Calculus Revisited by Robert W. Carroll (auth.)

By Robert W. Carroll (auth.)

In this ebook 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 thought, quantum integrable platforms, braiding, finite topological areas, a few points of geometry and quantum mechanics and gravity.

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PRELIMINARY IDEAS pair of Hopf algebras with 1> = Ad* rv left coadjoint action of H on H' and > h2 in [456]. The product can be defined as (ef. 123) = L = L b2a < h 1, (Sb 1)b3 > 0 < (Sh 2)h4, b4 > h3g = < h 1, Sb1 >< h2, b3 >< Sh 3, b4 >< h5, b5 > b2a 0 h4g "-+ "-+ L b2a 0 h2g < Sh1, b1 >< h3, b3 > rv Adh1 (b 1) = 'E b12 < hn(Sbn), b3 > and h 2 h 3· One recalls now e(x) = 'E X1 S (X2) = 'E S(X1)X2 with x = 'E X1 e(X2) = 'E X1 e(X2) and notes that < (S4>1)4>2, h >=< S4>l 0 4>2, tlh >=< S4>104>2, h 10 h 2 >=< 4>104>2, Sh10 h 2 >=< 4>, (Sh 1)h2 >=< 4>,1 > e(h) = e(4))e(h).

BASICS FOR QUANTUM GROUPS one notes that representations of the Ropf algebra have among themselves a tensor product operation. This phenomenon can be studied systematically and more conceptually via the language of category theory. For example the tensor product operation 0 : C xC - t C will be a functor (C '" a category) and this makes the category of representations of any bialgebra or Ropf algebra into a monoidal category (C, 0). Quasitriangularity turns out to be just what is needed to ensure isomorphisms W : V 0 W :::: W 0 V between any two representations in a coherent manner.

When Hand A are Hopf algebras so is H I>11III A with antipode (A52) S(h 0 a) = 2:(1 0 S(h 1a))(Sh 2 01). We recall now a few facts about HOP, Hcop , and H' (ef. 2 for * algebras). For a bialgebra H, HOP", H with h of 9 = gh is also a bialgebra. If H is a Hopf algebra one can show that HOP is Hopf if and only if S is invertible and then sop = S-l. Similarly for HCOP with 1. BASICS FOR QUANTUM GROUPS 1-26 5 COP = 5- 1 . Finally if 5 is invertible the map 5: H ~ HOp/cop (both opposites) is a Hopf algebra isomarphism (ef.

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