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Generalization Of A Lie Algebra
In mathematics, a Lie algebra has been generalized in several ways. Graded Lie algebra and Lie superalgebra A graded Lie algebra is a Lie algebra with grading. When the grading is \mathbb/2, it is also known as a Lie superalgebra. Lie-isotopic algebra A Lie-isotopic algebra is a generalization of Lie algebras proposed by physicist R. M. Santilli in 1978. Definition Recall that a finite-dimensional Lie algebra L with generators X_1, X_2, ..., X_n and commutation rules : _i X_j= X_i X_j - X_j X_i = C_^k X_k, can be defined (particularly in physics) as the totally anti-symmetric algebra A(L)^- attached to the universal enveloping associative algebra A(L)=\ equipped with the associative product X_i \times X_j over a numeric field F with multiplicative unit 1. Consider now the axiom-preserving lifting of A(L) into the form A^*(L^*)=\, called universal enveloping isoassociative algebra, with isoproduct :X_i\times X_j = X_i T^* X_j, verifying the isoassociative law :X_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) 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. 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 ,y/math>. 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, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |