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In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Lie group, Spin(''n'') therefore shares its
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
, , and its Lie algebra with the special orthogonal group. For , Spin(''n'') is simply connected and so coincides with the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of SO(''n''). The non-trivial element of the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −. Spin(''n'') can be constructed as a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the invertible elements in the Clifford algebra Cl(''n''). A distinct article discusses the
spin representation 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 two equ ...
s.


Motivation and physical interpretation

The spin group is used in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
. Strictly speaking, the spin group describes a fermion in a zero-dimensional space; but of course, space is not zero-dimensional, and so the spin group is used to define
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical ...
s on (pseudo-) Riemannian manifolds: the spin group is the structure group of a
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
. The
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
on a spinor bundle is the
spin connection In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz tr ...
; the spin connection is useful as it can simplify and bring elegance to many intricate calculations in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. The spin connection in turn enables the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
to be written in curved spacetime (effectively in the
tetrad Tetrad ('group of 4') or tetrade may refer to: * Tetrad (area), an area 2 km x 2 km square * Tetrad (astronomy), four total lunar eclipses within two years * Tetrad (chromosomal formation) * Tetrad (general relativity), or frame field ** Tetra ...
coordinates), which in turn provides a footing for quantum gravity, as well as a formalization of
Hawking radiation Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical a ...
(where one of a pair of entangled, virtual fermions falls past the event horizon, and the other does not). In short, the spin group is a vital cornerstone, centrally important for understanding advanced concepts in modern theoretical physics. In mathematics, the spin group is interesting in its own right: not only for these reasons, but for many more.


Construction

Construction of the Spin group often starts with the construction of a Clifford algebra over a real vector space ''V'' with a
definite quadratic form In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical de ...
''q''.Jürgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer Verlag ''(See Chapter 1.)'' The Clifford algebra is the quotient of the
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
T''V'' of ''V'' by a two-sided ideal. The tensor algebra (over the reals) may be written as :\mathrmV= \mathbb \oplus V \oplus (V\otimes V) \oplus \cdots The Clifford algebra Cl(''V'') is then the quotient algebra :\operatorname(V) = \mathrmV / \left( v \otimes v - q(v) \right) , where q(v) is the quadratic form applied to a vector v\in V. The resulting space is finite dimensional, naturally graded (as a vector space), and can be written as :\operatorname(V) = \operatorname^0 \oplus \operatorname^1 \oplus \operatorname^2 \oplus \cdots \oplus \operatorname^n where n is the dimension of V, \operatorname^0 = \mathbf and \operatorname^1 = V. The spin algebra \mathfrak is defined as :\operatorname^n =\mathfrak(V) = \mathfrak(n) , where the last is a short-hand for ''V'' being a real vector space of real dimension ''n''. It is a Lie algebra; it has a natural action on ''V'', and in this way can be shown to be isomorphic to the Lie algebra \mathfrak(n) of the special orthogonal group. The
pin group The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under ...
\operatorname(V) is a subgroup of \operatorname(V)'s Clifford group of all elements of the form :v_1 v_2 \cdots v_k , where each v_i\in V is of unit length: q(v_i) = 1. The spin group is then defined as :\operatorname(V) = \operatorname(V) \cap \operatorname^ , where \operatorname^\text=\operatorname^0 \oplus \operatorname^2 \oplus \operatorname^4 \oplus \cdots is the subspace generated by elements that are the product of an even number of vectors. That is, Spin(''V'') consists of all elements of Pin(''V''), given above, with the restriction to ''k'' being an even number. The restriction to the even subspace is key to the formation of two-component (Weyl) spinors, constructed below. If the set \ are an orthonormal basis of the (real) vector space ''V'', then the quotient above endows the space with a natural anti-commuting structure: :e_i e_j = -e_j e_i for i \ne j , which follows by considering v\otimes v for v=e_i+e_j. This anti-commutation turns out to be of importance in physics, as it captures the spirit of the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
for fermions. A precise formulation is out of scope, here, but it involves the creation of a
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
on
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
; the resulting spinor fields can be seen to be anti-commuting as a by-product of the Clifford algebra construction. This anti-commutation property is also key to the formulation of supersymmetry. The Clifford algebra and the spin group have many interesting and curious properties, some of which are listed below.


Double covering

For a quadratic space ''V'', a double covering of SO(''V'') by Spin(''V'') can be given explicitly, as follows. Let \ be an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
for ''V''. Define an
antiautomorphism In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From ...
t : \operatorname(V) \to \operatorname(V) by : \left(e_i e_j \cdots e_k\right)^t = e_k\cdots e_j e_i. This can be extended to all elements of a,b\in \operatorname(V) by linearity. It is an antihomomorphism since : (a b)^t = b^t a^t. Observe that Pin(''V'') can then be defined as all elements a \in \operatorname(V) for which :a a^t = 1. Now define the automorphism \alpha\colon \operatorname(V)\to\operatorname(V) which on degree 1 elements is given by :\alpha(v)=-v,\quad v\in V, and let a^* denote \alpha(a)^t, which is an antiautomorphism of Cl(''V''). With this notation, an explicit double covering is the homomorphism \operatorname(V)\to\operatorname O(V) given by :\rho(a) v = a v a^* , where v \in V. When ''a'' has degree 1 (i.e. a\in V), \rho(a) corresponds a reflection across the hyperplane orthogonal to ''a''; this follows from the anti-commuting property of the Clifford algebra. This gives a double covering of both O(''V'') by Pin(''V'') and of SO(''V'') by Spin(''V'') because a gives the same transformation as -a.


Spinor space

It is worth reviewing how spinor space and
Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
s are constructed, given this formalism. Given a real vector space ''V'' of dimension an even number, its
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
is V \otimes \mathbf. It can be written as the direct sum of a subspace W of spinors and a subspace \overline of anti-spinors: :V \otimes \mathbf = W \oplus \overline The space W is spanned by the spinors \eta_k = \left( e_ - ie_ \right) / \sqrt 2 for 1\le k\le m and the complex conjugate spinors span \overline. It is straightforward to see that the spinors anti-commute, and that the product of a spinor and anti-spinor is a scalar. The spinor space is defined as the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
\textstyle W. The (complexified) Clifford algebra acts naturally on this space; the (complexified) spin group corresponds to the length-preserving
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
s. There is a natural grading on the exterior algebra: the product of an odd number of copies of W correspond to the physics notion of fermions; the even subspace corresponds to the bosons. The representations of the action of the spin group on the spinor space can be built in a relatively straightforward fashion.


Complex case

The SpinC group is defined by the
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
:1 \to \mathrm_2 \to \operatorname^(n) \to \operatorname(n)\times \operatorname(1) \to 1. It is a multiplicative subgroup of the
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
\operatorname(V)\otimes \mathbf of the Clifford algebra, and specifically, it is the subgroup generated by Spin(''V'') and the unit circle in C. Alternately, it is the quotient :\operatorname^(V) = \left( \operatorname(V) \times S^1 \right) / \sim where the equivalence \sim identifies with . This has important applications in 4-manifold theory and Seiberg–Witten theory. In physics, the Spin group is appropriate for describing uncharged fermions, while the SpinC group is used to describe electrically charged fermions. In this case, the U(1) symmetry is specifically the
gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
.


Exceptional isomorphisms

In low dimensions, there are
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s among the classical Lie groups called ''
exceptional isomorphism In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
s''. For instance, there are isomorphisms between low-dimensional spin groups and certain classical Lie groups, owing to low-dimensional isomorphisms between the
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
s (and corresponding isomorphisms of
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s) of the different families of
simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of s ...
s. Writing R for the reals, C for the complex numbers, H for the quaternions and the general understanding that Cl(''n'') is a short-hand for Cl(R''n'') and that Spin(''n'') is a short-hand for Spin(R''n'') and so on, one then has that :Cleven(1) = R the real numbers :Pin(1) = :Spin(1) = O(1) =     the orthogonal group of dimension zero. -- :Cleven(2) = C the complex numbers :Spin(2) =
U(1) In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
= SO(2), which acts on ''z'' in R2 by double phase rotation .     dim = 1 -- :Cleven(3) = H the quaternions :Spin(3) = Sp(1) =
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
, corresponding to B_1 \cong A_1.     dim = 3 -- :Cleven(4) = H ⊕ H :Spin(4) = SU(2) × SU(2), corresponding to D_2 \cong A_1 \times A_1.     dim = 6 -- :Cleven(5)= M(2, H) the two-by-two matrices with quaternionic coefficients :Spin(5) = Sp(2), corresponding to B_2 \cong C_2.     dim = 10 -- :Cleven(6)= M(4, C) the four-by-four matrices with complex coefficients :Spin(6) = SU(4), corresponding to D_3 \cong A_3.     dim = 15 There are certain vestiges of these isomorphisms left over for (see
Spin(8) In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28. Spin(8) Like all special orthogonal groups of n > 2, SO(8) is n ...
for more details). For higher ''n'', these isomorphisms disappear entirely.


Indefinite signature

In indefinite signature, the spin group is constructed through Clifford algebras in a similar way to standard spin groups. It is a double cover of , the connected component of the identity of the
indefinite orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the p ...
. For , is connected; for there are two connected components. As in definite signature, there are some accidental isomorphisms in low dimensions: :Spin(1, 1) = GL(1, R) :Spin(2, 1) = SL(2, R) :Spin(3, 1) = SL(2, C) :Spin(2, 2) = SL(2, R) × SL(2, R) :Spin(4, 1) = Sp(1, 1) :Spin(3, 2) = Sp(4, R) :Spin(5, 1) = SL(2, H) :Spin(4, 2) = SU(2, 2) :Spin(3, 3) = SL(4, R) :Spin(6, 2) = SU(2, 2, H) Note that .


Topological considerations

Connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
and simply connected Lie groups are classified by their Lie algebra. So if ''G'' is a connected Lie group with a simple Lie algebra, with ''G''′ the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of ''G'', there is an inclusion : \pi_1 (G) \subset \operatorname(G'), with Z(''G''′) the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of ''G''′. This inclusion and the Lie algebra \mathfrak of ''G'' determine ''G'' entirely (note that it is not the case that \mathfrak and π1(''G'') determine ''G'' entirely; for instance SL(2, R) and PSL(2, R) have the same Lie algebra and same fundamental group Z, but are not isomorphic). The definite signature Spin(''n'') are all simply connected for ''n'' > 2, so they are the universal coverings of SO(''n''). In indefinite signature, Spin(''p'', ''q'') is not necessarily connected, and in general the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compo ...
, Spin0(''p'', ''q''), is not simply connected, thus it is not a universal cover. The fundamental group is most easily understood by considering the
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the class ...
of SO(''p'', ''q''), which is SO(''p'') × SO(''q''), and noting that rather than being the product of the 2-fold covers (hence a 4-fold cover), Spin(''p'', ''q'') is the "diagonal" 2-fold cover – it is a 2-fold quotient of the 4-fold cover. Explicitly, the maximal compact connected subgroup of Spin(''p'', ''q'') is :Spin(''p'') × Spin(''q'')/. This allows us to calculate the
fundamental groups In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of Spin(''p'', ''q''), taking ''p'' ≥ ''q'': :\pi_1(\mbox(p,q)) = \begin \mathrm_1 & (p,q)=(1,1) \mbox (1,0) \\ \mathrm_1 & p > 2, q = 0,1 \\ \mathbf & (p,q)=(2,0) \mbox (2,1) \\ \mathbf \times \mathbf & (p,q) = (2,2) \\ \mathbf & p > 2, q=2 \\ \mathrm_2 & p, q >2\\ \end Thus once the fundamental group is Z2, as it is a 2-fold quotient of a product of two universal covers. The maps on fundamental groups are given as follows. For , this implies that the map is given by going to . For , this map is given by . And finally, for , is sent to and is sent to .


Center

The center of the spin groups, for , (complex and real) are given as follows: :\begin \operatorname(\operatorname(n,\mathbf)) &= \begin \mathrm_2 & n = 2k+1\\ \mathrm_4 & n = 4k+2\\ \mathrm_2 \oplus \mathrm_2 & n = 4k\\ \end \\ \operatorname(\operatorname(p,q)) &= \begin \mathrm_2 & p \text q \text\\ \mathrm_4 & n = 4k+2, \text p, q \text\\ \mathrm_2 \oplus \mathrm_2 & n = 4k, \text p, q \text\\ \end \end


Quotient groups

Quotient groups can be obtained from a spin group by quotienting out by a subgroup of the center, with the spin group then being a
covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
of the resulting quotient, and both groups having the same Lie algebra. Quotienting out by the entire center yields the minimal such group, the projective special orthogonal group, which is centerless, while quotienting out by yields the special orthogonal group – if the center equals (namely in odd dimension), these two quotient groups agree. If the spin group is simply connected (as Spin(''n'') is for ), then Spin is the ''maximal'' group in the sequence, and one has a sequence of three groups, :Spin(''n'') → SO(''n'') → PSO(''n''), splitting by parity yields: :Spin(2''n'') → SO(2''n'') → PSO(2''n''), :Spin(2''n''+1) → SO(2''n''+1) = PSO(2''n''+1), which are the three compact real forms (or two, if ) of the
compact Lie algebra In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebr ...
\mathfrak (n, \mathbf). The homotopy groups of the cover and the quotient are related by the long exact sequence of a fibration, with discrete fiber (the fiber being the kernel) – thus all homotopy groups for are equal, but π0 and π1 may differ. For , Spin(''n'') is simply connected ( is trivial), so SO(''n'') is connected and has fundamental group Z2 while PSO(''n'') is connected and has fundamental group equal to the center of Spin(''n''). In indefinite signature the covers and homotopy groups are more complicated – Spin(''p'', ''q'') is not simply connected, and quotienting also affects connected components. The analysis is simpler if one considers the maximal (connected) compact and the component group of .


Whitehead tower

The spin group appears in a Whitehead tower anchored by the orthogonal group: :\ldots\rightarrow \text(n) \rightarrow \text(n)\rightarrow \text(n)\rightarrow \text(n) \rightarrow \text(n) The tower is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
s starting with an Eilenberg–MacLane space for the homotopy group to be removed. Killing the 3 homotopy group in Spin(''n''), one obtains the infinite-dimensional string group String(''n'').


Discrete subgroups

Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational point groups). Given the double cover , by the
lattice theorem In group theory, the correspondence theorem (also the lattice theorem,W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27. and variously and ambiguously the third and fourth isomorphism theorem ) states that if N is a normal subgroup of ...
, there is a Galois connection between subgroups of Spin(''n'') and subgroups of SO(''n'') (rotational point groups): the image of a subgroup of Spin(''n'') is a rotational point group, and the preimage of a point group is a subgroup of Spin(''n''), and the
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are de ...
on subgroups of Spin(''n'') is multiplication by . These may be called "binary point groups"; most familiar is the 3-dimensional case, known as
binary polyhedral group In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries ...
s. Concretely, every binary point group is either the preimage of a point group (hence denoted 2''G'', for the point group ''G''), or is an index 2 subgroup of the preimage of a point group which maps (isomorphically) onto the point group; in the latter case the full binary group is abstractly \mathrm_2 \times G (since is central). As an example of these latter, given a cyclic group of odd order \mathrm_ in SO(''n''), its preimage is a cyclic group of twice the order, \mathrm_ \cong \mathrm_ \times \mathrm_2, and the subgroup maps isomorphically to . Of particular note are two series: * higher
binary tetrahedral group In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of ...
s, corresponding to the 2-fold cover of symmetries of the ''n''-simplex; this group can also be considered as the double cover of the symmetric group, , with the alternating group being the (rotational) symmetry group of the ''n''-simplex. * higher
binary octahedral group In mathematics, the binary octahedral group, name as 2O or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group ''O'' or (2, ...
s, corresponding to the 2-fold covers of the
hyperoctahedral group In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a paramete ...
(symmetries of the hypercube, or equivalently of its dual, the cross-polytope). For point groups that reverse orientation, the situation is more complicated, as there are two
pin group The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under ...
s, so there are two possible binary groups corresponding to a given point group.


See also

* Clifford algebra *
Clifford analysis Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are ...
*
Spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
*
Spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
*
Spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical ...
* Table of Lie groups * Anyon * Orientation entanglement


Related groups

*
Pin group The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under ...
Pin(''n'') – two-fold cover of orthogonal group, O(''n'') * Metaplectic group Mp(2''n'') – two-fold cover of
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
, Sp(2''n'') * String group String(n) – the next group in the Whitehead tower


References


External links

* The essential dimension of spin groups is OEIS:A280191. * Grothendieck's "torsion index" is OEIS:A096336.


Further reading

* {{DEFAULTSORT:Spin Group Lie groups Topology of Lie groups Spinors