Calculus Revisited by Robert W. Carroll (auth.)

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 thought, quantum integrable platforms, braiding, finite topological areas, a few facets of geometry and quantum mechanics and gravity.

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Calculus Revisited

During 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 concept, quantum integrable platforms, braiding, finite topological areas, a few facets of geometry and quantum mechanics and gravity.

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89) (R) LXv ® E(XH) x x rv = l: xH ® XV) for right (resp. left) x; L(xv)v ® (XV)H ® XH = LXV ® (XH)' ® (XH)" x x x x for right (resp. left) comodules (here ~H(X) = ~(x) = l: x' ® x" E H ® H). There are other notations in [400, 456] but this approach seems most clear; commutative diagrams also help here (ef. [371, 400, 456]). We will discuss H-comodule algebras and coalgebras later. 90) (coasociativity in the form (id ®~) o ~ = (~® id) o ~ is checked in [456]). 1. 2, along with left (right) cross coproducts >0IIII (~).

Left) x; L(xv)v ® (XV)H ® XH = LXV ® (XH)' ® (XH)" x x x x for right (resp. left) comodules (here ~H(X) = ~(x) = l: x' ® x" E H ® H). There are other notations in [400, 456] but this approach seems most clear; commutative diagrams also help here (ef. [371, 400, 456]). We will discuss H-comodule algebras and coalgebras later. 90) (coasociativity in the form (id ®~) o ~ = (~® id) o ~ is checked in [456]). 1. 2, along with left (right) cross coproducts >0IIII (~). We want to say more about this, in clarifyirig various notations, and also introduce the ideas of double cross product (or bicrossed product) [Xl, left-right (resp.

144) (a 0 h)(b 0 g) = "L ab2 0 h2gu- 1(h 1 0 b1)u(h 3 0 b3) and cross relations (A 78) 2:: U(h1 0 adh2a2 = 2:: a1hw(h2 0 a2). 3. 15. 146) (~0 id) o a(a) = "L((id 0 (3) o a(a1))(10 a(a2)); (id 0~) o f3(h) = "L(f3(h 1) 01)((a 0 id) o f3(h 2)) and a(a)f3(h) = f3(h)a(a). Then there is a double cross coproduct bialgebra H ~ A with tensor product algebra structure and counit while ~(h 0 a) = 2:: h1 0 a(a1)f3(h2) 0 a2. If Hand A are Hopf algebras so is H ~ A. 3 via computations in terms of H' and A'. Thus (A80) < a(b), h 0 a >=< b, h and < f3(g) , h 0 a) >=< g, h [> a > and subsequently one renames H', A' as A, H.

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