MathGloss

Let $(V,F)$ be a vector space and let $\text{End}(V)$ be the set of linear transformations $V\to V$. This is a ring with resepect to the usual product operation. Define a new bracket operation by $[x,y] = xy-yx$. Then $\text{End}(V)$ is a Lie algebra over $F$. We call it the general linear algebra and denote it $\mathfrak{gl}(V)$.

If $V$ is of finite dimension, then $\text{dim}(\text{End}(V)) = \text{dim}(V)^2$.

The general linear algebra is a Lie algebra.

Wikidata ID: Q17521172

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