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  In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with a nonassociative operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finitedimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras.
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with a (typically nonassociative) operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not required to be (and typically will not be) associative. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity:
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not necessarily associative. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity:
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity, where brackets [,] represent the commutator. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not necessarily associative. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity:
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not necessarily associative. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity:

has abstract
  In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with a nonassociative operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finitedimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the noncommutativity between two rotations: since a rotation commutes with itself, we have the alternating property .
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with a (typically nonassociative) operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finitedimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the noncommutativity between two rotations: since a rotation commutes with itself, we have the alternating property .
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not required to be (and typically will not be) associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finitedimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the noncommutativity between two rotations: since a rotation commutes with itself, we have the alternating property .
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finitedimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the noncommutativity between two rotations: since a rotation commutes with itself, we have the alternating property .
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi identity, where brackets [,] represent the commutator. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finitedimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the noncommutativity between two rotations: since a rotation commutes with itself, we have the alternating property .
 In mathematics, a Lie algebra (pronounced "Lee") is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a nonassociative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finitedimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example is the space of three dimensional vectors with the bracket operation defined by the cross product This is skewsymmetric since , and instead of associativity it satisfies the Jacobi identity: This is the Lie algebra of the Lie group of rotations of space, and each vector may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the noncommutativity between two rotations: since a rotation commutes with itself, we have the alternating property .
