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1
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1answer
124 views

Equation for non-invertible elements in Clifford algebras

Suppose we have a Clifford algebra $Cl(V,q)$, $V\simeq \mathbb{R}^n$ and $q$ non-degenerate bilinear form. Then every non-zero element of $V\subset Cl(V,q)$ invertible, but they are not the only ones ...
10
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0answers
192 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 ...
23
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2answers
965 views

What are the “correct” conventions for defining Clifford algebras?

I have three related questions about conventions for defining Clifford algebras. 1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...
2
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0answers
87 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 ...
10
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2answers
136 views

Which real Pin groups agree?

In the Lie theory notes on my website it is claimed (Example 7.3.3.5) that $\mathrm{Pin}(4,0)$ and $\mathrm{Pin}(0,4)$ are not isomorphic. As Nigel Ray pointed out to me, this claim is not quite ...
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0answers
30 views

How can the Cayley-table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$ and $e_1=i$ and $e_2=j$ and so on. I'm looking for an ...
1
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0answers
55 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 ...
4
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0answers
117 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 ...
8
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2answers
208 views

Clifford algebras for quadratic modules over ringed spaces

What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ...
4
votes
1answer
376 views

Mathematica package for supergravity and string theory

I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
0
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2answers
191 views

Matrix algebra as Clifford algebra

Many kinds of Clifford algebras have corresponding sub-algebras of matrix algebras in sense of isomorphism. Say, quaternion, spacetime algebra and also Dirac algebra. Generally, Clifford algebra has ...
2
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0answers
129 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
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0answers
143 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 ...
16
votes
2answers
568 views

Conceptual explanation for the relationship between Clifford algebras and KO

Recall the following table of Clifford algebras: $$\begin{array}{ccc} n & Cl_n & M_n/i^{*}M_{n+1}\\ 1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\ 2 & \mathbb{H} & ...
6
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2answers
398 views

Weitzenböck Identity for $\Delta_{\bar{\partial}_E}$

This question is related to this MO question and this MSE question. Let $E$ be a hermitian holomorphic vector bundle over a hermitian manifold $X$. The bundle $\bigwedge^{\bullet,\bullet}X\otimes ...
14
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2answers
592 views

Karoubi versus Kasparov K-theory

I have the following, probably very elementary question: Let $Cl^{p,q}$ be the Clifford algebra on generators $e_i$, $i=1, \ldots, p+q$ with $e_i e_j = -e_j e_i$ and $e_{i}^{2}=-1$ for $i=1,\ldots,p$, ...
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0answers
132 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" ...
3
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1answer
206 views

What are the invariant definitions of spinorial quantities from mathematical physics?

When physicists write expressions involving spinors $\psi \in S \otimes V$, where $S=S_+ \oplus S_-$ is a complex spinor representation of a spin group $Spin(2d)$ and $V$ is a complex representation ...
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0answers
84 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 $f(v)f(v) = Q(v)$, $f$ ...
0
votes
1answer
313 views

Applications of the natural bilinear forms on the direct sum between a vector space and its dual

As is known, the vector space $V\oplus V^\ast$ admits the natural symmetric and skew-symmetric bilinear forms $$\langle X+\xi,Y+\eta\rangle|_\pm:=\frac 1 2 (\xi(Y) \pm \eta(X)).$$ I am interested in ...
1
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0answers
77 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 ...
4
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2answers
654 views

Clifford Lie Algebras

I'm studying the "Clifford Lie Algebra" (see http://arxiv.org/pdf/1007.2481.pdf page 30). It's basically a way to look at Clifford algebras and their properties in a Lie algebraic setting (which I ...
0
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1answer
370 views

Is the metaplectic group not a matrix group - counterexample

Is the statement below false? "The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite-dimensional representations." Possible "counterexample": Sp(2n,R) is a subgroup of ...
0
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1answer
179 views

2Pi and 4Pi rotations in the Pin(1,3) group

Hi everyone, I'm currently studying the construction of the $Pin(1,3)$ group and given the definition I'm using to find its elements I'm having some problems with the signs associated with $2\pi$ and ...
2
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0answers
122 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 ...
10
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0answers
298 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. ...
2
votes
0answers
178 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
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0answers
234 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 ...
5
votes
1answer
310 views

Spin and SO groups associated to a degenerate symmetric bilinear form

In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But the rest of the book ...
13
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0answers
647 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) = ...
1
vote
1answer
183 views

Three dimensional subalgebras of Clifford Algebras

Are there any three dimensional subalgebras of GA(n) where GA(n) is the geometric algebra corresponding to $R^n$? If yes, what about for GA(2)? Edit: a geometric algebra is a Clifford algebra.
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0answers
855 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
1answer
909 views

Koszul duality between Weyl and Clifford algebras?

Koszul duality Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can ...
8
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6answers
1k views

Clifford algebra non-zero

This should be a very easy question, but the proof in Lawson/Michelson (Spin geometry) is wrong and I do not find a really correct and complete argument: Let V be a nonzero real vector space with ...
3
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1answer
627 views

On matrix representations of the Clifford algebras of type $Cl(0,n)$

Can matrix representations of clifford algebras of type Cl(0,n) be found? Specifically for even orders
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6answers
3k views

Pythagorean 5-tuples

What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")? There are simple formulas describing Pythagorean n-tuples for n=3,4,6: n=3. The formula ...
12
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2answers
766 views

Clifford PBW theorem for quadratic form

Update: now with a question 2 which is much more elementary (and should be well-known!). Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:L\to k$ be a quadratic form, i. e., a ...
1
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2answers
191 views

Invariant space of lifted Chevalley automorphisms of the tensor algebra

Question. Let $k$ be a field of characteristic $0$. Let $L$ be a $k$-vector space. Consider the subspace $S$ of $L\otimes L\otimes L\otimes L$ spanned by all tensors of the form $\left[a,\left\lbrace ...
0
votes
1answer
756 views

Clifford Algebra and Gamma matrices: is this relation generally true for any dimension?

I expect the following relation to be vanishing. But it seems not that obvious. $\Gamma_{ab}^{\lambda}t^at^b \Gamma_{\lambda c(d)}t^c=0$ where $t^a$ are even ghosts, "$ab$" are indices for matrix ...
6
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5answers
989 views

Representations of Pin vs. Representations of Clifford

This may be total nonsense. But I need to know the answer quickly and I am too tired to think about it thoroughly. Let $k$ be a positive integer. Roe's "Elliptic Operators" claims that there is a ...
16
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3answers
1k views

How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best ...
7
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3answers
2k views

Clifford Algebra in Dirac Equation

I am wondering if there is any mathematical (or physical, besides the fact that classical quantum mechanics uses complex numbers) justification for why the complexified (1,3) Clifford algebra is used ...
12
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7answers
2k views

Why is Riemann-Roch an Index Problem?

I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..." Then right after that he argued in favour of such a sentence. ...
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6answers
2k views

Clifford algebra as an adjunction?

Background For definiteness (even though this is a categorical question!) let's agree that a vector space is a finite-dimensional real vector space and that an associative algebra is a ...