Let be a finite quantum group described by with an involutive antipode (I know this is true is the commutative or cocommutative case. I am not sure at this point how restrictive it is in general. The compact matrix quantum groups have this property so it isn’t a terrible restriction.) . Under the assumption of finiteness, there is a unique Haar state, on .

# Representation Theory

A *representation* of is a linear map that satisfies

The dimension of is given by . If has basis then we can define the *matrix elements* of by

One property of these that we will use it that .

Two representations and are said to be *equivalent*, , if there is an invertible *intertwiner* between them. An intertwiner between and is a map such that

We can show that every representation is equivalent to a unitary representation.

Timmermann shows that if is a maximal family of pairwise inequivalent irreducible representation that is a basis of . When we refer to “the matrix elements” we always refer to such a family. We define the span of as , the *space of matrix elements of* .

Given a representation , we define its *conjugate*, , where is the conjugate vector space of , by

so that the matrix elements of are .

Timmermann shows that the matrix elements have the following orthogonality relations:

- If and are inequivalent then for all and .
- If is such that the conjugate, , is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

This second relation is more complicated without the assumption and refers to the entries and trace of an intertwiner from to the coreprepresention with matrix elements . If , then this intertwiner is simply the identity on and so the the entries and the trace is .

Denote by the set of unitary equivalence classes of irreducible unitary representations of . For each , let be a representative of the class $\alpha$ where is the finite dimensional vector space on which acts.

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