Questions tagged [clifford-algebras]
The clifford-algebras tag has no usage guidance.
137
questions
4
votes
2
answers
602
views
Principled construction of the quaternions
Is there a construction of the quaternions that doesn't proceed through generators and relations, and which makes the connection with 3D rotations clear?
I'm not happy with Clifford Algebra as an ...
7
votes
0
answers
248
views
Enveloping von Neumann algebra of Clifford algebra
As explained in the book "Spinors in Hilbert Space" by Plymen and Robinson, if $V$ is a complex (separable) Hilbert space with a real structure, and $\mathrm{Cl}(V)$ the corresponding Clifford algebra,...
1
vote
0
answers
168
views
Simultaneous diagonalization of the tensor products of Dirac gamma matrices
Let $\gamma_i\ (i=1,2,\ldots N)$ be the Dirac gamma matrices satisfying the Clifford algebra
$$\gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij} I\ \ (i,j=1,2,\ldots,N).$$
Then the tensor products $\...
4
votes
0
answers
112
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 ...
12
votes
1
answer
509
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
votes
0
answers
156
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 $\...
1
vote
0
answers
103
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 ...
31
votes
4
answers
2k
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
votes
0
answers
98
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_{\...
1
vote
0
answers
99
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
votes
0
answers
67
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 ...
1
vote
0
answers
119
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 ...
1
vote
0
answers
63
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
votes
2
answers
251
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,
...
-1
votes
3
answers
288
views
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 $V$...
0
votes
1
answer
101
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 ...
5
votes
1
answer
376
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
votes
1
answer
125
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
$$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k}}...
4
votes
1
answer
497
views
Eigenvectors and Eigenvalues of the Clifford multiplication
I would like to get an answer to the following question: Let $\Delta_n$ be the vector space of complex $n$-spinors. A vector $X \in \mathbb{R}^n$ acts on $\Delta_n$ by Clifford multiplication.
We can ...
12
votes
1
answer
2k
views
Is there a way to embed Clifford algebras into the corresponding tensor algebra?
$\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}_q(V)}$$\newcommand{\qalg}{\mathcal{I}_q(V)}$Is there a way to embed Clifford algebras into the corresponding tensor algebra?
There ...
3
votes
1
answer
367
views
How does grade projection act on homogeneous multivectors in geometric algebra?
I'm reading Clifford Algebra to Geometric Calculus by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator.
It is noted that $A_r$ and $B_s$ are ...
4
votes
1
answer
498
views
Why does the Bogolyubov transformation work? - In language of Clifford Algebras?
Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l_{2k}\equiv Cl_{2k}\otimes_{\mathbb{R}}\mathbb{...
6
votes
0
answers
264
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 ...
18
votes
1
answer
1k
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
votes
1
answer
472
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
votes
1
answer
629
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
votes
1
answer
234
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}
\...
1
vote
0
answers
142
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 x_0 + \...
8
votes
1
answer
556
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
$$
\nabla_\mu \psi = (\partial_\mu - {i \over 4} \omega_\mu^{ab} \sigma_{ab}) \psi
\...
0
votes
5
answers
8k
views
Trace of six gamma matrices
I need to calculate this expression:
$$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
votes
0
answers
241
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
votes
3
answers
250
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
votes
0
answers
108
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
0
answers
208
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
vote
1
answer
135
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 $V$ ...
7
votes
0
answers
329
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 ...
5
votes
2
answers
1k
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
votes
1
answer
317
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 ...
2
votes
1
answer
316
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 $SO(7)<SO(8)$ ...
3
votes
1
answer
310
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
1
answer
516
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 ...
5
votes
2
answers
516
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 $\mathbb{R}$)....
19
votes
2
answers
4k
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 ...
3
votes
1
answer
418
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 (...
23
votes
0
answers
569
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 ...
33
votes
2
answers
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 ...
5
votes
0
answers
232
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 ...
14
votes
2
answers
293
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 ...
5
votes
2
answers
407
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
vote
0
answers
146
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 ...