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Questions tagged [clifford-algebras]

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38 views

What's the moment of inertia of a swept n-sphere?

I live in $n$ dimensions. I have a uniformly dense $n$-capsule $S$ of radius $r$ and semi-length $x$ - that is, I took an $n$-ball of radius $r$ from $[-x, 0, ..., 0]$ and swept it through to $[x, 0, ....
-2
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0answers
49 views

Are Clifford Fractions (defined via left and right reductions) well defined? Are they algebraically closed?

Given a Clifford algebra, it seems to me that we can define 2 different algebras of fractions via either left reduction and then right reduction, or vise versa (right reduction and then left reduction)...
4
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0answers
74 views

What minimal structure is required to define Clifford modules in a way as abstract as possible?

Start with a quadratic form $q$ on a vector space $V$. A module $M$ over the corresponding Clifford algebra is determined by a map $\cdot:V\otimes M\to M$ satisfying $v\cdot(v\cdot m)=-q(v)m$. Now ...
9
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1answer
154 views

Representations of degenerate Clifford algebras

Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ...
5
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0answers
66 views

Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here: Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\...
2
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0answers
68 views

A generalization of the Clifford algebra

Let $(E,g)$ be a vector space with a symmetric bilinear form, and $a,b$ be two endomorphisms of $E$. The generalized Clifford algebra is defined by the free algebra of $E$ with quotient by the ...
11
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2answers
476 views

Clifford algebras as deformations of exterior algebras

$\def\Cl{\mathcal C\ell} \def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$ I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question. A well ...
3
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0answers
87 views

Orthogonality of Clifford algebra's Fueter polynomial in Gaussian measure

In the article "Two integral operators in Clifford analysis" , https://www.sciencedirect.com/science/article/pii/S0022247X08012262, it said that $\langle V_{\alpha},V_{\alpha'}\rangle = \int_{\...
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0answers
59 views

Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?

In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
2
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0answers
54 views

evolution of Grassmannians along geodesic line

Let $p_0$, $p_1$ be two $n \times 2$ orthonormal matrices that represent two points on the real $Gr_{2,n}$, i.e. two 2-d subspaces in $\mathbb{R}^n$. Let $p(t): [0,1] \rightarrow Gr_{2,n}$ be a ...
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0answers
111 views

Characteristic classess of Cliford bundle of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold. Let $E$ be the Cliford bundle associated to $TM$. Does the structure of $E$, as a vector bundle depend on choosing the Riemannian metric $g$? How can we write ...
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0answers
40 views

symmetric polynomials for Super Hecke Clifford algebra

Fix a natural number $n$. In https://arxiv.org/abs/1107.1039, §3.5, Kang/Kashiwara/Tsuchioka define a (version of a) Hecke Clifford superalgebra. It is the superalgebra with the following generators: ...
4
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2answers
216 views

How should we define $\mathrm{PSL}_2$ of a Clifford group?

UPDATE - Feb. 9, 2017: The original title of this post was "The $\text{isometry}^+$ group of hyperbolic $n$-space as $\mathrm{PSL}_2$ of a Clifford group." The original question, which appears below, ...
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3answers
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natural embedding $V \to Cl(V,q)$ [closed]

(cf. LAWSON and MICHELSOHN's book on Spin Geometry page 8) The book proves there is an natural embedding from a vector space $V$ to its Clifford algebra $Cl(V,q)$, where $q$ is a quadratic form on $V$...
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1answer
82 views

New Clifford structure

For an $n$-dimensional space $V$ with a positive metric $g$, we can construct the Clifford algebra $Cl(V)$ and its representation space $S$, i.e. $$c(V):S\to S,~\forall v\in V.$$ Question: Under ...
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143 views

Intuition for Clifford Group

Clifford group $\Gamma$ of a Clifford algebra $C\ell (V,q)$ is defined to be the set of elements $g$ in $C\ell (V,q)$ for which there exists an inverse $g^{-1}$. This group can be represented by ...
5
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1answer
242 views

Exceptional isomorphism with Spin(6,2)?

There are all sorts of curios in low-dimensional Lie groups and Lie algebras, many of them due to the presence of the quaternions. There is, I have recently learned, an isomorphism $SO(6,2) \simeq SO(...
3
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1answer
116 views

Classification of $2k$-vectors modulo orthogonal transformations

Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}...
4
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1answer
217 views

Eigenvectors and Eigenvalues of the Clifford multiplication

I would like to get an answer to the following question: Let $\Delta_n$ be the vector space of complex $n$-spinors. A vector $X \in \mathbb{R}^n$ acts on $\Delta_n$ by Clifford multiplication. We can ...
9
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1answer
867 views

Is there a way to embed Clifford algebras into the corresponding tensor algebra?

$\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}_q(V)}$$\newcommand{\qalg}{\mathcal{I}_q(V)}$Is there a way to embed Clifford algebras into the corresponding tensor algebra? ...
3
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1answer
160 views

How does grade projection act on homogeneous multivectors in geometric algebra?

I'm reading Clifford Algebra to Geometric Calculus by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator. It is noted that $A_r$ and $B_s$ are ...
4
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1answer
391 views

Why does the Bogolyubov transformation work? - In language of Clifford Algebras?

Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l_{2k}\equiv Cl_{2k}\otimes_{\mathbb{R}}\mathbb{...
5
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0answers
170 views

Exceptional symmetric spaces embedded in exceptional Lie group

In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective ...
14
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1answer
748 views

$p$-adic Bott periodicity?

The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...
6
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1answer
433 views

What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero. If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...
9
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1answer
287 views

Generalized Dirac operators

So far I met three definitions of the so called generalized Dirac operator(or Dirac type operators. Everything takes place over Riemannian manifols $M$ and we have smooth hermitian vector bundle $S \...
3
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1answer
149 views

What is the correct generalization of the Wirtinger derivatives to arbitrary Clifford algebras?

In the complex numbers setting, the two Wirtinger derivatives are defined as: $\frac{\partial}{\partial z}= \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \...
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0answers
112 views

Does Feuter regularity imply derivability in all directions?

The standard type of regularity in Clifford Calculus is the one introduced by Feuter, namely: a function is Feuter regular iff it is in the zero set of the Clifford-Dirac operator $D= \partial ...
8
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1answer
516 views

What is the spin connection in 9 dimensions as opposed to 5 dimensions?

From Spin Connection in 5 dimensions I can define a massless fermion's covariant derivative on a curved manifold as $$ \nabla_\mu \psi = (\partial_\mu - {i \over 4} \omega_\mu^{ab} \sigma_{ab}) \psi \...
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3answers
2k views

Trace of six gamma matrices

I need to calculate this expression: $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$ I know that I can express this as: $$ Tr(\gamma^{\mu}\gamma^{\...
2
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0answers
160 views

What is the relation (if any) between Clifford algebras and Azumaya algebras?

Suppose the base field is $\mathbb{C}$ and the Clifford algebra is the classical one (i.e. associated to a quadratic form in $n$ variables). It seems that there are relations between Clifford algebras ...
4
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3answers
215 views

Is an associative division algebra required for this phenomenon?

For which integers $d \geq 1$ can we find real matrices $R_1, \dotsc, R_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R_1 v, \dotsc, R_d v$$ is an orthonormal basis? ...
3
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0answers
92 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 ...
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0answers
145 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} \gamma^{a_2}...
1
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1answer
102 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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0answers
227 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 ...
3
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2answers
773 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.
5
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1answer
263 views

Convention on Clifford Product [duplicate]

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 or ...
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1answer
206 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 $SO(7)<SO(8)$ ...
4
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1answer
268 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 ...
1
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1answer
244 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 To ...
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2answers
453 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 $\mathbb{R}$)....
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2answers
2k 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 ...
2
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1answer
256 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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0answers
453 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
2k 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 ...
4
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0answers
162 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
214 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 ...
2
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2answers
200 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 $e_0=1$, $e_1=i$, $e_2=j$ and so on. I'm looking for an ...
1
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0answers
124 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 ...