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| In [[mathematics]], the '''spin representations''' are particular [[projective representation]]s of the [[orthogonal group|orthogonal]] or [[special orthogonal group]]s in arbitrary [[dimension]] and [[metric signature|signature]] (i.e., including [[indefinite orthogonal group]]s). More precisely, they are [[representation of a Lie group|representations]] of the [[spin group]]s, which are [[Double covering group|double cover]]s of the special orthogonal groups. They are usually studied over the [[real number|real]] or [[complex number]]s, but they can be defined over other [[field (mathematics)|field]]s.
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| Elements of a spin representation are called [[spinor]]s. They play an important role in the [[physics|physical]] description of [[fermion]]s such as the [[electron]].
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| 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 representation]]s by introducing [[real structure]]s.
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| 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 (mathematics)|invariant]] [[bilinear form]]s, which can be used to [[Embedding|embed]] the spin groups into [[classical Lie group]]s. 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.
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| ==Set up==
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| Let ''V'' be a [[dimension (vector space)|finite dimensional]] real or complex [[vector space]] with a [[nondegenerate form|nondegenerate]] [[quadratic form]] ''Q''. The (real or complex) [[linear map]]s preserving ''Q'' form the [[orthogonal group]] O(''V'',''Q''). The identity component of the group is called the special orthogonal group SO(''V'',''Q''). (For ''V'' 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]], SO(''V'',''Q'') has a unique [[connected space|connected]] [[Double covering group|double cover]], the spin group Spin(''V'',''Q''). There is thus a [[group homomorphism]] Spin(''V'',''Q'') → SO(''V'',''Q'') whose [[kernel (group theory)|kernel]] has two elements denoted {1, −1}, where 1 is the [[identity element]].
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| O(''V'',''Q''), SO(''V'',''Q'') and Spin(''V'',''Q'') are all [[Lie groups]], and for fixed (''V'',''Q'') they have the same [[Lie algebra]], '''so'''(''V'',''Q''). If ''V'' is real, then ''V'' is a real vector subspace of its [[complexification]] ''V''<sub>'''C'''</sub> := ''V'' ⊗<sub>'''R'''</sub> '''C''', and the quadratic form ''Q'' extends naturally to a quadratic form ''Q''<sub>'''C'''</sub> on ''V''<sub>'''C'''</sub>. This embeds SO(''V'',''Q'') as a [[subgroup]] of SO(''V''<sub>'''C'''</sub>, ''Q''<sub>'''C'''</sub>), and hence we may realise Spin(''V'',''Q'') as a subgroup of Spin(''V''<sub>'''C'''</sub>, ''Q''<sub>'''C'''</sub>). Furthermore, '''so'''(''V''<sub>'''C'''</sub>, ''Q''<sub>'''C'''</sub>) is the complexification of '''so'''(''V'',''Q'').
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| In the complex case, quadratic forms are determined up to isomorphism by the dimension ''n'' of ''V''. Concretely, we may assume ''V''='''C'''<sup>''n''</sup> and
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| :<math>Q(z_1,\ldots z_n) = z_1^2+ z_2^2+\cdots+z_n^2.</math> | |
| The corresponding Lie groups and Lie algebra are denoted O(''n'','''C'''), SO(''n'','''C'''), Spin(''n'','''C''') and '''so'''(''n'','''C''').
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| In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers (''p'',''q'') where ''n'':=''p''+''q'' is the dimension of ''V'', and ''p''-''q'' is the [[Sylvester's law of inertia|signature]]. Concretely, we may assume ''V''='''R'''<sup>''n''</sup> and
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| :<math>q(x_1,\ldots x_n) = x_1^2+ x_2^2+\cdots+x_p^2-(x_{p+1}^2+\cdots +x_n^2).</math>
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| The corresponding Lie groups and Lie algebra are denoted O(''p'',''q''), SO(''p'',''q''), Spin(''p'',''q'') and '''so'''(''p'',''q''). We write '''R'''<sup>''p'',''q''</sup> in place of '''R'''<sup>''n''</sup> to make the signature explicit.
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| The spin representations are, in a sense, the simplest [[representation of a Lie group|representation]]s of Spin(''n'','''C''') and Spin(''p'',''q'') that do not come from representations of SO(''n'','''C''') and SO(''p'',''q''). A spin representation is, therefore, a real or complex vector space ''S'' together with a group homomorphism ''ρ'' from Spin(''n'','''C''') or Spin(''p'',''q'') to the [[general linear group]] GL(''S'') such that the element −1 is ''not'' in the kernel of ''ρ''.
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| If ''S'' 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 '''so'''(''n'',''C'') or '''so'''(''p'',''q'') to the Lie algebra '''gl'''(''S'') of [[linear map#Endomorphisms_and_automorphisms|endomorphisms]] of ''S'' with the [[commutator#Ring theory|commutator bracket]].
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| Spin representations can be analysed according to the following strategy: if ''S'' is a real spin representation of Spin(''p'',''q''), then its complexification is a complex spin representation of Spin(''p'',''q''); as a representation of '''so'''(''p'',''q''), it therefore extends to a complex representation of '''so'''(''n'','''C'''). Proceeding in reverse, we therefore ''first'' construct complex spin representations of Spin(''n'','''C''') and '''so'''(''n'','''C'''), then restrict them to complex spin representations of '''so'''(''p'',''q'') and Spin(''p'',''q''), then finally analyse possible reductions to real spin representations.
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| ==Complex spin representations==
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| Let ''V''='''C'''<sup>''n''</sup> with the standard quadratic form ''Q'' so that
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| :<math>\mathfrak{so}(V,Q) = \mathfrak{so}(n,\mathbb C).</math>
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| The [[symmetric bilinear form]] on ''V'' associated to ''Q'' by [[Polarization identity#Symmetric bilinear forms|polarization]] is denoted <.,.>.
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| ===Isotropic subspaces and root systems===
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| A standard construction of the spin representations of '''so'''(''n'','''C''') begins with a choice of a pair (''W'', ''W''<sup>∗</sup>)
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| of maximal [[isotropic subspace]]s of ''V'' with ''W'' ∩ ''W''<sup>∗</sup> = 0. Let us make such a choice. If ''n'' = 2''m'' or ''n'' = 2''m''+1, then ''W'' and ''W''<sup>∗</sup> both have dimension ''m''. If ''n'' = 2''m'', then ''V'' = ''W'' ⊕ ''W''<sup>∗</sup>, whereas if ''n'' = 2''m''+1, then ''V'' = ''W'' ⊕ ''U'' ⊕ ''W''<sup>∗</sup>, where ''U'' is the 1-dimensional orthogonal complement to ''W'' ⊕ ''W''<sup>∗</sup>. The bilinear form <.,.> induces a [[bilinear map|pairing]] between ''W'' and ''W''<sup>∗</sup>, which must be nondegenerate, because ''W'' and ''W''<sup>∗</sup> are [[isotropic subspace]]s and ''Q'' is nondegenerate. Hence ''W'' and ''W''<sup>∗</sup> are [[dual vector space]]s.
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| More concretely, let ''a''<sub>1</sub>, ... ''a''<sub>''m''</sub> be a basis for ''W''. Then there is a unique basis ''α''<sub>1</sub>, ... ''α''<sub>''m''</sub> of ''W''<sup>∗</sup> such that
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| :<math> \langle \alpha_i,a_j\rangle = \delta_{ij}.</math>
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| If ''A'' is a ''m'' × ''m'' matrix, then ''A'' induces a endomorphism of ''W'' with respect to this basis and the transpose ''A''<sup>T</sup> induces a transformation of ''W''<sup>∗</sup> with
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| :<math> \langle Aw, w^* \rangle = \langle w,A^T w^*\rangle</math> | |
| for all ''w'' in ''W'' and ''w''* in ''W''<sup>∗</sup>. It follows that the endomorphism ''ρ''<sub>''A''</sub> of ''V'', equal to ''A'' on ''W'', − ''A''<sup>T</sup> on ''W''<sup>∗</sup> and zero on ''U'' (if ''n'' is odd), is skew
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| :<math> \langle \rho_A v, w \rangle = -\langle v,\rho_A w\rangle</math>
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| and hence an element of '''so'''(''n'','''C''').
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| Using the diagonal matrices in this construction defines a [[Cartan subalgebra]] '''h''' of '''so'''(''n'','''C'''): the [[rank of a Lie group|rank]] of '''so'''(''n'','''C''') is ''m'', and the diagonal ''m'' × ''m'' matrices determine an ''m''-dimensional abelian subalgebra.
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| Let ''ε''<sub>1</sub>, ... ''ε''<sub>''m''</sub> be the basis of '''h'''<sup>∗</sup> such that, for a diagonal matrix ''A'', ''ε''<sub>''k''</sub>(''ρ''<sub>''A''</sub>) is the ''k''th diagonal entry of ''A''. Clearly this is a basis for '''h'''<sup>∗</sup>. Since the bilinear form identifies '''so'''(''n'','''C''') with <math>\wedge^2 V</math>, it is now easy to construct the [[root system]] associated to '''h'''. The [[root space]]s (simultaneous eigenspaces for the action of '''h''') are spanned by the following elements:
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| :<math> a_i\wedge a_j,\; i\neq j,</math> with [[root system|root]] (simultaneous eigenvalue) <math>\varepsilon_i + \varepsilon_j</math>
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| :<math> a_i\wedge \alpha_j</math> (which is in '''h''' if ''i'' = ''j'') with root <math> \varepsilon_i - \varepsilon_j</math>
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| :<math> \alpha_i\wedge \alpha_j,\; i\neq j,</math> with root <math> -\varepsilon_i - \varepsilon_j,</math>
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| and, if ''n'' is odd, and ''u'' is a nonzero element of ''U'',
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| :<math> a_i\wedge u,</math> with root <math> \varepsilon_i </math>
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| :<math> \alpha_i\wedge u,</math> with root <math> -\varepsilon_i.</math>
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| Thus, with respect to the basis ''ε''<sub>1</sub>, ... ''ε''<sub>''m''</sub>, the roots are the vectors in '''h'''<sup>∗</sup> that are permutations of
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| :<math>(\pm 1,\pm 1, 0, 0, \dots, 0)</math>
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| together with the permutations of
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| :<math>(\pm 1, 0, 0, \dots, 0)</math>
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| if ''n'' = 2''m''+1 is odd.
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| A system of [[positive root]]s is given by ''ε''<sub>''i''</sub>+''ε''<sub>''j''</sub> (''i''≠''j''), ''ε''<sub>''i''</sub>−''ε''<sub>''j''</sub> (''i''<''j'') and (for ''n'' odd) ''ε''<sub>''i''</sub>. The corresponding [[Simple root (root system)|simple root]]s are
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| :<math>\varepsilon_1-\varepsilon_2, \varepsilon_2-\varepsilon_3, \ldots, \varepsilon_{m-1}-\varepsilon_m, \left\{\begin{matrix}
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| \varepsilon_{m-1}+\varepsilon_m& n=2m\\
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| \varepsilon_m & n=2m+1.
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| \end{matrix}\right.</math>
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| The positive roots are nonnegative integer linear combinations of the simple roots.
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| ===Spin representations and their weights===
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| One construction of the spin representations of '''so'''(''n'','''C''') uses the [[exterior algebra]](s)
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| :<math>S=\wedge^\bullet W</math> and/or <math>S'=\wedge^\bullet W^*.</math>
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| There is an action of ''V'' on ''S'' such that for any element ''v'' = ''w''+''w''* in ''W'' ⊕ ''W''<sup>∗</sup> and any ''ψ'' in ''S'' the action is given by:
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| :<math> v\cdot \psi = 2^{\frac{1}{2}}(w\wedge\psi+\iota(w^*)\psi), </math>
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| where the second term is a contraction ([[interior multiplication]]) defined using the bilinear form, which pairs ''W'' and ''W''<sup>∗</sup>. This action respects the [[Clifford relation]]s ''v''<sup>2</sup> = ''Q''(''v'')'''1''', and so induces a homomorphism from the [[Clifford algebra]] Cl<sub>''n''</sub>'''C''' of ''V'' to End(''S''). A similar action can be defined on ''S''′, so that both ''S'' and ''S''′ are [[Clifford module]]s.
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| The Lie algebra '''so'''(''n'','''C''') is isomorphic to the complexified Lie algebra '''spin'''<sub>''n''</sub><sup>'''C'''</sup> in Cl<sub>''n''</sub>'''C''' via the mapping induced by the covering Spin(''n'') → SO(''n'')
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| :<math> v \wedge w \mapsto \tfrac14[v,w].</math>
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| It follows that both ''S'' and ''S''′ are representations of '''so'''(''n'','''C'''). They are actually [[isomorphism|equivalent]] representations, so we focus on ''S''.
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| The explicit description shows that the elements ''α''<sub>''i''</sub>∧''a''<sub>''i''</sub> of the Cartan subalgebra '''h''' act on ''S'' by
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| :<math> (\alpha_i\wedge a_i) \cdot \psi = \tfrac14 (2^{\tfrac12})^{2} ( \iota(\alpha_i)(a_i\wedge\psi)-a_i\wedge(\iota(\alpha_i)\psi))
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| = \tfrac12 \psi - a_i\wedge(\iota(\alpha_i)\psi).</math>
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| A basis for ''S'' is given by elements of the form
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| :<math> a_{i_1}\wedge a_{i_2}\wedge\cdots\wedge a_{i_k}</math>
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| for 0 ≤ ''k'' ≤ ''m'' and ''i''<sub>1</sub> < ... < ''i''<sub>''k''</sub>. These clearly span [[weight space]]s for the action of '''h''': ''α''<sub>''i''</sub>∧''a''<sub>''i''</sub> has eigenvalue -1/2 on the given basis vector if ''i'' = ''i''<sub>''j''</sub> for some ''j'', and has eigenvalue 1/2 otherwise.
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| It follows that the [[weight (representation theory)|weights]] of ''S'' are all possible combinations of
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| :<math>\bigl(\pm \tfrac12,\pm \tfrac12, \ldots \pm\tfrac12\bigr)</math>
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| and each [[weight space]] is one dimensional. Elements of ''S'' are called [[Dirac spinor]]s.
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| When ''n'' is even, ''S'' is not an [[irreducible representation]]: <math>S_+=\wedge^{\mathrm{even}} W</math> and <math>S_-=\wedge^{\mathrm{odd}} W</math> are invariant subspaces. The weights divide into those with an even number of minus signs, and those with an odd number of minus signs. Both ''S''<sub>+</sub> and ''S''<sub>−</sub> are irreducible representations of dimension 2<sup>''m''−1</sup> whose elements are called [[Weyl spinor]]s. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, the [[highest weight]]s of ''S''<sub>+</sub> and ''S''<sub>−</sub> are
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| :<math>\bigl(\tfrac12,\tfrac12, \ldots\tfrac12, \tfrac12\bigr)</math> and <math>\bigl(\tfrac12,\tfrac12, \ldots\tfrac12, -\tfrac12\bigr)</math>
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| respectively. The Clifford action identifies Cl<sub>''n''</sub>'''C''' with End(''S'') and the [[Clifford algebra#Grading|even subalgebra]] is identified with the endomorphisms preserving ''S''<sub>+</sub> and ''S''<sub>−</sub>. The other [[Clifford module]] ''S''′ is [[isomorphism|isomorphic]] to ''S'' in this case.
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| When ''n'' is odd, ''S'' is an irreducible representation of '''so'''(''n'','''C''') of dimension 2<sup>''m''</sup>: the Clifford action of a unit vector ''u'' ∈ ''U'' is given by
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| :<math> u\cdot \psi = \left\{\begin{matrix}
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| \psi&\hbox{if } \psi\in \wedge^{\mathrm{even}} W\\
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| -\psi&\hbox{if } \psi\in \wedge^{\mathrm{odd}} W | |
| \end{matrix}\right.</math>
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| and so elements of '''so'''(''n'','''C''') of the form ''u''∧''w'' or ''u''∧''w''* do not preserve the even and odd parts of the exterior algebra of ''W''. The highest weight of ''S'' is
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| :<math>\bigl(\tfrac12,\tfrac12, \ldots \tfrac12\bigr).</math>
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| The Clifford action is not faithful on ''S'': Cl<sub>''n''</sub>'''C''' can be identified with End(''S'') ⊕ End(''S''′), where ''u'' acts with the opposite sign on ''S''′. More precisely, the two representations are related by the [[parity (mathematics)|parity]] [[involution (mathematics)|involution]] ''α'' of Cl<sub>''n''</sub>'''C''' (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of Cl<sub>''n''</sub>'''C'''. In other words, there is a [[linear isomorphism]] from ''S'' to ''S''′, which identifies the action of ''A'' in Cl<sub>''n''</sub>'''C''' on ''S'' with the action of ''α''(''A'') on ''S''′.
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| ===Bilinear forms===
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| if ''λ'' is a weight of ''S'', so is −''λ''. It follows that ''S'' is isomorphic to the [[dual representation]] ''S''<sup>∗</sup>.
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| When ''n'' = 2''m''+1 is odd, the isomorphism ''B'': ''S'' → ''S''<sup>∗</sup> is unique up to scale by [[Schur's lemma]], since ''S'' is irreducible, and it defines a nondegenerate invariant bilinear form ''β'' on ''S'' via
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| :<math>\beta(\varphi,\psi) = B(\varphi)(\psi).</math>
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| Here invariance means that
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| : <math>\beta(\xi\cdot\varphi,\psi) + \beta(\varphi,\xi\cdot\psi) = 0</math>
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| for all ''ξ'' in '''so'''(''n'','''C''') and ''φ'', ''ψ'' in ''S'' — in other words the action of ''ξ'' is skew with respect to ''β''. In fact, more is true: ''S''<sup>∗</sup> is a representation of the [[opposite category|opposite]] Clifford algebra, and therefore, since Cl<sub>''n''</sub>'''C''' only has two nontrivial [[simple module]]s ''S'' and ''S''′, related by the parity involution ''α'', there is an [[antiautomorphism]] ''τ'' of Cl<sub>''n''</sub>'''C''' such that
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| : <math>\quad\beta(A\cdot\varphi,\psi) = \beta(\varphi,\tau(A)\cdot\psi)\qquad (1)</math>
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| for any ''A'' in Cl<sub>''n''</sub>'''C'''. In fact ''τ'' is reversion (the antiautomorphism induced by the identity on ''V'') for ''m'' even, and conjugation (the antiautomorphism induced by minus the identity on ''V'') for ''m'' odd. These two antiautomorphisms are related by parity involution ''α'', which is the automorphism induced by minus the identity on ''V''. Both satisfy ''τ''(''ξ'') = −''ξ'' for ''ξ'' in '''so'''(''n'','''C''').
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| When ''n'' = 2''m'', the situation depends more sensitively upon the parity of ''m''. For ''m'' even, a weight ''λ'' has an even number of minus signs if and only if −''λ'' does; it follows that there are separate isomorphisms ''B''<sub>±</sub>: ''S''<sub>±</sub> → ''S''<sub>±</sub><sup>∗</sup> of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphism ''B'': ''S'' → ''S''<sup>∗</sup>. For ''m'' odd, ''λ'' is a weight of ''S''<sub>+</sub> if and only if −''λ'' is a weight of ''S''<sub>−</sub>; thus there is an isomorphism from ''S''<sub>+</sub> to ''S''<sub>−</sub><sup>∗</sup>, again unique up to scale, and its [[Dual space#Transpose of a linear map|transpose]] provides an isomorphism from ''S''<sub>−</sub> to ''S''<sub>+</sub><sup>∗</sup>. These may again be combined into an isomorphism ''B'': ''S'' → ''S''<sup>∗</sup>.
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| For both ''m'' even and ''m'' odd, the freedom in the choice of ''B'' may be restricted to an overall scale by insisting that the bilinear form ''β'' corresponding to ''B'' satisfies (1), where ''τ'' is a fixed antiautomorphism (either reversion or conjugation).
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| ===Symmetry and the tensor square===
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| The symmetry properties of ''β'': ''S'' ⊗ ''S'' → '''C''' can be determined using Clifford algebras or representation theory. In fact much more can be said: the tensor square ''S'' ⊗ ''S'' must decompose into a direct sum of ''k''-forms on ''V'' for various ''k'', because its weights are all elements in '''h'''<sup>∗</sup> whose components belong to {−1,0,1}. Now [[equivariant]] linear maps ''S'' ⊗ ''S'' → ∧<sup>''k''</sup>''V''<sup>∗</sup> correspond bijectively to invariant maps ∧<sup>''k''</sup>''V'' ⊗ ''S'' ⊗ ''S'' → '''C''' and nonzero such maps can be constructed via the inclusion of ∧<sup>''k''</sup>''V'' into the Clifford algebra. Furthermore if ''β''(''φ'',''ψ'') = ''ε'' ''β''(''ψ'',''φ'') and ''τ'' has sign ''ε''<sub>''k''</sub> on ∧<sup>''k''</sup>''V'' then
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| :<math>\beta(A\cdot\varphi,\psi) = \varepsilon\varepsilon_k \beta(A\cdot\psi,\varphi)</math>
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| for ''A'' in ∧<sup>''k''</sup>''V''.
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| If ''n'' = 2''m''+1 is odd then it follows from Schur's Lemma that
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| :<math> S\otimes S \cong \bigoplus_{j=0}^{m} \wedge^{2j} V^*</math>
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| (both sides have dimension 2<sup>2''m''</sup> and the representations on the right are inequivalent). Because the symmetries are governed by an involution ''τ'' that is either conjugation or reversion, the symmetry of the ∧<sup>''2j''</sup>''V''<sup>∗</sup> component alternates with ''j''. Elementary combinatorics gives
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| :<math> \sum_{j=0}^m (-1)^j \dim \wedge^{2j} \C^{2m+1} = (-1)^{\frac12 m(m+1)} 2^m = (-1)^{\frac12 m(m+1)}(\dim \mathrm S^2S-\dim \wedge^2 S)</math>
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| and the sign determines which representations occur in S<sup>2</sup>''S'' and which occur in ∧<sup>2</sup>''S''.<ref>This sign can also be determined from the observation that if ''φ'' is a highest weight vector for ''S'' then ''φ''⊗''φ'' is a highest weight vector for ∧<sup>''m''</sup>''V'' ≅ ∧<sup>''m''+1</sup>''V'', so this summand must occur in S<sup>2</sup>''S''.</ref> In particular
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| :<math> \beta(\phi,\psi)=(-1)^{\frac12 m(m+1)}\beta(\psi,\phi),</math> and | |
| :<math> \beta(v\cdot\phi,\psi) = (-1)^m(-1)^{\frac12 m(m+1)}\beta(v\cdot\psi,\phi) = (-1)^m \beta(\phi,v\cdot\psi)</math>
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| for ''v'' ∈ ''V'' (which is isomorphic to ∧<sup>2''m''</sup>''V''), confirming that ''τ'' is reversion for ''m'' even, and conjugation for ''m'' odd.
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| If ''n''=2''m'' is even, then the analysis is more involved, but the result is a more refined decomposition: S<sup>''2</sup>''S''<sub>±</sub>, ∧<sup>''2</sup>''S''<sub>±</sub> and ''S''<sub>+</sub> ⊗ ''S''<sub>−</sub> can each be decomposed as a direct sum of ''k''-forms (where for ''k''=''m'' there is a further decomposition into selfdual and antiselfdual ''m''-forms).
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| The main outcome is a realisation of '''so'''(''n'','''C''') as a subalgebra of a classical Lie algebra on ''S'', depending upon ''n'' modulo 8, according to the following table:
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| {| class="wikitable"
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| |- style="text-align:center"
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| ! ''n'' mod 8
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| | 0
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| | 1
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| | 2
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| | 3
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| | 4
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| | 5
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| | 6
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| | 7
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| |-
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| ! Spinor algebra
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| | <math> \mathfrak{so}(S_+)\oplus\mathfrak{so}(S_-) </math>
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| | <math> \mathfrak{so}(S) </math>
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| | <math> \mathfrak{gl}(S_{\pm}) </math>
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| | <math> \mathfrak{sp}(S) </math>
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| | <math> \mathfrak{sp}(S_+)\oplus\mathfrak{sp}(S_-) </math>
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| | <math> \mathfrak{sp}(S) </math>
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| | <math> \mathfrak{gl}(S_{\pm}) </math>
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| | <math> \mathfrak{so}(S) </math>
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| |}
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| For ''n'' ≤ 6, these embeddings are isomorphisms (onto '''sl''' rather than '''gl''' for ''n''=6):
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| :<math> \mathfrak{so}(2,\mathbb C) \cong \mathfrak{gl}(1,\mathbb C)\qquad(=\mathbb C)</math>
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| :<math> \mathfrak{so}(3,\mathbb C) \cong \mathfrak{sp}(2,\mathbb C)\qquad(=\mathfrak{sl}(2,\mathbb C))</math>
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| :<math> \mathfrak{so}(4,\mathbb C) \cong \mathfrak{sp}(2,\mathbb C)\oplus\mathfrak{sp}(2,\mathbb C)</math>
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| :<math> \mathfrak{so}(5,\mathbb C) \cong \mathfrak{sp}(4,\mathbb C)</math>
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| :<math> \mathfrak{so}(6,\mathbb C) \cong \mathfrak{sl}(4,\mathbb C).</math>
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| | |
| ==Real representations==
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| | |
| The complex spin representations of '''so'''(''n'','''C''') yield real representations ''S'' of '''so'''(''p'',''q'') by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types.
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| # There is an invariant complex antilinear map ''r'': ''S'' → ''S'' with ''r''<sup>2</sup> = id<sub>''S''</sub>. The fixed point set of ''r'' is then a real vector subspace ''S''<sub>'''R'''</sub> of ''S'' with ''S''<sub>'''R'''</sub> ⊗ '''C''' = ''S''. This is called a '''real structure'''.
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| # There is an invariant complex antilinear map ''j'': ''S'' → ''S'' with ''j''<sup>2</sup> = −id<sub>''S''</sub>. It follows that the triple ''i'', ''j'' and ''k'':=''ij'' make ''S'' into a quaternionic vector space ''S''<sub>'''H'''</sub>. This is called a '''quaternionic structure'''.
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| # There is an invariant complex antilinear map ''b'': ''S'' → ''S''<sup>∗</sup> that is invertible. This defines a hermitian bilinear form on ''S'' and is called a '''hermitian structure'''.
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| | |
| The type of structure invariant under '''so'''(''p'',''q'') depends only on the signature ''p''−''q'' modulo 8, and is given by the following table.
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| {| class="wikitable"
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| |- style="text-align:center"
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| ! ''p''−''q'' mod 8
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| | 0
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| | 1
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| | 2
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| | 3
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| | 4
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| | 5
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| | 6
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| | 7
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| |-
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| ! Structure
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| | '''R''' + '''R'''
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| | '''R'''
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| | '''C'''
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| | '''H'''
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| | '''H''' + '''H'''
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| | '''H'''
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| | '''C'''
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| | '''R'''
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| |}
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| | |
| Here '''R''', '''C''' and '''H''' denote real, hermitian and quaternionic structures respectively, and '''R'''+'''R''' and '''H'''+'''H''' indicate that the half-spin representations both admit real or quaternionic structures respectively.
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| | |
| ===Description and tables===
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| To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms. Since ''n'' = ''p''+''q'' ≅ ''p'' - ''q'' mod 2, there are two cases: the dimension and signature are both even, and the dimension and signature are both odd.
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| | |
| The odd case is simpler, there is only one complex spin representation ''S'', and hermitian structures do not occur. Apart from the trivial case ''n'' = 1, ''S'' is always even dimensional, say dim ''S'' = 2''N''. The real forms of '''so'''(2''N'','''C''') are '''so'''(''K'',''L'') with ''K'' + ''L'' = 2''N'' and '''so'''*(''N'','''H'''), while the real forms of '''sp'''(2''N'','''C''') are '''sp'''(2''N'','''R''') and '''sp'''(''K'',''L'') with ''K'' + ''L'' = ''N''. The presence of a Clifford action of ''V'' on ''S'' forces ''K'' = ''L'' in both cases unless ''pq'' = 0, in which case ''KL''=0, which is denoted simply '''so'''(2''N'') or '''sp'''(''N''). Hence the odd spin representations may be summarized in the following table.
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| {| class="wikitable" style="text-align:center"
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| |-
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| !
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| ! ''n'' mod 8
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| ! 1, 7
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| ! 3, 5
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| |-
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| ! ''p''-''q'' mod 8
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| !
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| ! '''so'''(2''N'','''C''')
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| ! '''sp'''(2''N'','''C''')
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| |-
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| ! 1, 7
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| ! '''R'''
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| | '''so'''(''N'',''N'') or '''so'''(2''N'')
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| | '''sp'''(2''N'','''R''')
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| |-
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| ! 3, 5
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| ! '''H'''
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| | '''so'''*(''N'','''H''')
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| | '''sp'''(''N''/2,''N''/2)<sup>†</sup> or '''sp'''(''N'')
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| |}
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| (†) ''N'' is even for ''n''>3 and for ''n''=3, this is '''sp'''(1).
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| | |
| The even dimensional case is similar. For ''n''>2, the complex half-spin representations are even dimensional. We have additionally to deal with hermitian structures and the real forms of '''sl'''(2''N'','''C'''), which are '''sl'''(2''N'','''R'''), '''su'''(''K'',''L'') with ''K'' + ''L'' = 2''N'', and '''sl'''(''N'','''H'''). The resulting even spin representations are summarized as follows.
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| {| class="wikitable" style="text-align:center"
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| |-
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| !
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| ! ''n'' mod 8
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| ! 0
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| ! 2, 6
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| ! 4
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| |-
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| ! ''p''-''q'' mod 8
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| !
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| ! '''so'''(2''N'','''C''')+'''so'''(2''N'','''C''')
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| ! '''sl'''(2''N'','''C''')
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| ! '''sp'''(2''N'','''C''')+'''sp'''(2''N'','''C''')
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| |-
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| ! 0
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| ! '''R'''+'''R'''
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| | '''so'''(''N'',''N'')+'''so'''(''N'',''N'')<sup>*</sup>
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| | '''sl'''(2''N'','''R''')
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| | '''sp'''(2''N'','''R''')+'''sp'''(2''N'','''R''')
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| |-
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| ! 2, 6
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| ! '''C'''
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| | '''so'''(2''N'','''C''')
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| | '''su'''(''N'',''N'')
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| | '''sp'''(2''N'','''C''')
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| |-
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| ! 4
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| ! '''H'''+'''H'''
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| | '''so'''*(''N'','''H''')+'''so'''*(''N'','''H''')
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| | '''sl'''(''N'','''H''')
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| | '''sp'''(''N''/2,''N''/2)+'''sp'''(''N''/2,''N''/2)<sup>†</sup>
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| |}
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| (*) For ''pq''=0, we have instead '''so'''(2''N'')+'''so'''(2''N'')
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| | |
| (†) ''N'' is even for ''n''>4 and for ''pq''=0 (which includes ''n''=4 with ''N''=1), we have instead '''sp'''(''N'')+'''sp'''(''N'')
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| | |
| The low dimensional isomorphisms in the complex case have the following real forms.
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| {| class="wikitable"
| |
| |- style="text-align:center"
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| | Euclidean signature
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| | Minkowskian signature
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| | colspan=2 | Other signatures
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| |-
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| | <math>\mathfrak{so}(2)\cong \mathfrak{u}(1)</math>
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| | <math>\mathfrak{so}(1,1)\cong \mathbb R</math>
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| | colspan=2 |
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| |-
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| | <math>\mathfrak{so}(3)\cong \mathfrak{sp}(1)</math>
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| | <math>\mathfrak{so}(2,1)\cong \mathfrak{sl}(2,\mathbb R)</math>
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| | colspan=2 |
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| |-
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| | <math>\mathfrak{so}(4)\cong \mathfrak{sp}(1)\oplus\mathfrak{sp}(1)</math>
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| | <math>\mathfrak{so}(3,1)\cong \mathfrak{sl}(2,\mathbb C)</math>
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| | <math>\mathfrak{so}(2,2)\cong \mathfrak{sl}(2,\mathbb R)\oplus\mathfrak{sl}(2,\mathbb R)</math>
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| |
| |
| |-
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| | <math>\mathfrak{so}(5)\cong \mathfrak{sp}(2)</math>
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| | <math>\mathfrak{so}(4,1)\cong \mathfrak{sp}(1,1)</math>
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| | <math>\mathfrak{so}(3,2)\cong \mathfrak{sp}(4,\mathbb R)</math>
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| |-
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| | <math>\mathfrak{so}(6)\cong \mathfrak{su}(4)</math>
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| | <math>\mathfrak{so}(5,1)\cong \mathfrak{sl}(2,\mathbb H)</math>
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| | <math>\mathfrak{so}(4,2)\cong \mathfrak{su}(2,2)</math>
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| | <math>\mathfrak{so}(3,3)\cong \mathfrak{sl}(4,\mathbb R)</math>
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| |}
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| The only special isomorphism of real Lie algebras missing from this table is
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| <math>\mathfrak{so}^*(3,\mathbb H) \cong \mathfrak{su}(3,1).</math>
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| | |
| ==Notes==
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| {{reflist}}
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| | |
| ==References==
| |
| * {{citation|last1=Brauer|first1=Richard|author1-link=Richard Brauer|last2=Weyl|first2= Hermann|author2-link=Hermann Weyl|title=Spinors in n dimensions|journal= American Journal of Mathematics|volume=57|issue= 2|year=1935|pages= 425–449|doi=10.2307/2371218|jstor=2371218|publisher=American Journal of Mathematics, Vol. 57, No. 2}}.
| |
| * {{citation|last=Cartan|first=Élie|authorlink=Élie Cartan|year=1966|title=The theory of spinors|publisher = Paris, Hermann (reprinted 1981, Dover Publications)| isbn= 978-0-486-64070-9}}.
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| * {{citation|last= Chevalley|first= Claude|authorlink=Claude Chevalley|title=The algebraic theory of spinors and Clifford algebras|publisher=Columbia University Press (reprinted 1996, Springer)|year=1954|isbn=978-3-540-57063-9}}.
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| * {{citation|first=Pierre|last=Deligne|authorlink=Pierre Deligne|chapter=Notes on spinors|title= Quantum Fields and Strings: A Course for Mathematicians|editor= P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, E. Witten|publisher=American Mathematical Society|place= Providence|year=1999|pages=99–135}}. See also [http://www.math.ias.edu/QFT the programme website] for a preliminary version.
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| * {{citation| last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | last2=Harris | first2=Joe | author2-link=Joe Harris (mathematician) | title=Representation theory. A first course | publisher=[[Springer-Verlag]] | location=New York | series=[[Graduate Texts in Mathematics]], Readings in Mathematics| isbn=0-387-97495-4 | mr=1153249 | year=1991 | volume=129 }}.
| |
| * {{citation|title=Spinors and Calibrations|last=Harvey|first= F. Reese|authorlink=F. Reese Harvey|publisher=Academic Press|year=1990|isbn=978-0-12-329650-4}}.
| |
| * {{citation|last1=Lawson|first1= H. Blaine|author1-link=H. Blaine Lawson|last2=Michelsohn|first2=Marie-Louise|author2-link=Marie-Louise Michelsohn|title=Spin Geometry|publisher= Princeton University Press|year=1989|isbn= 0-691-08542-0}}.
| |
| * {{citation|title=The Classical Groups: Their Invariants and Representations|first=Hermann|last= Weyl|authorlink=Hermann Weyl|year=1946|edition=2nd|publisher = Princeton University Press (reprinted 1997)| isbn= 978-0-691-05756-9}}.
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| [[Category:Spinors]]
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| [[Category:Representation theory of Lie groups]]
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