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Bott Periodicity Flavors

Lie group embeddings

{$O(16r)\supset U(8r)\supset Sp(4r)\supset Sp(2r)\times Sp(2r) \supset Sp(2r) \supset U(2r) \supset O(2r) \supset O(r)\times O(r) \supset O(r)$}

{$U(2r)\supset U(r)\times U(r)\supset U(r)$}

Symmetric spaces

There are 10=3+7 infinite series of compact Riemannian symmetric spaces according to Cartan's classification. Three are {$O(n)\times O(n)/O(n)$}, {$U(n)\times U(n)/U(n)$}, {$Sp(n)\times Sp(n)/Sp(n)$}. Seven are quotients {$G/K$}. Six of these are found in real Bott periodicity and {$SU(p+q)/SU(p)\times SU(q)$} in complex Bott periodicity. In Wikipedia the "special" forms are given, without reflection, whereas in Stone-Chiu-Roy they are given in the general form. Perhaps they are the same if the reflections in the quotients cancel out.

A *-algebra is a real associative algebra {$A$} with unit {$1\in A$} and operation {$*:A\rightarrow A$} (like complex conjugation or quaternionic conjugation) with {$a^{**}=a$}, {$(a+b)^*=a^*+b^*$}, {$(\alpha a)^*=\alpha a^*)$}, {$(ab)^*=b*a*$} for all {$a,b\in A$}, {$\alpha\in\mathbb{R}$}.

John Baez forgetful functor {$F:\mathbf{Rep}(Cl_{0,n+1})\rightarrow\mathbf{Rep}(Cl_{0,n})$} restricts a *-representation of {$Cl_{0,n+1}$} to a *-representation of {$Cl_{0,n}$}.

{$F^{-1}(H)$} is a disjoint sum of compact symmetric spaces. Is this the adjoint functor, the free functor?

From this point of view, the most basic are the split real Hilbert spaces {$\mathbf{Rep}(Cl_{0,7})$} from which {$F^{-1}$} takes us by direct sum? to real Hilbert spaces {$\mathbf{Rep}(Cl_{0,6})$}, then by complexification to complex Hilbert spaces {$\mathbf{Rep}(Cl_{0,5})$}, by quaterniofication to quaternionic Hilbert spaces {$\mathbf{Rep}(Cl_{0,4})$}, by direct sum? to split quaternionic Hilbert spaces {$\mathbf{Rep}(Cl_{0,3})$}, at the opposite end. Then it takes us back by doubling? to quaternionic Hilbert spaces {$\mathbf{Rep}(Cl_{0,2})$}, then to the underlying complex Hilbert spaces {$\mathbf{Rep}(Cl_{0,1})$}, then to the underlying real Hilbert spaces {$\mathbf{Rep}(Cl_{0,0})$}, and then by doubling? to the split real Hilbert spaces {$\mathbf{Rep}(Cl_{0,7})$}.

Representations of Clifford algebras

The distinction is made between irreducible representations of {$Cl_{0,n-1}$} that can be extended to irreducible representations of {$Cl_{0,n}$} and those that can't. This can be done by way of symmetric spaces as quotients or by way of the adjoint to Baez's forgetful functor. I think this distinguishes those that involve a reflection and those that don't.

Given the inclusion {$i:Cl_{0,k}\rightarrow Cl_{0,k+1}$}, we see that the representations of the larger Clifford algebra are also representations of the smaller Clifford algebra, and so we can define {$i^*:N(C_{k+1})\rightarrow N(C_k)$} as a map on the Grothendieck groups for the monoids of the representations of the Clifford algebras. We can then compare and calculate the quotient group {$N(Cl_k)/i^*(N(Cl_{k+1}))=A_k$}. Crucially, not every representation of {$Cl_{0,k}$} may be gotten from a representation of {$Cl_{0,k+1}$}. A representation that was irreducible in {$Cl_{0,k+1}$} may be reducible in the context of {$Cl_{0,k}$}. This means that not every representation of {$Cl_{0,k}$} is extendible to a representation of {$Cl_{0,k+1}$}.

In particular, there are two cases. One is where a {$2\times 2$} matrix encoding a complex number gets sent to two {$1\times 1$} matrices encoding real numbers. Similarly, with the quaternions and the complex numbers. This yields the group {$\mathbb{Z}_2$}. The other case is where a representation of {$2\times 2$} matrices of real numbers gets sent to the representation of a direct sum of {$1\times 1$} matrices of real numbers. And similarly with the quaternions. This yields the group {$\mathbb{Z}$}.

Spinor algebras and Spin representations

The basic properties of real spin representations repeat with period 8, and of complex spin representations repeat with period 2.

{$\textrm{Spin}(n,0)$} is the unique connected double cover of {$SO(n)$}. In this case, real spin representations are the simplest representations of {$\textrm{Spin}(n,0)$} that do not come from representations of {$SO(n)$}. A real spin representation is a real vector space {$S$} together with a group homomorphism {$\rho:\textrm{Spin}(n,0)\rightarrow GL(S)$} such that {$-1$} is not in the kernel of {$\rho$}.

Spinor algebras are the complex Lie algebras {$\frak{so}$}{$(n,\mathbb{C})=\frak{o}$}{$(n,\mathbb{C})=\frak{spin}$}{$(n,\mathbb{C})$}. These are subalgebras of the following classical Lie algebras on the Clifford module {$S$} which depend on {$n \mod 8$}. The elements of {$S$} are the Dirac spinors.

For {$n\leq 5$} these are not just embeddings but isomorphisms.

The dimension of {$\frak{so}$}{$(n,\mathbb{C})$} grows as {$n(n-1)/2$}. Whereas with the Clifford algebras, the number of generators grows as {$n$} and the number of basis elements grows as {$2^n$}.

Structures invariant under the action of the real Lie algebras

Restricting the action of the complex spin representations of {$\mathbf{so}(n,\mathbb{C})$} to the real subalgebras yields real representations {$S$} of {$\mathbf{so}(n,0)$}.

The action of the real Lie algebras yields the following kinds of invariant structures:

• Real structure R: Invariant complex antilinear map {$r:S\rightarrow S$} with {$r^2=id_S$}. The fixed point set of {$r$} is a real vector space {$S_R$} of {$S$} with {$S_R\otimes\mathbb{C}=S$}.
• Quaternionic structure H: Invariant complex antilinear map {$j:S\rightarrow S$} with {$j^2=-id_S$}. The triple {$i, j, ij=k$} make {$S$} into a quaternionic vector space {$S_H$}.
• Hermitian structure C: Invariant complex antilinear map {$b:S\rightarrow S^*$} that is invertible. This defines a pseudohermitian bilinear form on {$S$}.

The structure invariant depends on {$n\mod 8$} as follows:

CPT symmetry

Wigner showed that, in quantum mechanics, a symmetry operation {$S$} of a Hamiltonian is represented either by a unitary operator {$S=U$} or an antiunitary operator {$S=UK$} where {$K$} is complex conjugation. These are the only operations that preserve the length of the projection of a state-vector onto another state-vector. (See: T-symmetry)

• Parity reverses direction of position {$x$} and direction of momentum {$p$}, thus must be unitary.
• Time does not reverse the direction of position, but does reverse the direction of momentum, thus must be anti-unitary.

Topological invariants

Super division algebras

Super division algebras explain how a Clifford algebra splits into an even and odd part distinguished by a generator and related by an automorphism.

• Todd Trimble. The Super Brauer Group and Super Division Algebras. Classifies real super division algebras and relates them to the super Brauer group and to Clifford algebras. There are two {$\mathbb{R}$}-automorphisms from {$\mathbb{C}$} to {$\mathbb{C}$}. The identity automorphism generates the super Brauer group over {$C$}. The conjugate automorphism generates the super Brauer group over {$R$}.
  • Brauer-Wall group Super Brauer Group
  • Brauer group

Clifford algebras

{$\textrm{Cl}_{0,0}=\mathbb{R}$}, {$\textrm{Cl}_{0,1}=\mathbb{C}$}, {$\textrm{Cl}_{0,2}=\mathbb{H}$}, {$\textrm{Cl}_{0,3}=\mathbb{H}\oplus\mathbb{H}$}, {$\textrm{Cl}_{0,4}=M_2(\mathbb{H})$}, {$\textrm{Cl}_{0,5}=M_4(\mathbb{C})$}, {$\textrm{Cl}_{0,6}=M_8(\mathbb{R})$}, {$\textrm{Cl}_{0,7}=M_8(\mathbb{R})\oplus M_8(\mathbb{R})$}, {$\textrm{Cl}_{0,8}=M_{16}(\mathbb{R})$}

{$\mathbf{Cl}_0=\mathbb{C}$}, {$\mathbf{Cl}_1=\mathbb{C}\oplus\mathbb{C}$}

Clifford modules

Linearly independent vector fields

Vector fields on spheres, John Baez. Octonions, divison algebras, Bott periodicity. {$\textrm{Cl}_{0,n}$} has a representation on a {$k$}-dimensional real vector space if and only if the unit sphere in that vector space, {$S_{k-1}$}, admits {$n$} linearly independent smooth vector fields.

• Unit real numbers {$S_0$} admit {$0$} linearly independent vector fields. Thus {$\textrm{Cl}_{0,0}$} has a representation on {$1$}-dimensional real space.
• Unit complex numbers {$S_1$} admit {$1$} linearly independent vector fields. Thus {$\textrm{Cl}_{0,1}$} has a representation on {$2$}-dimensional real space.
• Unit quaternions {$S_3$} admit {$3$} linearly independent vector fields. Thus {$\textrm{Cl}_{0,2}$}, {$\textrm{Cl}_{0,3}$} have representations on {$4$}-dimensional real space.
• Unit octonions {$S_7$} admit {$7$} linearly independent vector fields. Thus {$\textrm{Cl}_{0,4}$}, {$\textrm{Cl}_{0,5}$}, {$\textrm{Cl}_{0,6}$}, {$\textrm{Cl}_{0,7}$}, have representations on {$8$}-dimensional real space.

Octonionic line bundles

John Baez. OP1 and Bott Periodicity. Canonical line bundles {$L_\mathbb{R}$}, {$L_\mathbb{C}$}, {$L_\mathbb{H}$}, {$L_\mathbb{O}$} give elements {$ [L_\mathbb{R}]$}, {$ [L_\mathbb{C}]$}, {$ [L_\mathbb{H}]$}, {$ [L_\mathbb{O}]$} that generate, respectively, {$\widetilde{KO}(S^1)\cong\mathbb{Z}_2$},{$\widetilde{KO}(S^2)\cong\mathbb{Z}_2$},{$\widetilde{KO}(S^4)\cong\mathbb{Z}$},{$\widetilde{KO}(S^8)\cong\mathbb{Z}$}. The isomorphism {$\widetilde{KO}(S^n)\rightarrow \widetilde{KO}(S^{n+8})$} given by {$x\rightarrow [L_\mathbb{O}]$} yields Bott periodicity. The canonical octonionic line bundle over {$\mathbb{O}P^1$} generates Bott periodicity.

Loop spaces and classifying spaces

Loop space functor {$\Omega$} is right adjoint to suspension {$\Sigma$} and left adjoint to classifying space construction.

The loop spaces in Bott periodicity are homotopy equivalent to the symmetric spaces of successive quotients, with additional discrete factors of {$\mathbb{Z}$}.

The Grassmannian {$BU$} of n-planes in an infinite-dimensional complex Hilbert space.

{$\Omega^2BU\cong\mathbb{Z}\times BU$} or equivalently {$\Omega^2U=U$}

The Grassmannian {$BO$} of n-planes in an infinite-dimensional real space {$\mathbb {R}^{\infty}$}

{$\Omega^8BO\cong\mathbb{Z}\times BO$} or equivalently {$\Omega^8O=O$}

Using {$\pi_0$}, {$O$} and {$O/U$} have two components, {$KO = BO × \mathbb{Z}$} and {$KSp = BSp × \mathbb{Z}$} have countably many components, and the rest are connected.

• {$\pi_0(KO) = \mathbb{Z}$} is about dimension
• {$\pi_1(KO) = \mathbb{Z}/2\mathbb{Z}$} is about orientation
• {$\pi_2(KO) = \mathbb{Z}/2\mathbb{Z}$} is about spin
• {$\pi_4(KO) = \mathbb{Z}$} is about topological quantum field theory.

Homotopy groups of {$O(\infty)$}d

These can be derived from the Clifford algebra periodicity by way of Dirac operators {$D$}, where we have {$\sigma_D(\epsilon):S^k\to GL(n,\mathbb{R})$} and {$\pi_k(GL(n,\mathbb{R}))=\pi_k(O(n))$}.

• {$\pi_0(O(\infty))=\mathbb{Z}_2$}
• {$\pi_1(O(\infty))=\mathbb{Z}_2$}
• {$\pi_2(O(\infty))=0$}
• {$\pi_3(O(\infty))=\mathbb{Z}$}
• {$\pi_4(O(\infty))=0$}
• {$\pi_5(O(\infty))=0$}
• {$\pi_6(O(\infty))=0$}
• {$\pi_7(O(\infty))=\mathbb{Z}_2$}
• {$\pi_{n+8}(O(\infty))=\pi_n(O(\infty))$}
• {$\pi_0(U(\infty))=0$}
• {$\pi_1(U(\infty))=\mathbb{Z}$}
• {$\pi_{n+2}(U(\infty))=\pi_n(U(\infty))$}

Homotopy groups of {$GL_\mathbb{R}$}

nLab: Bott periodicity

• {$\pi_n(GL_\mathbb{R})=\pi_n(O(\infty)$}
• {$\pi_n(GL_\mathbb{C})=\pi_n(U(\infty)$}

Stable homotopy groups of spheres

This is seen from the J-isomorphism of homotopy groups {$J:\pi_k(SO(n))\rightarrow\pi_{n+k}(S_n)$}. When {$n\geq k+2$}, {$\pi_k(SO(n))$} only depend on {$k\;(\textrm{mod}\;8)$}.

Equivariant K-theory

KO-theory of the point

{$R_\mathbb{R}(G)≃KO^0_G(*)$}

K-theory of the integers

See: Quillen–Lichtenbaum_conjecture