A module over a Lie algebra is a vector space equipped with a linear action of the Lie algebra that satisfies certain properties, allowing for the representation of the Lie algebra's structure. This concept is essential for understanding how Lie algebras can act on various mathematical objects, leading to important constructions like direct sums and semidirect products. Modules provide a framework for analyzing representations, revealing relationships between different algebras and their actions.
congrats on reading the definition of Module over a Lie algebra. now let's actually learn it.