By Samuel M. Vovsi (auth.)
The building thought of in those notes relies on an easy thought. permit (A, G ) and (B, G ) be crew representations, for definiteness devoted and finite 1 2 dimensional, over an arbitrary box. we will say trustworthy illustration (V, G) is an extension of (A, G ) through (B, G ) if there's a G-submodule W of V such that 1 2 the clearly coming up representations (W, G) and (V/W, G) are isomorphic, modulo their kernels, to (A, G ) and (B, G ) respectively. 1 2 query. between all of the extensions of (A, G ) by way of (B, G ), does there exist 1 2 one of these "universal" extension which includes an isomorphic reproduction of the other one? the answer's within the affirmative. rather, permit dim A = m and dim B = n, then the teams G and G could be regarded as matrix teams of levels m and n 1 2 respectively. If (V, G) is an extension of (A, G ) by way of (B, G ) then, below yes 1 2 collection of a foundation in V, all components of G are represented by way of (m + n) x (m + n) mat rices of the shape (*) ~1-~ ~-J lh I g2 I .
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