geometric representation theory
representation, 2-representation, ∞-representation
group algebra, algebraic group, Lie algebra
vector space, n-vector space
affine space, symplectic vector space
module, equivariant object
bimodule, Morita equivalence
induced representation, Frobenius reciprocity
Hilbert space, Banach space, Fourier transform, functional analysis
orbit, coadjoint orbit, Killing form
geometric quantization, coherent state
module algebra, comodule algebra, Hopf action, measuring
Geometric representation theory
D-module, perverse sheaf,
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
geometric function theory, groupoidification
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
A projective representation of a group is a representation up to a central term: a group homomorphism , to the projective general linear group of some -vector space .
The group extension and its cocycle
By construction, there is a short exact sequence
which exhibits as a group extension of by the group of units of the ground field. This is classified by a 2-cocycle in group cohomology .
It is useful to re-express this equivalently in terms of homotopy theory via the discussion at looping and delooping, by which group homomorphisms are equivalently maps between their delooping groupoids.
In terms of this the above group extension and its classifying cocycle is exhibited by a homotopy fiber sequence of deloopings of the form
Relation to genuine representations
Via the projection , every linear representation of induces a projective representation.
By the universal property of the homotopy pullback, the discussion above means that the obstruction to lift a given projective representation to a linear representation
is the class of the 2-cocycle in the group cohomology class .
As twisted linear representations
By the above and the discussion at group extension – Central group extensions – Formulation in homotopy theory the cocycle map of homotopy types may be represented by a zigzag (“2-anafunctor”) of crossed complexes as
Here the 2-groupoid looks schematically like
This shows that a map may equivalently be represented by two functions (not group homomorphisms in general!)
such that for all
is a group 2-cocycle on with values in representing the above cohomology class of .
This is the form in which projective representations are often discussed in the literature.
As genuine representations after extensions
Alternatively, one may consider the above diagram
and form the further pullback along . By the pasting law this is (the delooping of) the group extension of which is classified by :
This way the projective representation of induces a genuine linear representation of . One finds (this is a special case of the general discussion at twisted infinity-bundle) that this constitutes an equivalence between projective representations of and genuine representations of .