In this lecture we will see that there are, as the astronomer Carl Sagan used to say, ‘billions upon billions’ of truly quantum groups, or at least noncommutative and noncocommutative bialgebras. Just as every reasonable space has an associated ‘diffeomorphism group’, so every finitedimensional algebra has a ‘diffeomorphism or automorphism’ quantum group or comeasuring bialgebra associated to it. We use the latter technical term to avoid confusion with the usual automorphism group, which also exists, but which is too restrictive to serve in the correct geometrical role (of diffeomorphisms) in noncommutative geometry.

Definition 4.1Let A be an algebra. A comeasuring of A is a pair (B, β)

where

1. B is an algebra.

2. β : A → A ⊗ B is an algebra map.

This is like a right comodule algebra but we only require B to be an algebra and hence do not require the comodule property itself. Morphisms between comeasurings are, by definition, maps between their underlying algebras connecting the corresponding β. We define (M(A), βu), when it exists, to be the initial universal object in the category of comeasurings of A, i.e. a comeasuring such that for any other comeasuring (B = β), there exists a unique algebra map π : M(A) → B such that β = (id ⊗ π) ∘ βU. Like all universal objects, if it exists it is unique up to unique isomorphism.

Proposition 4.2Let A be an algebra. Then M(A), if it exists, is a bialgebra and βUmakes A an M(A)-comodule algebra. Any other coaction of a bialgebra on A as an algebra is the push-out of this one.