MathGloss

Let $\mathfrak g$ be a Lie algebra over the field $F$. A $\mathfrak g$-module over a ring is a pair $(V,\cdot)$ of a vector space $V$ over $F$ and $\cdot:\mathfrak g \times V \to V$ a map such that

1. $(\lambda x + \mu y)\cdot v = \lambda (x\cdot v) + \mu(y\cdot v)$
2. $x\cdot (\lambda v + \mu w) = \lambda (x\cdot v) + \mu(y\cdot v)$
3. $[x,y]\cdot v = x\cdot (y\cdot v) - y\cdot (x\cdot v)$

for all $x,y \in \mathfrak g$, $v,w\in V$, and $\lambda,\mu \in F$.

Wikidata ID: Q18848

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