
Isotypic component
Encyclopedia
The Isotypic component of weight
of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight
.
This defines the isotypic component of weight
of V:
where
is maximal.


Definition
- A finite dimensional moduleModule (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
of a reductive Lie algebra
(or of the correspondign Lie group
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
) can be decomposed into irreducible submodules.
- Each finite dimensional irreducible representation of
is uniquely identified (up to isomorphism) by its highest weight
, where
denotes the highest weight module with highest weight
.
- In the decomposition of
, a certain isomorphism class might appear more than once, hence
.
This defines the isotypic component of weight


