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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} ...
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1answer
75 views

Cayley Subspaces in a Calibrated 8-Space

Suppose we are given $(\mathbb{R}^8,\Phi)$, where $\Phi$ is the self-dual 4-form that defines $Spin(7)\subset SO(8)$ (Cayley calibration, see Notes on the Octonians, page 23). Now some 4-subspaces $V$ ...
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79 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 ...
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1answer
306 views

Automorphisms of Clifford Algebras

What are the automorphisms of real Clifford algebras $Cl_{n,0}$? Of course, I'm interested in the case where they are not central simple.
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0answers
53 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 ...
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1answer
117 views

$Spin(7)$ as stabilizer of a $4$-form revisited

For a better understanding of this question, please see the question and answer here. In $Spin(8)$ there are plenty of copies of $Spin(7)$; consider, for instance, the antiimage of $SO(7)<SO(8)$ ...
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0answers
137 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 ...
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1answer
147 views

Isomorphisms of Positive and Negative Spinor Bundles

Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf, 1998): 2.6 Spin Bundles and the Dirac Operator ...
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1answer
231 views

Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...
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2answers
798 views

Exact Definition of Dirac Operator

Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ...
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1answer
147 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 ...
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231 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 ...
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2answers
1k 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 ...
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0answers
98 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 ...
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2answers
149 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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1answer
55 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 ...
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0answers
109 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 ...
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125 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 ...
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2answers
223 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
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1answer
452 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 ...
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2answers
221 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 ...
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0answers
151 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 ...
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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 ...
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2answers
651 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} & ...
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2answers
454 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 ...
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2answers
662 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
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" ...
3
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1answer
207 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
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 $f(v)f(v) = Q(v)$, $f$ ...
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1answer
399 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 ...
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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 ...
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2answers
742 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 ...
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1answer
403 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 ...
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1answer
201 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 ...
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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 ...
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312 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. ...
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192 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 ...
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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 ...
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1answer
316 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 ...
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0answers
685 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) = ...
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1answer
186 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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938 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
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1answer
1k 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 ...
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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 ...
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1answer
670 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 ...
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2answers
781 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 ...
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2answers
194 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 ...
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1answer
777 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 ...
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5answers
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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 ...