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In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special〔Here, ''special'' means the subgroup of the full automorphism group whose elements have determinant 1.〕 automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.〔 p. 94.〕 Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph ''The Classical Groups''. The classical groups form the deepest and most useful part of the subject of linear Lie groups.〔 p. 91.〕 Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group is a symmetry group of spacetime of special relativity. The special unitary group is the symmetry group of quantum chromodynamics and the symplectic group finds application in hamiltonian mechanics and quantum mechanical versions of it. ==The classical groups== The classical groups are exactly the general linear groups over and together with the automorphism groups of non-degenerate forms discussed below.〔 p, 94〕 These groups are usually additionally restricted to the subgroups whose elements have determinant 1. The classical groups, with the determinant 1 condition, are listed in the table below. In the sequel, the determinant 1 condition is ''not'' used consistently in the interest of greater generality. The complex classical groups are , and . A group is complex according to whether its Lie algebra is complex. The real classical groups refers to all of the classical groups since any Lie algebra is a real algebra. The compact classical groups are the compact real forms of the complex classical groups. These are, in turn, , and . One characterization of the compact real form is in terms of the Lie algebra . If , the complexification of , then if the connected group generated by is a compact, is a compact real form.〔 p. 103.〕 The classical groups can uniformly be characterized in a different way using real forms. The classical groups (here with the determinant 1 condition, but this is not necessary) are the following: :The complex linear algebraic groups , and together with their real forms.〔 See end of chapter 1.〕 For instance, is a real form of , is a real form of , and is a real form of . Without the determinant 1 condition, replace the special linear groups with the corresponding general linear groups in the characterization. The algebraic groups in question are Lie groups, but the "algebraic" qualifier is needed to get the right notion of "real form". 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「Classical group」の詳細全文を読む スポンサード リンク
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