Projective representations and twisted algebra modules

On basis elements, $(u_q{u}_r)u_s=\alpha (q,r)\alpha (qr,s)u_{qrs}$ and $u_q(u_r{u}_s)=\alpha (r,s)\alpha (q,rs)u_{qrs}$ by [F1], and these scalars are equal; since the product is bilinear and the $u_q$ span $A$, the product is associative.

On basis elements $u_1{u}_q=\alpha (1,q)u_q=u_q$ and $u_q{u}_1=\alpha (q,1)u_q=u_q$ by [F1]; by bilinearity $u_{1}a=a=a{u}_1$ for every $a\in A$, so $u_1$ is a two-sided identity.

Conversely, let $P$ be a normalized projective representation of $Q$ on a nonzero finite-dimensional space $V$ with factor set $\alpha$. Define $a\cdot v$ for $a={\sum }_q{\lambda }_q{u}_q$ by $a\cdot v:={\sum }_q{\lambda }_{q}P(q)v$. This is well defined and $\mathbb{C}$-bilinear by [A1], and it satisfies $u_1\cdot v=P(1)v=v$ and $(u_q{u}_r)\cdot v=\alpha (q,r)u_{qr}\cdot v=\alpha (q,r)P(qr)v=P(q)P(r)v=u_q\cdot (u_r\cdot v)$ by [F2]; both sides extend by linearity to arbitrary $a\in A$, so [F5] makes $V$ a left $A$-module.

Suppose $M$ has finite dimension over $\mathbb{C}$ and let $W\le M$ be an $A$-submodule. On $M$ define $P(q)m:=u_{q}m$, using the scalar action $\lambda m=(\lambda u_1)m$. These operators are $\mathbb{C}$-linear since $u_q(\lambda u_1)=(\lambda u_1)u_q$. The module law gives $P(1)={\mathrm{id}}_M$ and $P(q)P(r)=\alpha (q,r)P(qr)$; [F1] and the cocycle identity at $(q,q^{-1},q)$ give $P(q)P(q^{-1})=P(q^{-1})P(q)=\alpha (q,q^{-1}){\mathrm{id}}_M$, so $P(q{)}^{-1}=\alpha (q,q^{-1}{)}^{-1}P(q^{-1})$. This holds also for $M=0$. Choose a $\mathbb{C}$-linear projection $\pi :M\to W$, which exists by [A1] applied to a basis of $W$ extended to a basis of $M$. Averaging its conjugates, define $\bar{\pi }:=\frac{1}{\mathrm{\mid }Q\mathrm{\mid }}{\sum }_{q\in Q}P(q)\pi P(q{)}^{-1}$.

Steps 1.1 and 1.2 make $A$ a unital associative ring, and $\lambda \mapsto \lambda u_1$ is a unital ring homomorphism $\mathbb{C}\to A$ with central image, so $A$ is a $\mathbb{C}$-algebra in the sense of [F4]; by [F1] and [F3] each $u_q$ has the two-sided inverse $\alpha (q,q^{-1}{)}^{-1}{u}_{q^{-1}}$.

Each $P(q)\pi P(q{)}^{-1}$ maps $M$ into $W$, because $W$ is $A$-stable and $P(q)$ is invertible; hence $\bar{\pi }(M)\subseteq W$, and $\bar{\pi }$ fixes $W$ pointwise because $P(q)\pi P(q{)}^{-1}w=P(q)P(q{)}^{-1}w=w$ for $w\in W$.

For every $r\in Q$, $P(r)\bar{\pi }P(r{)}^{-1}=\frac{1}{\mathrm{\mid }Q\mathrm{\mid }}{\sum }_{q}P(r)P(q)\pi P(q{)}^{-1}P(r{)}^{-1}=\frac{1}{\mathrm{\mid }Q\mathrm{\mid }}{\sum }_q\alpha (r,q)\alpha (r,q{)}^{-1}P(rq)\pi P(rq{)}^{-1}=\frac{1}{\mathrm{\mid }Q\mathrm{\mid }}{\sum }_{q^{\mathrm{\prime }}}P(q^{\mathrm{\prime }})\pi P(q^{\mathrm{\prime }}{)}^{-1}=\bar{\pi }$ by step 2.1 and reindexing $q^{\mathrm{\prime }}=rq$, so $\bar{\pi }$ commutes with every $P(r)$ and hence with the action of $A$.

Let $M\mathrm{\ne }0$ be a finite-dimensional left $A$-module and let $P(q)$ be the action of $u_q$ on $M$, which is a $\mathbb{C}$-linear map by [F5]. Then $P(1)={\mathrm{id}}_M$ by [F5] and [F1], and $P(q)P(r)=\alpha (q,r)P(qr)$ because the action respects products and $u_q{u}_r=\alpha (q,r)u_{qr}$; each $P(q)$ has inverse $\alpha (q,q^{-1}{)}^{-1}P(q^{-1})$ by step 2.2, so $P(q)\in \mathrm{GL}(M)$. Thus every nonzero finite-dimensional $A$-module gives a normalized projective representation with factor set $\alpha$.

Steps 3.1 and 3.2 show that $M=W\oplus \ker \bar{\pi }$: the average is a surjection onto $W$ fixing $W$, so it is a module map with image $W$ and kernel a complement. Therefore every submodule of a finite-dimensional $A$-module has a complementary submodule.

The two constructions of steps 3.3 and 1.3 are mutually inverse: starting from $M$, the module built from $P$ acts as $u_q\cdot m=P(q)m$, which is the original action, and starting from $P$, the representation of the built module sends $q$ to the action of $u_q$, which is $P(q)$. Hence nonzero finite-dimensional left $A$-modules correspond bijectively to normalized projective $Q$-representations with factor set $\alpha$.

A finite-dimensional $A$-module has finite length: a strictly increasing chain of submodules has strictly increasing complex dimensions, so no chain of submodules can be longer than $1+{\dim }_{\mathbb{C}}M$ terms, and a maximal chain is a composition series in the sense of [F6]. By [F9], the complement property of step 4.1 makes $M$ a direct sum of simple submodules, that is, semisimple in the sense of [F7].

The zero space carries the unique $A$-module structure, all operators being zero, and it is the zero object of the category of left $A$-modules: the zero map is the only map from it and the only map to it, and both are module maps. It is therefore a module for every coefficient cocycle $\alpha$, while it determines no factor set, which is why step 4.2 is stated for nonzero modules.

Applying step 5.1 to the left regular module ${}_{A}A$, which is finite-dimensional of dimension $\mathrm{\mid }Q\mathrm{\mid }$, shows that $A$ is a semisimple ring in the sense of [F8], and step 5.1 shows in addition that every finite-dimensional left $A$-module is semisimple.