A module is semisimple, or completely reducible, when it is a direct sum of simple modules. Equivalently, every submodule has a direct-sum complement.
A finite-length semisimple module has a canonical decompositionin which is the sum of all simple submodules isomorphic to a fixed simple module . Each is isomorphic to a finite direct sum .
The decomposition is intrinsic to the semisimple module: the isomorphism types of its simple submodules determine the summands.
In a complex representation of a finite group , the -isotypic component is the sum of all irreducible submodules with character . The character idempotent projects onto it. Every -submodule is preserved by these projections and decomposes into its intersections with the isotypic components. If , conjugation permutes them according to .
Articles by others on the same topic
In the context of module theory and representation theory in algebra, a **semisimple module** is a specific type of module that has a particular structure. A module \( M \) over a ring \( R \) is said to be **semisimple** if it satisfies the following equivalent conditions: 1. **Direct Sum Decomposition**: \( M \) can be expressed as a direct sum of simple modules.