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Concept# Lie algebra

Summary

In mathematics, a Lie algebra (pronounced liː ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \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 x and y is denoted [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 [x,y] = x y - yx correctly defines a Lie bracket in addition to the already existing multiplication operation.
Lie algebras are closely related

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