If and are -modules over a field (i.e. representations of in and ), then a map is a -map if is -linear and preserves the -action, i.e. if
-maps have subrepresentations, also called -submodules, as their kernel and image. To see this, let be a -map; let and be the kernel and image respectively of . is a submodule of if it is stable under the action of , but
is a submodule of if it is stable under the action of , but
Finally, we define the intuitive concept of a quotient -module. Suppose is a -submodule. Then is a finite-dimensional vector space. We can define an action of on via , so that is well-defined under the action and is a -module.
|Date of creation||2013-03-22 16:37:59|
|Last modified on||2013-03-22 16:37:59|
|Last modified by||rm50 (10146)|