Today we’ll resolve half the puzzle of why the cohomology group appears both when classifying projective representations of a group over a field and when classifying -linear actions of on the category of -vector spaces by describing a functor from the former to the latter.
(There is a second half that goes in the other direction.)
For the remainder of this post we’ll restrict our attention to finite-dimensional projective representations .
The first observation is that naturally acts on the matrix algebra . More precisely, the conjugation action of on naturally factors through . In fact, is precisely the automorphism group of . This is guaranteed by the following theorem.
Theorem (Noether-Skolem): Let be a finite-dimensional simple -algebra, let be a central simple -algebra (meaning is finite-dimensional, simple, and ), and let be -algebra homomorphisms. Then there is an invertible element such that for all .
Corollary: Every automorphism of a central simple algebra is inner, and hence the automorphism group of a central simple algebra is .
Proof. are finite-dimensional simple algebras, and hence have unique simple modules . The restriction of to an -module along either or produces an -module which (by simplicity of ) is a direct sum of copies of ; in particular, the isomorphism type of this module is determined by its dimension. Hence these two restrictions are isomorphic. An isomorphism between them is some -linear automorphism of , so by the Jacobson density theorem (and centrality of ) it is given by multiplication by some invertible . This is the desired .
Hence an equivalent description of (finite-dimensional) projective representations of over is that they are actions of on matrix algebras .
Next, it will be useful to describe something about the functoriality of the construction associating to a -algebra the category of (right) -modules. This is both a covariant and a contravariant functor, although we’ll only be interested in the covariant version, which sends a morphism of -algebras to the extension of scalars functor
where acquires the structure of an -bimodule over as follows: the left action of is given by left multiplication by , while the right action is given by right multiplication as usual.
This construction is in fact a 2-functor from the category of -algebras to the Morita 2-category of -algebras, -bimodules, and bimodule homomorphisms, or equivalently of module categories , cocontinuous -linear functors, and natural transformations.
Lemma: Let be two morphisms of -algebras. Natural transformations of functors can be identified with elements acting by left multiplication on such that
Proof. By Eilenberg-Watts, natural transformations of functors can be identified with morphisms of bimodules, where the first bimodule has its left -module structure coming from and the second coming from . By the Yoneda lemma, morphisms of right -modules can be identified with elements acting by left multiplication, and then the additional compatibility with the left -module structures is precisely the above condition.
In other words, the 2-functor factors through a 2-category whose
- objects are -algebras ,
- morphisms are -algebra morphisms ,
- 2-morphisms are elements such that ,
and the resulting 2-functor is fully faithful on hom categories.
Now, recall that is Morita equivalent to . Hence an action of on gives rise not only to an action of on (by applying the above 2-functor), but also to an action on (since one payoff of writing down the 2-categorical notion of action is that it transports across equivalences of categories). In fact, we can be more precise about which action we get.
Theorem: Let be a finite-dimensional projective representation of over . It induces a -linear action of on , and the isomorphism class of this action is the Schur class .
Proof. First, we observe that a -linear Morita equivalence induces a -linear isomorphism on centers
We’ll be talking about 2-cocycles with values in , but because of the Morita equivalence above this is equivalent to talking about corresponding 2-cocycles with values in , which will allow us to match up the 2-cocycles that are about to appear with the 2-cocycles that appeared when we classified -linear actions of on . Apart from this observation we will no longer need to explicitly talk about the Morita equivalence.
Now, given , consider the corresponding automorphism . Either because of the Morita equivalence to or by Noether-Skolem, we know that is equivalent to the identity. Explicitly, we know that admits a lift to some , and that , thought of as acting on by left multiplication, furnishes a natural isomorphism between and the identity.
Hence we can describe natural isomorphisms between various composites of the by first trivializing them using composites of the lifts , then describing natural automorphisms of the identity. The identity is as an -bimodule, and hence its natural automorphisms can naturally be identified with invertible elements in the center acting by left multiplication.
There are natural isomorphisms relating and which, after trivializing both using the lift and the product of lifts , can be described as multiplication by scalars such that
and so, up to making sure our conventions are consistent, we find that writing down the 2-cocycle representing the Schur class of corresponds precisely to writing down the 2-cocycle representing the corresponding action of on , as desired.
The Schur class as a characteristic class
The Schur class can be thought of as a characteristic class of projective representations, and in the same way that characteristic classes of vector bundles in classical algebraic topology come from universal cohomology classes of classifying spaces, the Schur class comes from a universal characteristic class
which classifies the universal projective representation . In other words, the universal Schur class is a map
of 2-groupoids, and the content of the above discussion is that it admits a functorial description in terms of the 2-functor sending a matrix algebra to its category of modules.