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

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**Additional resources for Calculus Revisited**

**Example text**

Let R be a bialgebra or Ropf algebra and let HM be the category of algebra representations as before. Then 0, defined using module notation by h 1> (v 0 w) = 2: h 1 1> v 0 h2 1> w (via the coproduct) • makes H M into a monoidal category. Since the constructions have dual counterparts, if A is a bialgebra or a Ropf algebra, then the category MA of right A-comodules is a monoidal cateory. The tensor product comodule V 0 W is defined via the coaction (A88) ,8Vl8>w(v 0 w) = 2:v 1 0 w 1 0 v 2w 2 in terms of the coactions on V, W and the product of A.

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

15)). Indeed U = v and we recover S-2h = UhU-1. • There are also dual versions of alI this (ef. [456]). We mention next the idea of quasi Ropf algebra. The idea is that along with relaxing cocommutativity up to conjugation (via 'R) one can also relax the coassociativity of ~ up to conjugation. cfJ, 'R) where R is a unital algebra and ~ : H ~ H ® H satisfies (A96) (id ®~) o ~ = cfJ«(~ ® id) o ~( )cfJ-l. The axioms for E are as usual and cfJ E H ® H ® H, which controls the nonassociativity, is invertible and is required to be a counital 3-cocycle.