📂Representation Theory

Li subalgebra

Definition¹

Lie algebra $\mathfrak{g}$ If a subspace $\mathfrak{h}$ is closed under the bracket $[\cdot, \cdot]$ of $\mathfrak{g}$, then $\mathfrak{h}$ is called a Lie subalgebra of $\mathfrak{g}$.

$$ \text{the subspace $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$, $\quad$ if } [H_{1}, H_{2}] \in \mathfrak{h} \quad \forall H_{1}, H_{2} \in \mathfrak{h} $$

Ideal

If the subalgebra $\mathfrak{h}$ satisfies the following, it is called an ideal of $\mathfrak{g}$.

$$ [X, H] \in \mathfrak{h} \quad \forall X \in \mathfrak{g}, \forall H \in \mathfrak{h} $$

In set notation,

$$ [\mathfrak{g}, \mathfrak{h}] \subset \mathfrak{h} $$

Center

For the Lie algebra $\mathfrak{g}$, the set of elements that commute with every element of $\mathfrak{g}$ (with respect to the Lie product) is called the center of $\mathfrak{g}$.

$$ \text{center} = \left\{ X \in \mathfrak{g} : [X, Y] = 0, \quad \forall Y \in \mathfrak{g} \right\} $$

Properties

(a) Let $\mathfrak{g}$ be a Lie algebra. The center of $\mathfrak{g}$ is an ideal of $\mathfrak{g}$.

Let $G$ and $H \subset G$ be matrix Lie groups. Then the Lie algebra $\mathfrak{h}$ of $H$ is a subalgebra of the Lie algebra $\mathfrak{g}$ of $G$.

(b) If $H$ is a normal subgroup of $G$, then $\mathfrak{h}$ is an ideal of $\mathfrak{g}$.