By Robert W. Carroll (auth.)

In this booklet the main points of many calculations are supplied for entry to paintings in quantum teams, algebraic differential calculus, noncommutative geometry, fuzzy physics, discrete geometry, gauge concept, quantum integrable platforms, braiding, finite topological areas, a few points of geometry and quantum mechanics and gravity.

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H then q; = 8F is an ad-invariant 3-cocycle of the form required. To see this one notes first that in general. (when H is noncocommutative) the ad-invariance conditions used here are not the same as the requirement of invariance under the quantum adjoint action Adh(g) = I: h1gSh 2, extended to tensor powers. This is the reason behind the quotation marks here. 17) does hold, and q; is a counital 3-cocycle, then we have a quasibialgebra. For the quasi Hopf algebra structure we have to verify the antipode and 3-cocycle conditions.

113) is (((h 2)*hS- 1(h 1)*) 1> b* ® ((h2)*h) = ((h*hS- 1(h*h) 1> b* ® (h*)J = b* ® h* (ef. (A59) and recall S(h*)* = S-lh with h1 1> h2 '" h1h2 noting €(h) = L: S(h 1 )h2 implies that €(h) = h2S- 1hi while €(h 1)h2 = h*). • Further details are in [456]. 13. Given A an H-module algebra the crossproduct A>

More generally if C is a coalgebra and B an algebra then H om( e, B) has a convolution algebra structure via (A73) (