#
nLab

semidirect product Lie algebra

# Contents

## 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 $\mathfrak{a}$.

## References

Created on August 27, 2015 at 04:30:17.
See the history of this page for a list of all contributions to it.