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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 :PGL(''V'', ''F'') = GL(''V'', ''F'') / ''F''∗, where GL(''V'', ''F'') is the general linear group of invertible linear transformations of ''V'' over ''F'' and ''F''∗ is the normal subgroup consisting of multiplications of vectors in ''V'' by nonzero elements of ''F'' (that is, scalar multiples of the identity; scalar transformations).〔.〕 ==Linear representations and projective representations== One way in which a projective representation can arise is by taking a linear group representation of on and applying the quotient map : which is the quotient by the subgroup 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 a conventional ''linear representation''. In general, given a projective representation it cannot be lifted to a linear representation , and the obstruction to this lifting can be understood via group homology, as described below. However, one ''can'' lift a projective representation of to a linear representation of a different group , which will be a central extension of . To understand this, note that is a central extension of , meaning that the kernel is central (in fact, is exactly the center of ). One can pull back the projective representation along the quotient map, obtaining a ''linear'' representation and will be a central extension of because it is a pullback of a central extension. Thus projective representations of can be understood in terms of linear representations of (certain) central extensions of . Notably, for a perfect group there is a single universal perfect central extension of that can be used. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Projective representation」の詳細全文を読む スポンサード リンク
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