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In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields. Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures. The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions. ==Set up== Let be a finite-dimensional real or complex vector space with a nondegenerate quadratic form . The (real or complex) linear maps preserving form the orthogonal group . The identity component of the group is called the special orthogonal group . (For real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, has a unique connected double cover, the spin group . There is thus a group homomorphism whose kernel has two elements denoted }, where is the identity element. The groups and are all Lie groups, and for fixed they have the same Lie algebra, . If is real, then is a real vector subspace of its complexification , and the quadratic form extends naturally to a quadratic form on . This embeds as a subgroup of , and hence we may realise as a subgroup of . Furthermore, is the complexification of . In the complex case, quadratic forms are determined up to isomorphism by the dimension of . Concretely, we may assume and : The corresponding Lie groups and Lie algebra are denoted and . In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers where is the dimension of , and is the signature. Concretely, we may assume and : The corresponding Lie groups and Lie algebra are denoted and . We write in place of to make the signature explicit. The spin representations are, in a sense, the simplest representations of and that do not come from representations of and . A spin representation is, therefore, a real or complex vector space together with a group homomorphism from or to the general linear group such that the element is ''not'' in the kernel of . If is such a representation, then according to the relation between Lie groups and Lie algebras, it induces a Lie algebra representation, i.e., a Lie algebra homomorphism from or to the Lie algebra of endomorphisms of with the commutator bracket. Spin representations can be analysed according to the following strategy: if is a real spin representation of , then its complexification is a complex spin representation of ; as a representation of , it therefore extends to a complex representation of . Proceeding in reverse, we therefore ''first'' construct complex spin representations of and , then restrict them to complex spin representations of and , then finally analyse possible reductions to real spin representations. 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「Spin representation」の詳細全文を読む スポンサード リンク
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