By Robert W. Carroll (auth.)

In this publication 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 conception, quantum integrable structures, braiding, finite topological areas, a few features of geometry and quantum mechanics and gravity.

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**Extra resources for Calculus Revisited**

**Sample text**

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. r We will see below (ef. 3) that (H'OP, H) form a matched 1-31 1.

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

129) (b 0 h)