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Concept
Projective representation
Summary
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group
where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation).
In more concrete terms, a projective representation of is a collection of operators satisfying the homomorphism property up to a constant:
for some constant . Equivalently, a projective representation of is a collection of operators , such that . Note that, in this notation, is a set of linear operators related by multiplication with some nonzero scalar.
If it is possible to choose a particular representative in each family of operators in such a way that the homomorphism property is satisfied on the nose, rather than just up to a constant, then we say that can be "de-projectivized", or that can be "lifted to an ordinary representation". More concretely, we thus say that can be de-projectivized if there are for each such that . This possibility is discussed further below.
One way in which a projective representation can arise is by taking a linear group representation of G on V and applying the quotient map
which is the quotient by the subgroup F∗ of scalar transformations (diagonal matrices with all diagonal entries equal). The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to an ordinary linear representation. A general projective representation ρ: G → PGL(V) cannot be lifted to a linear representation G → GL(V), and the obstruction to this lifting can be understood via group cohomology, as described below.
However, one can lift a projective representation of G to a linear representation of a different group H, which will be a central extension of G. The group is the subgroup of defined as follows:
where is the quotient map of onto .
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In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since this group is noncompact, these unitary representations are infinite-dimensional.) It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states.
Projective representation
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). In more concrete terms, a projective representation of is a collection of operators satisfying the homomorphism property up to a constant: for some constant .
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In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group.
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