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In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and . The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the matrices used to represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name "symplectic group" is due to Hermann Weyl (details) as a replacement for the previous confusing names of (line) complex group and Abelian linear group, and is the Greek analog of "complex". ==== The symplectic group of degree over a field , denoted , is the group of symplectic matrices with entries in , and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant , the symplectic group is a subgroup of the special linear group . More abstractly, the symplectic group can be defined as the set of linear transformations of a -dimensional vector space over that preserve a non-degenerate, skew-symmetric, bilinear form, see classical group for this definition. Such a vector space is called a symplectic vector space. The symplectic group of an abstract symplectic vector space is also denoted . Typically, the field is the field of real numbers, , or complex numbers, . In this case is a real/complex Lie group of real/complex dimension . These groups are connected but non-compact. The centre of consists of the matrices and as long as the characteristic of the field is not .〔("Symplectic group" ), ''Encyclopedia of Mathematics'' Retrieved on 13 December 2014.〕 Here denotes the identity matrix. The non-triviality of the centre of and its relation to the simplicity of the group is discussed here. The real rank of the Lie Algebra, and hence, the Lie Group for is . The condition that a symplectic matrix preserves the symplectic form can be written as : where ''A''T is the transpose of ''A'' and : The Lie algebra of is given by the set of matrices ''A'' (with entries in ''F'') that satisfy : When , the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that . For , there are additional conditions, i.e. is then a proper subgroup of . 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「Symplectic group」の詳細全文を読む スポンサード リンク
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