Approaches to Quantum Gravity

498 J. Kowalski-Glikmanformer reduces to the latter in an appropriate limit. The other structures of Hopfalgebras, like co-product and antipode, are also relevant in the context of DSR, andI will introduce them in the next section.In the case of quantum algebra SO q (3, 1) the algebraic part looks as follows (theparameter z used below is related to q by z = ln q)[M 2,3 , M 1,3 ]= 1 z sinh(zM 1,2) cosh(zM 0,3 )[M 2,3 , M 1,2 ]=M 1,3[M 2,3 , M 0,3 ]=M 0,2[M 2,3 , M 0,2 ]= 1 z sinh(zM 0,3) cosh(zM 1,2 )[M 1,3 , M 1,2 ]=−M 2,3[M 1,3 , M 0,3 ]=M 0,1[M 1,3 , M 0,1 ]= 1 z sinh(zM 0,3) cosh(zM 1,2 )[M 1,2 , M 0,2 ]=−M 0,1[M 1,2 , M 0,1 ]=M 0,2[M 0,3 , M 0,2 ]=M 2,3[M 0,3 , M 0,1 ]=M 1,3[M 0,2 , M 0,1 ]= 1 z sinh(zM 1,2) cosh(zM 0,3 ). (25.5)Observe that on the right hand sides we do not have linear functions generators,as in the Lie algebra case, but some (analytic) functions of them. However, westill assume that the brackets are antisymmetric and, it is easy to show, that Jacobiidentity holds. Note that in the limit z → 0 the algebra (25.5) becomes the standardalgebra SO(3, 1), and this is the reason for using the term SO q (3, 1).The SO(3, 1) Lie algebra is the 2+1 dimensional de Sitter algebra and it is wellknown how to obtain the 2+1 dimensional Poincaré algebra from it. First of allone has to single out the energy and momentum generators of the right physicaldimension (note that the generators M μν of (25.5) are dimensionless): one identifiesthree-momenta P μ ≡ (E, P i ) (μ = 1, 2, 3, i = 1, 2) as appropriately rescaledgenerators M 0,μ and then one takes the Inömü–Wigner contraction limit. In thequantum algebra case, the contraction is a bit more tricky, as one has to convinceoneself that after the contraction the structure one obtains is still a quantum algebra.Such contractions have been discussed in [13].Let us try to contract the algebra (25.5). To this aim, since momenta are dimensionful,while the generators M in (25.5) are dimensionless, we must first rescale