One way in which such representations can arise is using the homomorphism GL(V) to PGL(V), tkaing the quotient by the subgroup K^{x}. The interest for algebra is in the process in the other direction: given a *projective representation*, try to 'lift' it to a conventional *linear group representation*. This brings in questions of group cohomology.

In fact if one introduces for g in G a lifted element L(g), and a scalar matrix c(g) for g in G representing the freedom in lifting from PGL(V) back to GL(V), and then looks at the condition for lifted images to satisfy the homomorphism condition L(gh) = L(g)L(h) after modification by c(g), c(h) and c(gh), one finds a cocycle equation. This need not come down to a coboundary: that is, projective representations may not lift. It is shown, however, that this leads to an extension problem for G. If G is correctly extended we can speak of a linear representation of the extended group, which gives back the initial projective representation on factoring by K^{x} and the extending subgroup.

See also linear representation, affine representation, group action.

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