nLab
semidirect product Lie algebra

Contents

Idea

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

ρ:𝔤𝔡𝔢𝔯(𝔞) \rho \colon \mathfrak{g}\longrightarrow \mathfrak{der}(\mathfrak{a})

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

𝔤^=𝔤𝔞 \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]+ρ(x 1)(t 2)ρ(x 2)(t 1))). [(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 𝔤\mathfrak{g} with 𝔞\mathfrak{a}.

References

Created on August 27, 2015 at 04:30:15. See the history of this page for a list of all contributions to it.