: Mathématiques: Lie bracket

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Lie bracket

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space ${\\displaystyle {\\mathfrak {g}}}$ together with an operation called the Lie bracket, an alternating bilinear map ${\\displaystyle {\\mathfrak {g}}\ imes {\\mathfrak {g}}\ ightarrow {\\mathfrak {g}}}$ , that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors ${\\displaystyle x}$ and ${\\displaystyle y}$ is denoted ${\\displaystyle [x,y]}$ . The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative. Given an associative algebra (like for example the space of square matrices), a Lie bracket can be and is often defined through the commutator, namely defining ${\\displaystyle [x,y]=xy-yx}$ correctly defines a Lie bracket in addition to the already existing multiplication operation.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Lie_algebra)