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Serre's Theorem On A Semisimple Lie Algebra
In abstract algebra, specifically the theory of Lie algebras, Serre's theorem states: given a (finite reduced) root system \Phi, there exists a finite-dimensional semisimple Lie algebra whose root system is the given \Phi. Statement The theorem states that: given a root system \Phi in a Euclidean space with an inner product (, ), \langle \beta, \alpha \rangle = 2(\alpha, \beta)/(\alpha, \alpha), \beta, \alpha \in E and a base \ of \Phi, the Lie algebra \mathfrak g defined by (1) 3n generators e_i, f_i, h_i and (2) the relations :[h_i, h_j] = 0, :[e_i, f_i] = h_i, \, [e_i, f_j] = 0, i \ne j, :[h_i, e_j] = \langle \alpha_i, \alpha_j \rangle e_j, \, [h_i, f_j] = -\langle \alpha_i, \alpha_j \rangle f_j, :\operatorname(e_i)^(e_j) = 0, i \ne j, :\operatorname(f_i)^(f_j) = 0, i \ne j. is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by h_i's and with the root system \Phi. The square matrix [\langle \alpha_i, \alpha_j \rangle]_ is called the Cartan matri ... [...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. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative 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 finite-dimensional 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |