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In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. In fact, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan. == Lie algebras == A generalized Cartan matrix is a square matrix with integral entries such that # For diagonal entries, ''aii'' = 2. # For non-diagonal entries, . # if and only if # ''A'' can be written as ''DS'', where ''D'' is a diagonal matrix, and ''S'' is a symmetric matrix. For example, the Cartan matrix for ''G''2 can be decomposed as such: : The third condition is not independent but is really a consequence of the first and fourth conditions. We can always choose a ''D'' with positive diagonal entries. In that case, if ''S'' in the above decomposition is positive definite, then ''A'' is said to be a Cartan matrix. The Cartan matrix of a simple Lie algebra is the matrix whose elements are the scalar products : (sometimes called the Cartan integers) where ''ri'' are the simple roots of the algebra. The entries are integral from one of the properties of roots. The first condition follows from the definition, the second from the fact that for is a root which is a linear combination of the simple roots ''ri'' and ''rj'' with a positive coefficient for ''rj'' and so, the coefficient for ''ri'' has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let and . Because the simple roots span a Euclidean space, S is positive definite. Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (See Kac–Moody algebra for more details). 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「Cartan matrix」の詳細全文を読む スポンサード リンク
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