Spin group
Encyclopedia
In mathematics
the spin group Spin(n) is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie group
s
As a Lie group Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra
with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).
The non-trivial element of the kernel is denoted
, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted 
.
Spin(n) can be constructed as a subgroup
of the invertible elements in the Clifford algebra
Cℓ(n).
s among the classical Lie groups called accidental isomorphisms. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups, due to low dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Specifically, we have
There are certain vestiges of these isomorphisms left over for n = 7,8 (see Spin(8) for more details). For higher n, these isomorphisms disappear entirely.
Note that Spin(p,q) = Spin(q,p).
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 of G, there is an inclusion
with Z(G′) the centre
of G′. This inclusion and the Lie algebra
of G determine G entirely (note that it is not the fact that 
and 
determine G entirely; for instance SL(2,R) and PSL(2,R) have the same Lie algebra and same fundamental group 
, but are not isomorphic).
The definite signature Spin(n) are all simply connected for (n>2), so they are the universal coverings for SO(n).
In indefinite signature, Spin(p,q) is not connected, and in general the identity component
, 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
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
This allows us to calculate the fundamental groups of Spin(p,q), taking
:
Thus once
the fundamental group is 
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 p=2, q>2, this map is given by 
. And finally, for p=q=2, 
is sent to 
and 
is sent to 
.
s 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 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 {±1} yields the special orthogonal group – if the center equals {±1} (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,
which are the three compact real forms (or two, if SO=PSO) of the compact Lie algebra
The homotopy group
s 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 
and 
may differ.
For Spin(n) with
Spin(n) is simply connected (
is trivial), so SO(n) is connected and has fundamental group 
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 Spin(p,q).
s).
Given the double cover
by the lattice theorem
, 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
on subgroups of Spin(n) is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as binary polyhedral groups.
Concretely, every binary point group is either the preimage of a point group (hence denoted
for the point group 
), 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 
(since 
is central). As an example of these latter, given a cyclic group of odd order 
in SO(n), its preimage is a cyclic group of twice the order, 
and the subgroup 
maps isomorphically to 
Of particular note are two series:
For point groups that reverse orientation, the situation is more complicated, as there are two pin group
s, so there are two possible binary groups corresponding to a given point group.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
the spin group Spin(n) is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s
As a Lie group Spin(n) therefore shares its dimension, n(n − 1)/2, and its Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
with the special orthogonal group. For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n).
The non-trivial element of the kernel is denoted
, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted .Spin(n) can be constructed as a subgroup
Subgroup
In group theory, 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 G if the restriction of * to H x H is a group operation on H...
of the invertible elements in the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
Cℓ(n).
Accidental isomorphisms
In low dimensions, there are isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s among the classical Lie groups called accidental isomorphisms. For instance, there are isomorphisms between low dimensional spin groups and certain classical Lie groups, due to low dimensional isomorphisms between the root systems (and corresponding isomorphisms of Dynkin diagrams) of the different families of simple Lie algebras. Specifically, we have
- Spin(1) = O(1)Orthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
- Spin(2) = U(1)Unitary groupIn mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
= SO(2) which acts on
by double phase rotation 
→
- Spin(3) = Sp(1)Symplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
= SU(2)Special unitary groupThe special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
, corresponding to
- Spin(4) = Sp(1)Symplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
× Sp(1)Symplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, corresponding to
- Spin(5) = Sp(2)Symplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, corresponding to
- Spin(6) = SU(4)Special unitary groupThe special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
, corresponding to
There are certain vestiges of these isomorphisms left over for n = 7,8 (see Spin(8) for more details). For higher n, these isomorphisms disappear entirely.
Indefinite signature
In indefinite signature, the spin group Spin(p,q) is constructed through Clifford algebras in a similar way to standard spin groups. It is a connected double cover of SO0(p,q), the connected component of the identity of the indefinite orthogonal group SO(p,q) (there are a variety of conventions on the connectedness of Spin(p,q); in this article, it is taken to be connected for p+q>2). As in definite signature, there are some accidental isomorphisms in low dimensions:- Spin(1,1) = GL(1,R)General linear groupIn mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
- Spin(2,1) = SL(2,R)
- Spin(3,1) = SL(2,C)Special linear groupIn mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
- Spin(2,2) = SL(2,R) × SL(2,R)
- Spin(4,1) = Sp(1,1)Symplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
- Spin(3,2) = Sp(4,R)Symplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
- Spin(5,1) = SL(2,H)Special linear groupIn mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
- Spin(4,2) = SU(2,2)Special unitary groupThe special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
- Spin(3,3) = SL(4,R)Special linear groupIn mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
Note that Spin(p,q) = Spin(q,p).
Topological considerations
ConnectedConnected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
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 of G, there is an inclusion
with Z(G′) the centre
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
of G′. This inclusion and the Lie algebra
of G determine G entirely (note that it is not the fact that and determine G entirely; for instance SL(2,R) and PSL(2,R) have the same Lie algebra and same fundamental group , but are not isomorphic).The definite signature Spin(n) are all simply connected for (n>2), so they are the universal coverings for SO(n).
In indefinite signature, Spin(p,q) is not connected, and in general the identity component
Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
, 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
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....
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
This allows us to calculate the fundamental groups of Spin(p,q), taking
:Thus once
the fundamental group is 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 p=2, q>2, this map is given by . And finally, for p=q=2, is sent to and is sent to .Center
The center of the spin groups (complex and real) are given as follows:Quotient groups
Quotient groupQuotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
s 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 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 {±1} yields the special orthogonal group – if the center equals {±1} (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),
which are the three compact real forms (or two, if SO=PSO) of the compact Lie algebra
Compact Lie algebra
In the mathematical field of Lie theory, a Lie algebra is compact if it is the Lie algebra of a compact Lie group. Intrinsically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite, though this definition does not quite agree with the previous...
The homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s 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 and may differ.For Spin(n) with
Spin(n) is simply connected ( is trivial), so SO(n) is connected and has fundamental group 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 Spin(p,q).Discrete subgroups
Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational point groupPoint group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
s).
Given the double cover
by the lattice theoremLattice theorem
In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G such that A contains N, onto the set of all subgroups...
, there is a Galois connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
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
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
on subgroups of Spin(n) is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as binary polyhedral groups.
Concretely, every binary point group is either the preimage of a point group (hence denoted
for the point group ), 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 (since is central). As an example of these latter, given a cyclic group of odd order in SO(n), its preimage is a cyclic group of twice the order, and the subgroup maps isomorphically to Of particular note are two series:
- higher binary tetrahedral groups, 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 groupCovering groups of the alternating and symmetric groupsIn the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups...
,
with the alternating group being the (rotational) symmetry group of the n-simplex.
- This group can also be considered as the double cover of the symmetric group
- higher binary octahedral groups, corresponding to the 2-fold covers of the hyperoctahedral groupHyperoctahedral groupIn 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. Groups of this type are identified by a parameter n, the dimension of the hypercube....
(symmetries of the hypercubeHypercubeIn geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
, or equivalently of its dual, the cross-polytopeCross-polytopeIn geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...
).
For point groups that reverse orientation, the situation is more complicated, as there are two pin group
Pin group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....
s, so there are two possible binary groups corresponding to a given point group.
See also
- Clifford algebraClifford algebraIn mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
- Clifford analysisClifford analysisClifford 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...
- SpinorSpinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
- Spinor bundle
- Spin structureSpin structureIn 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....
- AnyonAnyonIn physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
- Orientation entanglementOrientation entanglementIn mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected....
- Complex Spin Group
Related groups
- Pin groupPin groupIn mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....
Pin(n) – two-fold cover of orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
, O(n) - Metaplectic groupMetaplectic groupIn mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers...
Mp(2n) – two-fold cover of symplectic groupSymplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, Sp(2n)