By A. T. Fomenko

ISBN-10: 1904868320

ISBN-13: 9781904868323

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**Example text**

10 (Fixed Points) Let G be a k-group scheme and M a G-module. Set (1) M G = { m ~ M I g ( m O 1 ) = m O 1 forall g E G ( A ) andall A } . This is a k-submodule of M and its elements are called the fixedpoints of G on M . If we take g = id,[,] E G(k[G]) in (l), then we get (2) M G = { m e MlA,(m) = m @ l } . 34 Representations of Algebraic Groups This description of M Gas kernel of AM - idMQ 1 yields (3) Let k' be a k-algebra which is flat as a k-module. Then (M Q k')Gk' = M G Q k'. In case k is a field, this implies, of course, = (MG),.

X G, ( n factors) and k[M,] with the polynomial ring k[T,, T',. . ,T,]. The multiplicatioe group over k is the k-group functor G, with G,(A) = A" = {units of A ) for all A. It is an algebraic k-group with k[G,] = k[T, 7-11. Any k-module M defines a k-group functor G L ( M ) with G L ( M ) ( A )= (End,(M 0A))" called the general linear group of M. In case M = k", we may identify G L ( M ) with GL, where G L , ( A ) is the group of all invertible (n x n)-matrices over A . Obviously, GL, is an algebraic k-group with k[GL,] isomorphic to the localization of the polynomial ring k [ q j , 1 Ii, j In] with respect to {(det)"I n E N}.

9). In the case where our ground ring k is a field we can be more precise. Then the injective G-modules are determined up to isomorphism by their socle and any semi-simple G-module M occurs as a socle of such an injective G-module; the injective hull of M . The indecomposable injective G-modules are just the injective hulls of the simple G-modules. We get especially a decomposition of k[G] generalizing the decomposition of the regular representation of a finite group into principal indecomposable modules.

### A Short Course in Differential Geometry and Topology by A. T. Fomenko

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