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Lie algebra - Wikipedia

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Join the Wiki­data contest and help improve geo­graphi­cally located items in 27 coun­tries and regions! Coordinate Me  ❭  MAY 2025 Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space 𝑔 together with an operation called the Lie bracket, an alternating bilinear map 𝑔 × 𝑔 → 𝑔 , that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors 𝑥 and 𝑦 is denoted [ 𝑥 , 𝑦 ] . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, [ 𝑥 , 𝑦 ] = 𝑥 𝑦 − 𝑦 𝑥 . Lie algebras are closely related to Lie groups, which

Lie algebra - Wikipedia Jump to content From Wikipedia, the free encyclopedia Algebraic structure used in analysis "Lie bracket" redirects here. For the operation on vector fields, see Lie bracket of vector fields . This article has multiple issues. Please help improve it or discuss these issues on the talk page . ( Learn how and when to remove these messages ) This article needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed. Find sources:   "Lie algebra"  –  new

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