The Isotypic component of weight
λ
of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight
λ
.
A finite-dimensional module
V
of a reductive Lie algebra
g
(or of the corresponding Lie group) can be decomposed into irreducible submodules
V
=
⊕
i
=
1
N
V
i
.
Each finite-dimensional irreducible representation of
g
is uniquely identified (up to isomorphism) by its highest weight
∀
i
∈
{
1
,
…
,
N
}
∃
λ
∈
P
(
g
)
:
V
i
≃
M
λ
, where
M
λ
denotes the highest weight module with highest weight
λ
.
In the decomposition of
V
, a certain isomorphism class might appear more than once, hence
V
≃
⊕
λ
∈
P
(
g
)
(
⊕
i
=
1
d
λ
M
λ
)
.
This defines the isotypic component of weight
λ
of V:
λ
(
V
)
:=
⊕
i
=
1
d
λ
V
i
≃
C
d
λ
⊗
M
λ
where
d
λ
is maximal.
