nLab semidirect product Lie algebra

Idea

Consider a Lie algebra action of a Lie algebra $\mathfrak{g}$ on Lie algebra $\mathfrak{a}$, hence a Lie algebra homomorphism

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

to the derivation Lie algebra of $\mathfrak{a}$.

There is a Lie algebra extension of $\mathfrak{g}$ by $\mathfrak{a}$ whose underlying vector space is the direct sum (product in Vect)

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

and whose Lie bracket is given by the formula

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

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

Related concepts

• Lie algebra extension

• semidirect product group

References

• Tadeusz Ostrowski, A note on semidirect sum of Lie algebras, Discussiones Mathematicae - General Algebra and Applications (2013) 33 2 (2013) 233-247 [eudml:270203]

• Mohammad Reza Alemi, Farshid Saeedi, Derivation algebra of semi-direct sum of Lie algebras, Asian-European Journal of Mathematics 15 04 2250063 (2022) [doi:10.1142/S1793557122500632]

See also:

• Wikipedia, Lie algebra extension – Construction