Three days ago we stated the following puzzle: we can compute that isomorphism classes of -linear actions of a group on the category of vector spaces over a field correspond to elements of the cohomology group
This is the same group that appears in the classification of projective representations of over , and we asked whether this was a coincidence.
Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.
The Schur class of a projective representation
Definition: A projective representation of a group over a field is a homomorphism , where is the projective linear group of a -vector space .
A natural question to ask about a projective representation is when it can be lifted to a linear representation . We can certainly individually lift each to some ; what we can’t necessarily do is guarantee that . Since , we instead have
for some scalars (uniquely determined by our choices of lifts ). These scalars can’t be arbitrary, though, since can be decomposed in two different ways as a product, which gives
on the one hand and
on the other hand. Hence must satisfy
which is just the 2-cocycle condition. If we also decided that we might as well lift to , then this is even a unital 2-cocycle.
We made some arbitrary choices when choosing the lifts , so now let’s see what happens if we choose different lifts . These must differ from our original lifts by some scalars (that is, ), and after some computation the relationship between our old 2-cocycle and our new 2-cocycle is that
which is just the condition that differ by the 2-coboundary .
Altogether we’ve proven the following.
Theorem: Associated to any projective representation is a cohomology class , the Schur class. It can be constructed using any choice of lifts as above and does not depend on the choice. lifts to a linear representation iff the Schur class vanishes.
This is perhaps the simplest interesting example of a cohomology class being the obstruction to a lifting problem.
Although there is no natural way to take the direct sum of two projective representations, it is possible to make sense of the tensor product of two projective representations, and Schur classes are multiplicative with respect to tensor product: that is, if is the Schur class of a projective representation , then
Categories of projective representations
There’s a natural notion of isomorphism of projective representations: namely, a projective isomorphism between two projective representations is an element such that
for every . However, it’s less clear how to generalize this definition to describe a notion of not-necessarily-invertible morphism between projective representations.
Rather than do this directly, we’ll observe that the Schur class is a projective isomorphism invariant of projective representations, so the study of projective representations naturally breaks up into the study of projective representations of a fixed Schur class , for each . This suggests the following refined definition.
Definition: Fix and fix a 2-cocycle representing . An -projective representation of is a map such that
A morphism of -projective representations is a -linear map such that
for every .
Note that an -projective representation has more structure than a projective representation with Schur class , and that it really is necessary to pick a 2-cocycle in order to state this definition. There is a functor from the groupoid of -projective representations to projective representations, but it is not faithful: the notion of morphism above does not involve quotienting by scalar multiplication.
Unlike ordinary projective representations, -projective representations admit a direct sum, and in fact the category of -projective representations is about as well-behaved as possible in the following sense.
Theorem: The category of -projective representations is equivalent to the category of modules over the twisted group algebra . This algebra is as a -vector space, but with the modified multiplication
If are cohomologous, then are isomorphic, and hence the categories of -projective and -projective representations are equivalent.
This theorem guarantees that -projective representations exist, such as the regular representation of the twisted group algebra on itself. Note that there is no analogue of the trivial representation here, so this isn’t obvious. In other words:
Corollary: The Schur class map from projective representations to is surjective.