nLab semidirect product Lie algebra

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Contents

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

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

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

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

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)). \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 𝔤\mathfrak{g} with 𝔞\mathfrak{a}.

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:

Last revised on December 4, 2023 at 07:42:29. See the history of this page for a list of all contributions to it.