This is a study note that spells out @Konstantinos's answer
explicitly.

### Preface

Our goal is to classify all finite dimensional representations over
the complex number field for the quantum double $D(G)$ for a fixed
finite group $G$, with proofs. We will use [G] as out main reference,
while auxiliary results can be found in [S] and [CR].

For other considerations, see [W], [L], and [B]. For the
representation theory of $D(G)$ over other fields, see [W]. For
the representation theory of other Hopf algebras, see [L], which
deals with a class of (possibly infinite dimensional) Hopf algebra
with a technical condition: co-semi-simple + involutive). For more
applications, see [B].

### Abstract

In what follows, semi-simplicity allows us to focus on the simple modules. We can get lots of them by induction from the underlying group $G$. Character theory for $D(G)$ distinguishes the simple modules we get from induction, showing the abundance. The structure theorem of $D(G)$ predicts how many non-isomorphic simple modules there should be. Examining how many different simple modules we've got from allows us to complete the proof.

### Semi-simplicity of $D(G)$

Following [G] and its notations, the first main result is the
semi-simplicity of $D(G)$. Theorem 2.3 says that any finite
dimensional Hopf algebra $A$ is semi-simple if and only if there
exists a left integral $x \in A$, this is a powerful criterion for
semi-simplicity. A left integral of $D(G)$ is given in [G. (16)],
where $x = E_\iota 1^*$, so $D(G)$ is semi-simple.

The proof of theorem 2.3, the powerful semi-simplicity criterion, can
be found in [S. Theorem 5.18]. There, Sweedler first defined the left
integrals for $H^*$. As $H$ is finite-dimensional, $H$ is isomorphic
to $H^{**}$ naturally, whose left integrals can be doubly-dualed back
to $H$. This definition coincides with that of [G]. Anyway, one can
use a left integral to "average" an arbitrary linear projection and
get a Hopf linear projection from any larger module to any smaller
submodule, proving semi-simplicity. An explicit averaging formula is
given in the proof of [S. Theorem 5.18]. The other side is easy: if
$H$ is semi-simple, than the complement of $ker(\epsilon)$ is the set
of left integrals. A few immediate corollaries are

$D(G)$, $\operatorname{Fun}(G)$, and $\mathbb{C}[G]$ are all
semisimple.

$k[G]$ is semisimple if and only if $\epsilon(x=\Sigma g) = |G|$ is
not zero, which in turn is equivalent to that $|G|$ is not
divisible by $\operatorname(char)k.

$k[X]/<X^p>$ is not semi-simple, since $\epsilon(x^{p-1})$ is zero.

$k[X]/<X^p - X>$ is semi-simple, since $\epsilon(x^{p-1})$ is -1.

### Unitarity of representations and orthogonality of matrix elements

Every finite dimensional $D(G)$-module is equivalent to a unitary one
[G. Lemma 4.1], so in particular $D(G)$ is proven again to be
semi-simple. Routine arguments show the orthogonality of matrix
elements [G. Theorem 4.1]. Applying this to characters, we get the
orthogonal relations among them [G. Theorem 5.1]. Note that this can
be generalized to a larger class of Hopf algebras (possibly infinite
dimensional), which are co-semi-simple and involutive [L]. The rest of
chapter 5 in [G] exhibits the character theory for $D(G)$ and finds an
explicit basis for the center of $D(G)$ [G. (25) -- Thm 5.2]. This
basis is in 1-1 correspondence to the number of $G$-equivalence
classes of $Q$, and is also in 1-1 correspondence to the set of
non-isomorphic irreducible $D(G)$-modules by the structure theorem for
Artinian semisimple rings [G. Theorem 5.2]. We will justify the last
statement later.

### Enumeration of representations of $D(G)$

The representations of $D(G)$ can be obtained by induction from the
centralizer subgroups of $G$. This is done in chapter 6. The character
theory developed in chapter 5 distinguishes one from another, showing
the abundance of the results. Since we have known how large
$\operatorname{Irrep}(D(G))$ is, we will be done by showing the
structure theorem for $D(G)$.

### Structure theorem for $D(G)$

In this section, our reference is [CR. section 23 to 26]. From now on,
we will assume $R$ to be a unital Artinian ring (associative, but not
necessarily commutative). We will show the structure theorem for $R$
if it is semisimple. Since $D(G)$ obviously satisfies all the
conditions, we will then be done.

Since $R$ is Artinian, any left ideal $I$ is nilpotent if and only if
it has no idempotent elements. It is then easy to show that the set of
nilpotent left ideals is closed under finite sum. More interestingly,
the sum of **all** nilpotent left ideals is a nilpotent **two-sided**
ideal, called the radical $\sqrt(R)$ of $R$. If the radical is zero,
we call $R$ semisimple. It is easy to show that $R/\sqrt(R)$ is
semisimple.

If $R$ is semisimple, then any minimal left ideal $L$ is not nilpotent
and thus have an idempotent element $e$. Minimality guarantees that
$L$ is generated by that idempotent element. Note that the generator
is not unique in general. In this case, $R = Re \oplus R(1-e) = L
\oplus L'$. One can furthur decompose $R$ into $R = Re_1 \oplus \cdots
Re_n$, where the $e_i$'s are orthonormal idempotents. It is easy to
show the uniqueness of decomposition, and also that any $R$ with this
decomposition is in fact semisimple. The decomposition breaks the unit
$1$ into the sum of the $e_i$'s, this is the key. Using this key, it
is not hard to show that every left $R$ ideals are completely
reducible [CR. 25.8], and also that any irreducible $R$-module is
isomorphic to some minimal left ideal in $R$.

Therefore, the complete set of non-isomorphic simple modules can be
found in the decomposition of $_RR$ as a left $R$-module! The
Wedderburn structure theorem shows that the number of them is the same
as the size of the center of $R$ (TODO: needs clarification). This
completes the argument.

### References

[G]. Quantum double finite group algebras and their representations,
Bull. Austr. Math. Soc., 48, 1993, p.275-301, by M.D. Gould.

[S]. Hopf algebras (Benjamin, New York, 1969), by M.E. Sweedler.

[CR]. Representation theory of finite groups and associative algebras,
by C.W. Curtis and I. Reiner.

[W] The representation ring of the quantum double of a finite group,
J. of Algebra, 179, p.305-329, (1996), by S.J. Witherspoon.

[L] Characters of Hopf algebras, J. Algebra 17 (1971), 352-368, by
R.G. Larson.

[B] Exactly solved models in statistical mechanics (Academic press,
1982), by R.J. Baxter.

`\text`

or`\operatorname`

: $\text{$\operatorname{Rep} D(G)$ and some text}$`$\text{$\operatorname{Rep} D(G)$ and some text}$`

. I have edited accordingly. $\endgroup$ – LSpice Dec 10 '19 at 15:51