nLab semidirect product Lie algebra

Idea

Given by a Lie action of a Lie algebra $\mathfrak{g}$ on another Lie algebra $\mathfrak{a}$, hence a Lie algebra homomorphism

$$\rho \colon \mathfrak{g}\longrightarrow \mathfrak{der}(\mathfrak{a})$$

to the derivations on $\mathfrak{a}$, then there is a Lie algebra extension of $\mathfrak{g}$ by $\mathfrak{a}$ whose underlying vector space is

$$\hat \mathfrak{g} = \mathfrak{g} \oplus \mathfrak{a}$$

and whose Lie bracket is given by the formula

$$[(x_1,t_1), (x_2,t_2)] = ( [x_1,x_2], \;([t_1,t_2] + \rho(x_1)(t_2) - \rho(x_2)(t_1)) ) \,.$$

This is the semidirect product of $\mathfrak{g}$ with $\mathfrak{a}$.

Related concepts

• semidirect product group

References