projective representation



A projective representation of a group GG is a representation up to a central term: a group homomorphism GPGL(V)G\to PGL(V), where VV is a 𝕂\mathbb{K}-vector space. Via the projection GL(V)PGL(V)=GL(V)/𝕂 *GL(V)\to PGL(V)=GL(V)/\mathbb{K}^*, every linear representation of GG induces a projective representation.

By the fibration sequence

B𝕂 * BGL(V) * BPGL(V) \array{ \mathbf{B}\mathbb{K}^*&\to & \mathbf{B}GL(V)\\ \downarrow&& \downarrow\\ *&\to&\mathbf{B}PGL(V) }

the obstruction to lift a projective representation of GG to a linear representation is represented by an element c ρc_\rho in H 2(G,𝕂 *)H^2(G,\mathbb{K}^*).

A refinement of this idea consists in looking at the 2-groupoid BGL(V)//K *\mathbf{B}GL(V)//\mathbf{K}^*. Then a functor (ρ,λ):𝔹GBGL(V)//K *(\rho,\lambda):\mathbb{B}G\to \mathbf{B}GL(V)//\mathbf{K}^* consists of two maps ρ:GGL(V)\rho:G\to GL(V) and λ:G×G𝕂 *\lambda:G\times G\to \mathbb{K}^* such that ρ(g)ρ(h)=λ(g,h)ρ(gh)\rho(g)\rho(h)=\lambda(g,h)\rho(g h), and λ\lambda is a 2-cocycle on GG with values in 𝕂 *\mathbb{K}^* representing the cohomology class c ρc_\rho.

Revised on July 17, 2013 21:38:29 by Urs Schreiber (