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For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term “Lie algebra” (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name “infinitesimal group” is used.

An example of a Lie algebra is $\mathbb{R}^3$ endowed with the cross product.

For a family of examples, let $A$ be any associative algebra and define, for every $v,w\in A$, $[v,w]=v.w-w.v$. Then $\bigl(A,[\cdot,\cdot]\bigr)$ is a Lie algebra.

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