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See [70][71] 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[11][27][33][19][25] and U.

131) (132) Some authors[57] 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 [11],[28]. Some interesting new results about Ω(B(R)) from the point of view of braided groups are in [40][42]. 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[63], 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.

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