| Triangular Products of Group Representations and Their Applications Softcover Repri Edition Contributor(s): Vovsi, S. M. (Author) |
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ISBN: 1468467239 ISBN-13: 9781468467239 Publisher: Birkhauser OUR PRICE: $52.24 Product Type: Paperback Published: February 2012 |
| Additional Information |
| BISAC Categories: - Gardening - Mathematics | Algebra - Abstract - Mathematics | Group Theory |
| Dewey: 512.2 |
| Series: Progress in Mathematics |
| Physical Information: 0.31" H x 6.14" W x 9.21" (0.47 lbs) 132 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: The construction considered in these notes is based on a very simple idea. Let (A, G ) and (B, G ) be two group representations, for definiteness faithful and finite- 1 2 dimensional, over an arbitrary field. We shall say that a faithful representation (V, G) is an extension of (A, G ) by (B, G ) if there is a G-submodule W of V such that 1 2 the naturally arising representations (W, G) and (V/W, G) are isomorphic, modulo their kernels, to (A, G ) and (B, G ) respectively. 1 2 Question. Among all the extensions of (A, G ) by (B, G ), does there exist 1 2 such a "universal" extension which contains an isomorphic copy of any other one? The answer is in the affirmative. Really, let dim A = m and dim B = n, then the groups G and G may be considered as matrix groups of degrees m and n 1 2 respectively. If (V, G) is an extension of (A, G ) by (B, G ) then, under certain 1 2 choice of a basis in V, all elements of G are represented by (m + n) x (m + n) mat- rices of the form (*) 1- -J lh I g2 I . |