# Questions tagged [clifford-algebras]

The clifford-algebras tag has no usage guidance.

121
questions

2
votes

1
answer

150
views

### Another formula for the Schwinger term — problems with a calculation

$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...

6
votes

1
answer

139
views

### CCR vs. CAR vs. Clifford algebras, infinite tensor products and type of the corresponding von Neumann algebra

$\newcommand\CAR{\mathit{CAR}}\newcommand\Cl{\mathbb C\mathit l}$This question will be rather long and it will be my attempt to finally clarify many issues concerning CCR, CAR and Clifford algebras ...

2
votes

0
answers

214
views

### Proving that a product of reflections and an orthogonal matrix is in $\mathrm{SO}_*(V)$

Let $V=(V,b)$ be a finite-dimensional vector space equipped with $b$ a symmetric and positive definite bilinear form. And let $\{e_1,\dotsc,e_n\}$ be a orthonormal basis for the subspace $\ker((P_A)^t)...

3
votes

0
answers

98
views

### Bott periodicity in characteristic p via Clifford algebras

I am currently reading Husemoller's wonderful book on fibre bundles, specifically the section on Clifford algebras. He defines these groups $L_k$ as follows. Let $M_k$ denote the free abelian groups ...

5
votes

1
answer

203
views

### Example of nice isomorphism between Cl$_{p,q}(\mathbb R)$ and matrix algebras over $\mathbb R,\mathbb C,\mathbb H,\mathbb R^2,\mathbb C^2,\mathbb H^2$

$\DeclareMathOperator\Cl{Cl}$It is known that every Clifford Algebra $\Cl(Q)$ over the real numbers where $Q: \mathbb R^n \to \mathbb R$ is a non-degenerate quadratic form is isomorphic to a matrix ...

3
votes

1
answer

219
views

### Clifford Algebra - axiomatic definition by Hestenes

I am reading David Hestenes' book "Clifford Algebra to Geometric Calculus" and am already getting stuck on the first few pages.
My university math is rusty and I've never studied Clifford ...

2
votes

0
answers

89
views

### Non-associative Clifford algebra

Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$.
My question is if there exists an analog of a Clifford algebra in ...

2
votes

0
answers

51
views

### Decomposition of a bivector of a Lorentzian manifold [closed]

In the case of the $(1, -1, -1, -1)$ metric, a bivector (or a 2-form) $F$ decomposes uniquely as the sum of two orthogonal simple bivectors if $F^2 \ne 0$.
I have the impression that it is very little ...

3
votes

0
answers

65
views

### Integral representation of cochains and a theorem of Hopf

The classical theorem of Hopf asserts that for any
n-dimensional CW-complex $K$, there is an isomorphism
between homotopy classes from $K$ to the sphere $S^n$ and
the nth singular cohomology group:
$$
...

1
vote

0
answers

60
views

### Relationship with between Clifford multiplication and pullback

Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...

1
vote

1
answer

81
views

### Problem concerning about an $n$-subspace of $ A_{n}(F) $

Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...

4
votes

0
answers

100
views

### Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...

1
vote

0
answers

28
views

### Can one extend the norm function to the symmetric square of a (complexified) Clifford algebra?

Let $A = Cl_{r,s} \otimes \mathbb{C}$ be the complexification of the real Clifford algebra $Cl_{r,s}$ associated to a non-degenerate quadratic form on $\mathbb{R}^n$, with $n = r+s$, with signature $(...

4
votes

1
answer

263
views

### Identifying a $4$-form on a $6$-dimensional manifold

Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms ...

4
votes

1
answer

273
views

### Finding inverses in Clifford Algebras

Let $C = \operatorname{Cl}(V,q)$ be a Clifford algebra where $V$ is an $N$-dimensional space with basis $B = \{e_1,e_2, \dotsc, e_N\}$. I'm looking for a way to invert elements.
What I've already ...

6
votes

1
answer

240
views

### Euclidean and Minkowski Majorana spinors - inconsistency with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and
Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf
Somehow I find some inconsistency with his Tables of Euclidean and ...

0
votes

0
answers

117
views

### Clifford multiplication formula for $d\beta$ where $\beta$ is a $3$-form

Say $X$ be a $6$-dimensional compact Riemannian manifold which admits a $Spin^{\mathbb{C}}$ structure. Now I want to have a Clifford multiplication formula by $d\beta$ in terms of $\beta$ where $\beta$...

2
votes

1
answer

342
views

### Gamma matrices are irreducible

For $\mu=0,1,2,3$, let $\gamma^{\mu}$ the set of Dirac gamma matrices. What does it mean to say that $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$ is irreducible?
From my previous question, I know ...

3
votes

1
answer

315
views

### What is the relationship between the Dirac algebra and the Clifford algebra?

While I'm still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is ...

12
votes

2
answers

635
views

### How are Clifford algebras and spinors used to study the Ising model?

I've heard Clifford algebras and spinors are useful tools to study the Ising model, but I've never find any good discussion on this matter. Also, as far as I know, in the original solutions of the 2-D ...

6
votes

1
answer

152
views

### Hasse invariant and the Clifford algbera

Let
$$q = a_1 x_1^2 + \cdots + a_n x_n^2$$
be a quadratic form over some $p$-adic field $\mathbb{Q}_p$. We thus have its Hasse invariant
$$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...

0
votes

0
answers

113
views

### Central extensions of orthogonal group by $C_2$

Suppose $(V,Q)$ is a quadratic space for definite quadratic form $Q$. It is stated in Pin groups that there are two central extensions of the orthogonal group $O(V)$ by the cyclic group $C_2$, ...

9
votes

1
answer

240
views

### Efficient computation of scalar part in Clifford algebra

$\DeclareMathOperator\Cl{Cl}$Problem: Let $\Cl(d)$ be the Clifford algebra corresponding to the vector space $\mathbb{R}^d$ with the usual inner product. Given $v_1, \dotsc, v_k \in \mathbb{R}^d$, ...

8
votes

1
answer

189
views

### What is the inverse in K-theory represented by Clifford module extensions?

I am working on a model for topological KO-theory which is represented by explicit spaces of orthogonal Clifford module extensions. That is, assuming $M$ compact, $KO^{-n+1}(M) := [M,X_n]$ where the ...

1
vote

0
answers

89
views

### Control and observability of Clifford algebra and quaternion valued systems?

Good evening everybody, I'm asking you if there is a mathematical theory about control and observability of differential systems within the quaternionic and Clifford (geometric) algebras. And if so, ...

6
votes

1
answer

618
views

### Explicit computation of spinor norm

I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow.
Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\...

8
votes

0
answers

310
views

### Atiyah-Singer theorem in heat kernels and Dirac operators

I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne.
I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the ...

5
votes

0
answers

137
views

### Invariant theory for the orthogonal group and Clifford algebras

The first fundamental theorem of invariant theory for
the orthogonal group $O_n(k)$ asserts that the
ring of invariants is generated by the scalar products:
a polynomial function of $m$ vectors $v_1,.....

3
votes

1
answer

430
views

### The inner product of a Clifford Algebra

Any Clifford algebra $\operatorname{Cl}(k, p)$ carries an induced inner product, which is the "trace" on its 0-blade: $\langle AB\rangle_0$ for given elements $A, B$ of the algebra.
This inner ...

5
votes

0
answers

152
views

### Is there a list of all real unital subalgebras of M(2,C)?

Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex ...

28
votes

6
answers

3k
views

### What's "geometric algebra"?

Sometimes one bumps into the name "geometric algebra" (henceforth "GA"), in the sense of this Wikipedia article. Other names appear in that context such as "vector manifold", "pseudoscalar", and "...

1
vote

0
answers

40
views

### Maximal orders in Clifford algebras

Let
$$
\mathcal{C}_n(R)=R\langle e_1,\ldots,e_n\rangle/(\{e_i^2+1\}, \{e_ie_j+e_je_i:i\neq j\})
$$
be the Clifford algebra for the negative definite quadratic form $-\sum_ix_i^2$ obtained by adjoining ...

4
votes

1
answer

186
views

### Bott periodicity homeomorphisms for spaces of Clifford extensions

I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules).
Let $W = \mathbb{R}^{\...

4
votes

2
answers

563
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 ...

6
votes

0
answers

185
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

123
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

108
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 ...

11
votes

1
answer

366
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

137
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

91
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 ...

28
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

92
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

75
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

65
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

117
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

54
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

243
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,
...

0
votes

3
answers

222
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

97
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 ...

4
votes

0
answers

177
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 ...