k | field
A | |
finite dimensional k-algebra (associative, with 1, usually not semisimple).
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Modules are usually finite dimensional left A-modules
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An A-module is just a k-space V endowed with some linear transformations V → V
which are labelled α1,...,αm, and which satisfy the given
relations.
Example: consider the algebra A with one generator α and the relation
αt = 0.
Then an A-module is a vector space V with a nilpotent
endomorphism α with αt = 0
(or in case V is finite-dimensional, we may just think of α as a
square matrix with αt = 0).
A submodule U of (V,αi) is a subspace of V such that αi(U) is contained in U, for all i.
If U, U' are submodules of (V,αi) such that U∩U' = 0 and U+U' = V,
then V is said to be the direct sum of these submodules, and U and U' are
direct summands
of V.
Usually not all submodules will be direct summands!
An A-module V ≠ 0 is said to be indecomposable, if its only direct summands are 0 und V.
Krull-Remak-Schmidt: If one writes a finite dimensional A-module as a direct sum of indecomposable direct summands, then these summands are unique up to isomorphism.
Thus, in order to classify fin.dim. A-modules, it suffices to look at indecomposable ones. |