions obeyus = (u⊲s)(u⊳s), ∀u ∈ G, s ∈ M. (24.6)u⊳e = u, e⊲s = s, u⊲e = e, e⊳s = e(u⊳s)⊳t = u⊳(st), u⊲(v⊲s) = (uv)⊲su⊲(st) = (u⊲s)((u⊳s)⊲t)(uv)⊳s = (u⊳(v⊲s))(v⊳s) (24.7)for all u,v ∈ G, s, t ∈ M. Heree denotes the relevant group unit element. A pairof groups equipped with such actions is said to be a ‘matched pair’ (M, G). Onecan then define a ‘double cross product group’ M ⊲⊳ G with product(s, u).(t,v)= (s(u⊲t), (u⊳t)v) (24.8)and with M, G as subgroups. Since it is built on the direct product space, the biggergroup factorizes into these subgroups and in fact one recovers X in this way.These notions were known for finite groups since the 1910s but in a Lie group setting[12; 15] one has the similar notion of a ‘local factoristion’ X ≈ MG and a