---- > [!definition] Definition. ([[character of a Lie algebra representation]]) > Let $\mathfrak{g}$ be a (say, [[semisimple Lie algebra|semisimple]]) [[Lie algebra]] $V$ a finite-dimensional [[Lie algebra representation|representation]] of $\mathfrak{g}$. Let $\mathfrak{t}$ be the [[Cartan subalgebra|CSA]] of $\mathfrak{g}$. Write $V=\bigoplus_{\lambda \in \mathfrak{t^{*}}} V_{\lambda}$ for the [[on factorizing a Lie algebra representation into weight spaces|weight space]] [[on the weights of a representation|decomposition]] of $V$. The **character** of $V$ is the element $\text{ch}(V)=\sum_{\lambda \in X} \text{dim}(V_{\lambda})e^{\lambda}$ of the [[character ring of a root system|character ring]] $\mathbb{Z}[X]$. An arbitrary element of $\mathbb{Z}[X]$ is called a **formal character**. ^definition > [!note] Remark. > If $V$ is [[irreducible Lie algebra representation|irreducible]], $V=V(\nu)$ [[the classification of complex irreducible semisimple Lie algebra representations|for some]] $\nu \in \mathfrak{t}^{*}$,[^1] then $\text{ch}(V)$ is given by the [[Weyl character formula]]. ^note [^1]: Here the notation $V(\nu)$ is for the [[highest weight module|unique irreducible]] [[quotient representation|quotient]] of the [[Verma module]] $M(\nu)$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```