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In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semi-simple Lie algebras. Here the construction of a root system of a semi-simple Lie algebraand, conversely, the construction of a semi-simple Lie algebra from a reduced crystallographic root systemare shown. == Associated root system == Let g be a semi-simple complex Lie algebra. Let further h be a Cartan subalgebra of g. Then h acts on g via simultaneously diagonalizable linear maps in the adjoint representation. For λ in h * define the subspace gλ ⊂ g by : We call a non-zero ''λ'' in h * a root if the subspace g''λ'' is nontrivial. In this case g''λ'' is called the root space of ''λ''. The definition of Cartan subalgebra guarantees that g0 = h. One can show that each non-trivial g''λ'' (i.e. for ''λ''≠0) is one-dimensional. Let ''R'' be the set of all roots. Since the elements of h are simultaneously diagonalizable, we have : The Cartan subalgebra h inherits an inner product from the Killing form on g. This induces an inner product on h *. One can show that with respect to this inner product ''R'' is a reduced crystallographic root lattice. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Root system of a semi-simple Lie algebra」の詳細全文を読む スポンサード リンク
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