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19
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0answers
947 views

Why do Clifford algebras determine $KO$ (and $K$-)-theory?

In the paper "Clifford modules" by Atiyah-Bott-Shapiro, they construct a family of Clifford algebras $C_k$ over the real numbers, so that $C_k$ is the algebra associated to a negative definite form on ...
13
votes
0answers
689 views

What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = ...
11
votes
0answers
232 views

What is the symmetric monoidal functor from Clifford algebras to invertible K-module spectra?

There ought to be a symmetric monoidal functor from the symmetric monoidal $2$-groupoid whose objects are Morita-invertible real superalgebras (precisely the Clifford algebras), morphisms are ...
10
votes
0answers
316 views

Examples of Clifford Modules

For a Riemannian manifold $M$, a Clifford module is a bundle over $M$ that is, fiberwise, a representation of the Clifford algebra bundle $Cl(TM)$ and has a connection compatible with this action. ...
4
votes
0answers
81 views

Finitely generated projective modules over the algebra of sections of the Clifford bundle

Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...
4
votes
0answers
126 views

Element in spin group

I've got the following question: why is it true (if it really is?), that if I have a unitary element $u$ in the (real) Clifford algebra $Cl(V,g)$ which is even and the operator $\varphi(u)$ defined ...
2
votes
0answers
64 views

$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here. Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and ...
2
votes
0answers
139 views

Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...
2
votes
0answers
99 views

Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$. Q1) Is it ...
2
votes
0answers
152 views

What is the Atiyah-Bott-Shapiro map for a bundle of *complex* quadratic forms?

In order to ask the question in the title more precisely, let me recall some standard stuff introduced in [1; Atiyah, Bott, Shapiro]. Suppose $X$ is a compact CW complex and $V \to X$ is an oriented ...
2
votes
0answers
146 views

Discriminants of Clifford algebras

I have a Clifford algebra defined over a field of characteristic not equal to 2. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...
2
votes
0answers
124 views

Clifford algebra is graded separable

Let $D$ be an algebra of odd differential operators on a free module $V$, this algebra is isomorphic to the Clifford algebra $Cl(V^* \oplus V)$. Let $m$ denote multiplication map $$m : D\otimes D \to ...
2
votes
0answers
193 views

fast multipole method and geometric algebra

Hello, I just learned about fast multipole method(FMM) from this article http://math.nyu.edu/faculty/greengar/shortcourse_fmm.pdf and I really liked the use of complex numbers in 2d. But I didn't ...
1
vote
0answers
97 views

Generalization of De Rham cohomology for spinor fields

Is there a generalization of De Rham cohomology for spinors fields? I can see that one can construct p form fields out of spinor field by contraction of the type $\bar{\psi} \gamma^{a_1} ...
1
vote
0answers
54 views

Convention on Clifford Product

When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign: $vv=Q(v)$ (see, for instance, Wikipedia) $vv=-Q(v)$ (see, for instance, MathWorld ...
1
vote
0answers
110 views

Norm of the operator acting on spinor bundle

Please forgive me if the question is too elementary, but however I was unable to manage by myself. The question comes from J.Varilly, H.Figueroa and J. Gracia-Bondia book "Elements of noncommutative ...
1
vote
0answers
82 views

Homomorphism of algebra of “Clifford-valued continuous functions”

I am interested in the general question of When is a map between algebra of "Clifford-valued continuous functions" homomorphism? As a starter, I would like to first understand the case for ...
1
vote
0answers
239 views

Properties of Clifford Algebras

It is a well known fact that Clifford algebras, $Cl(p,q)$, have similar properties depending on $(p-q)\mod 8$. In most of the places I have found a proof of the theorem, explicit representations of ...
0
votes
0answers
134 views

Need help determining whether a certain map is a $C^\ast$ homomorphism

Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" ...
0
votes
0answers
93 views

General criterion for homomorphism between Clifford Algebras

I understand that there is a universal property which tells me that given a Clifford algebra $Cl(V, q)$, for a linear map $f: V \to A$ ($A$ any associative algebra) satisfying $f(v)f(v) = Q(v)$, $f$ ...