Isotypic component

The Isotypic component of weight $\lambda$ of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight $\lambda$.

Definition

• A finite-dimensional module $V$ of a reductive Lie algebra $\mathfrak{g}$ (or of the corresponding Lie group) can be decomposed into irreducible submodules

$V = \oplus_{i=1}^N V_i$.

• Each finite-dimensional irreducible representation of $\mathfrak{g}$ is uniquely identified (up to isomorphism) by its highest weight

$\forall i \in \{1,\ldots,N\} \exists \lambda \in P(\mathfrak{g}) : V_i \simeq M_\lambda$, where $M_\lambda$ denotes the highest weight module with highest weight $\lambda$.

• In the decomposition of $V$, a certain isomorphism class might appear more than once, hence

$V \simeq \oplus_{\lambda \in P(\mathfrak{g})} ( \oplus_{i=1}^{d_\lambda} M_{\lambda})$.

This defines the isotypic component of weight $\lambda$ of V: $\lambda(V) := \oplus_{i=1}^{d_\lambda} V_i \simeq \mathbb{C}^{d_\lambda} \otimes M_{\lambda}$ where $d_\lambda$ is maximal.

See also

• Lie algebra representation
• Weight (representation theory)