By Robert W. Carroll (auth.)

In this e-book 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 idea, quantum integrable platforms, braiding, finite topological areas, a few features of geometry and quantum mechanics and gravity.

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Next one notes that the axioms of a quasitriangular Hopf algebra are not self dual and a dual formulation is of possible interest. In this spirit one would want an invertible map A®A - t k (in place ofR: H®H - t k (here A dual Hopf algebra) and we look first at the idea of convolution algebra structure of Hom(A ® A, k). More generally if C is a coalgebra and B an algebra then H om( e, B) has a convolution algebra structure via (A73) (

In which ta build representations), and to show that the category MONIV, consisting of monoidal categories equipped with functors to V, is self dual in the representation theoretic sense. Morphisms in M O N IV are monoidal functors compatible with the given functors ta V. Thus let F : C --+ V be a monoidal functor between monoidal categories and define a representation of C in V to be a pair (V, AV), where V E V and AV E Nat(V 181 F, F 181 V) is a natural equivalence, Le. 1) AV,l = id; AV,Y o AV,X = c x: y o AV,X®Y o CX,Y The collection of such representations forms a monoidal category CO (the dual of C over V) and explicitly the morphisms (V, AV) --+ (w, AW) between representations are morphisms

Sgl, CI >< g3, Cs >< Sh l , C2 bl >< h3, C4 b3 >= ®h3g2! 121) S(a ® h) = (1 ® Sh)(S-la ® 1) = L S-la2 ® Sh 2 < h l , al >< Sh3, a3 > if Hand H'op are to be sub-Hopf algebras. This is checked in [456J. 122) LUî ® 1) ® U2 ® 1) ® (1 ® ea) = Lua ® 1)Ub ® 1) ® (1 ® eaeb); r Lua ® 1) ® (1 ® eal) ® (1 ® ea2) = LU b ® 1) ® (1 ® eb) ® (1 ® ea) which is easily checked by evaluation against general elements. 60) via a calculat ionyielding < g, n~(a®h) >=< g, (7 o ~(a ® h))n > and computes n- l = I: S-l ® 1 ® 1 ® ea.

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