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See  for recent applications in connection with the differential calculus and Koszul complex of U q (g). 7 q-Deformed spacetime ¯ q (2) At various points in our tour of braided geometry we have mentioned that the examples M and BMq (2) make natural q-Euclidean and q-Minkowski spaces when q is real. We just have ¯ to specialise our general A(R) and B(R) constructions to the case of R given by the standard quantum plane or Jones polynomial R-matrix in the Hecke normalisation. We have seen above many results about these general algebras, all of which constitutes the braided approach to qspacetime due to the author and U.

131) (132) Some authors prefer the second form for Ω(B(R)) but let us stress that they are just the Wess-Zumino construction (127) in the matrix notation and not mathematically new once the required matrices R , R in (36) had been introduced by the author and U. Meyer in ,. Some interesting new results about Ω(B(R)) from the point of view of braided groups are in . 7. So ada = q2da a, bdb = q 2 db b, adb = qdb a, bda = qda b + (q 2 − 1)db a, etc. 7. The relations among the 1-forms are of just the same form with an extra minus sign, forcing a finite-dimensional algebra as shown.

3] is that of a dual-quasitriangular structure or ‘universal R-matrix functional’ R : A ⊗ A → C on a Hopf algebra or quantum group A. It is characterised by axioms which are the dual of those of Drinfeld, namely R(a ⊗ bc) = R(a(1) ⊗ c)R(a(2) ⊗ b), R(ab ⊗ c) = R(a ⊗ c(1) )R(b ⊗ c(2) ) (143) b(1) a(1) R(a(2) ⊗ b(2) ) = R(a(1) ⊗ b(1) )a(2) b(2) for all a, b, c ∈ A and ∆a = a(1) ⊗ a(2) the coproduct. The main theorem we need is the result due to the author in [60, Sec. , R(t1 ⊗ t2) = R (144) on the matrix generators.