Consider a toy example of a diffeomorphism group – the group of diffeomorphisms of a 1-dimensional manifold with a disconnected boundary (2 points).

The group is a group of monotonically increasing functions defined on the interval $[0..1]$ such that

$$ f(0) = 0, \quad f(1) = 1. $$

The group product is given by the composition of functions, and the group inverse is the function inverse. The group identity element is

$$ e(\tau) = \tau. $$

Now consider a suitably defined algebra of nonvanishing functionals over the diffeomorphism group. Because it is an algebra of functions over a group, it is a Hopf algebra:

  • Algebraic product and inverse are given by point-wise multipliction and inverse.
  • Comultiplication and the antipode are given by duals of the group product and inverse (function composition and function inverse).

I'm interested in nontrivial examples of deformations of the mentioned above Hopf algebra in the category of Hopf algebras, such that commutativity of the algebraic product is restored in the $\hbar \rightarrow 0$ limit. In other words, I'm interested in quantizations of the diffeomorphism group.

Do such deformations exist? Have they been studied?


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