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Questions tagged [clifford-algebras]

Clifford algebras are associative algebras constructed from quadratic forms on vector spaces. They can be viewed as generalizations of the real numbers, complex numbers, and quaternions. These algebras have applications in geometry and theoretical physics.

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Clifford algebra and exterior algebra

Let $E$ be a finite dimensional real vector space with $E^*$ its dual, and let $\langle \; , \; \rangle$ be an inner product on $E$. For any $e \in E$, denote by $e^* = \langle e, \; \rangle \in E^*$ ...
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How to determine if two algebra automorphisms of $M_n(\mathbb{R})$ are similar?

Given two algebra automorphisms $\phi, \psi: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ of the real algebra $M_n(\mathbb{R})$, we say they are similar if there exists an algebra automorphism $\alpha: M_n(\...
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Quaternification of a complex vector space

Let $V$ be a vector space over $\mathbb{R}$. A complex structure on $V$ is a linear map $J:V\to V$ such that $J^2=-\text{Id}$. Remark 1: if $V$ has a complex structure, then $V$ is even dimensional....
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The commutator product as a replacement for the cross and wedge product in geometric algebra?

From an axiomatic approach to geometric algebra, the wedge product of two vectors $a$ and $b$ is typcially defined as the antisymmetric $a \wedge b = \frac{1}{2}(ab - ba)$, where $ab$ is the geometric ...
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Inner product structure on geometric algebra?

I understand that geometric algebra equips itself with the contraction operators $\rfloor$ and $\lfloor$. While these are awesome when one wishes to project a subspace onto another, it is not an inner ...
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Write a bivector as the exterior product of two vectors

The Wikipedia article https://en.wikipedia.org/wiki/Bivector#Simple_bivectors states that "A bivector that can be written as the exterior product of two vectors is simple. In two and three ...
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Anticommuting sets of Dirac $\gamma$-matrices

At the end of this webpage, it is said that there exist 6 maximal anticommuting sets each consisting of 5 Dirac $\gamma$-matrices. I couldn't find anything more in the book cited there, either. But ...
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How to Take Dot Product of a Vector and a Bivector

I'm new to geometric Algebra and im trying to take geometric product of a bivector and a vector I can understand wedge product but I can't get the meaning of a dot product between bivector and a ...
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Matrix anticommuting with four or five Dirac $\Gamma$-matrices

Consider the following four or five Dirac $\Gamma$-matrices\begin{gather}σ_{x,y}\otimes τ_0,σ_{x,z}\otimes\tau_z,\tag{*1}\\σ_{x,y}\otimes τ_0,σ_{x,y,z}\otimes τ_z,\tag{*2}\end{gather} where $\sigma$ ...
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Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the Clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
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How to compute if a multivector inverse exists in Clifford Algebra

Suppose we have a 4 dimension positive signature clifford algebra. In Calculating the inverse of a multivector and Inverse of a general nonfactorizable multivector, the inverse of a multivector is ...
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Why is clifford group a group?

Let $C(Q)$ denote the clifford algebra of vector space $Q$ with respect to a quadratic form $q:V \rightarrow \Bbb R$. Hence we have the relation $w^2 = Q(w) \cdot 1$ for $w \in V$. Let $\alpha:C(Q) \...
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Deriving reduction formula in Geometric Algebra

I am trying to learn Geometric Algebra by going through the book "New Foundations for Classical Mechanics" by David Hestenes. I was reading the part about reduction formula (shown below) but couldn't ...
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Equivalent definitions of Clifford algebra, verification

Let $(V,B)$ be a finite dimensional $k$ vector space $V$ with an associated quadratic form $Q$. $char \, k \not= 2$. Let $X:= \{e_i \}_{i=1}^n$ be a set of basis for $V$. Construct $k\langle X \...
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On the algebraic formulation of the Clifford algebra

I apologize in advance for a rather wordy question. As a physicist trying to learn new mathematics, I figured this was the place to ask. I am having trouble understanding the algebraic formulation of ...
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Clifford algebra from a bunch of commutation and anti-commutation relations

When reading the paper by Kitaev (arXiv:0901.2686), it seems to me there is a certain kind of theorem roughly like this: "Consider an algebra formed by $B_i$, $i=1,2...n$ with $[ B_i, B_j]_{s_{ij}} =...
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Double cross product in 2D

Hello I have a question about a double cross product, appearing in centrifugal force \begin{align*} \mathbf{F}_{centrifugal} = -m \boldsymbol{\omega} \times [\boldsymbol{\omega} \times \mathbf{r}] \, ....
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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 $\...
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wedging and contracting operators generate a clifford algebra

Im working on a paper and having trouble proving the following: The wedging and contracting operators $\phi(e_i), \phi^*(e_i)$, $1\leq i\leq n$ generate a Clifford Algebra with $W = V\oplus V^*$ and $...
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Definition of Clifford Algebra

Cliffard algebra defined by relation: $x*y+y*x=g(x,y)1$, where g(x,y) is bilinear symmetric form. What does mean $g(x,y)1$, why it's not just $g(x,y)$, without the identity?
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Trace of chiral matrix in higher odd dimension

Suppose we have $D$ gamma marices $$\gamma_{a}\gamma_{b}+\gamma_{b}\gamma_{a}=2\delta_{ab}I$$ with $a,b=1,2,...,D$. Chiral matrix is $$\gamma_{chir}=\gamma_{1}\gamma_{2}...\gamma_{D}$$ in even ...
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Rotation around a whole sphere by multiplying a single hypercomplex number forever?

In quaternion number system, any unit quaternion $\mathbf{q}\in\mathbb{H}$ can be written as $$ \mathbf{q} = \cos \theta + (v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k})\sin \theta $$ for some $\...
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Combining the Laws of classical mechanics into a single equation using multi vectors

Maxwell's laws can be combined into a single equation using multi vectors. What about the Laws of classical mechanics? Can they be combined into a single equation? https://en.wikipedia.org/wiki/...
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Clifford algebra is isomorphic to exterior algebra, proof in Lawson's

This Proposition 1.2, pg 10 of Lawson's Spin Geometry. Context: We have a homomorphism of graded algebra $$ \wedge^*(V) \rightarrow G^*$$ induced on each component from the map $$\wedge^r(V) \...
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Clifford Multiplication and Spinor Bundles

I am trying to follow the discussion of Clifford multiplication on page 384 of The Wild World of 4-Manifolds, by Alexandru Scorpan (link, although I hope this will be totally self-contained), and I'm ...
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Geometric product of 2 vectors derived from Pauli matrices

The 3 Pauli matrices are: ${\color{blue}{\sigma_1}}$ = $ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ ${\color{blue}{\sigma_2}}$ = $ \begin{pmatrix} 0 & -i \\ i & 0 \...
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Identity matrix times i

Why is the psuedoscalar in 5 dimensions the identity matrix? In 3 dimensions its $\begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}$ In 3 dimensions when you multiply all 3 vectors (Pauli ...
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Is the $\mathbb{Z}$-grading of a Clifford algebra basis independent?

Let $V$ be a finite dimensional vector space over a field $K$ of characteristic $\neq 2$, and let $q \colon V \to K$ be a quadratic form. One of the first things to show when learning the theory of ...
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How do the symmetries of a Lie manifold manifest in the metric tensor of that manifold?

Suppose we have some Lie manifold (ie a Lie group) that is also a Riemannian manifold endowed with a metric tensor How does the Lie group symmetry manifest in the properties of the metric tensor? I ...
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What is the motivation for the development of dual numbers?

I searched the backlog of this website for something along the lines of my question to no avail. Not to mention a myriad of PDFs online which just drop the definition and allege more details can be ...
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Clifford algebra Cliff(0) as reals, Cliff(1) as complex…

In this paper by John Baez: http://math.ucr.edu/home/baez/octonions/node6.html he says that assuming a Clifford algebra with $vw + wv = -2<v, w>$, you can see that Cliff(0) = $\mathbb{R}$, Cliff(...
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Wedge Product on $C\ell^+(0,3,0)$ Relationship to Quaternion Cross Product

The even Clifford sub-algebra $C\ell^+(0,3,0)$ is isomorphic to the quaternion algebra. The mapping between terms is $e_0 \mapsto 1$, $e_{23} \mapsto i$, $e_{31} \mapsto j$, $e_{12} \mapsto k$. In ...
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What does the notation $[,]$, $\{,\}$ in the Clifford algebra mean?

From Charbonneau, Harland: Deformations of nearly Kähler instantons: It is explained in the previous paragraph that the authors use the canonical identification $\operatorname{Cl}(V,g)=\Lambda^*V$. ...
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How to perform wedge product

I have heard all kinds of great things about Clifford/Geometric algebra, but I can't find any good resources. I have been looking EVERYWHERE for just one actual example of a wedge product being ...
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Proof of these equalities for Lorentz transformations

I'm working on Lorentz transformations for elements of Dirac algebra $\mathcal{D}$ (the Clifford algebra generated by $\mathbb{R}^{1,3}$). In this algebra we write an element $$ A=A_0+A_1+A_2+A_3+A_4 $...
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Transformation of metric tensor in clifford algebra for nonorthogonal transformations

I study physics so apologies for any nonstandard notation/terminology. In geometric (aka spacetime) algebra one speaks of basis $\gamma^{\mu}$ ( possibly represented by matrices) as transforming in a ...
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Transformation of metric tensor under clifford algebra?

I've had to rewrite the question as i made agrievious errors in the first go-round. Given a (pseudo)Riemannian metric g, we can identify its components with the symmetric product of gamma matrices: ...
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cross product not associative, outer product associative

The cross product is not associative. If $i=(1,0,0)$, $j=(0,1,0)$ and $k=(0,0,1)$, then \begin{eqnarray} i \times (i \times j) = i \times k = -j \\ (i \times i) \times j = 0 \end{eqnarray} However ...
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How to decompose a bivector into a sum of _orthogonal_ blades?

In Geometric Algebra, any bivector $B\in\Lambda^2\mathbb R^n$ is a sum of blades: $$B = B_1 + B_2 + \cdots$$ $$= \vec v_1\wedge\vec w_1 + \vec v_2\wedge\vec w_2 + \cdots$$ Each blade's component ...
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Primitive idempotents of Cl(1,3) over the complex numbers

Simply put, I need to find all primitive idempotents of the Clifford Algebra $Cl(1,3)$ over the complex numbers. I have found some general results but they're only applicable over the real numbers. I ...
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Clifford algebras of even rank, unique irrep.

I would like to prove that there is one and only one (non trivial) irreducible representation (up to equivalence) of the Clifford algebra $Cl(n)$ with $n=2p$ and $p\in\mathbb{Z}$. I have seen a proof ...
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K-theory and Clifford modules

I'm trying to wrap my head around the "Clifford modules" definition of K-theory. Let's just deal with K-theory of a point. One common definition of the $-n^\text{th}$ K-group is the quotient$$K^{-n}=M(...
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Orthogonality of Clifford algebra's Fueter polynomial in certain 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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“Square root” of a decomposition of a homogeneous polynomial to harmonic and $x^2 q$.

It is well known that any homogeneous polynomial $f \in \mathbb R[x_1, \ldots, x_n]$ can be uniquely split as $f = f_0 + x^2 f_1$, where $x^2 \equiv (x_1)^2 + \ldots + (x_n)^2$ and $f_0$ harmonic: $\...
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Geometric Calculus, Clifford Algebra, and Calculus of Variations

It has always bothered me that I was told in my Calculus of Variations class that it's only possible to minimize a function with respect to one objective. Obviously sometimes it is possible to ...
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Spinor chiral transformation by $\psi \to \gamma^5 \psi$

Let $\psi$ be a spinor. Let $\gamma^0,\gamma^1, \gamma^2, \gamma^3$ be the usual gamma matrices and the fifth $\gamma^5 : = i\gamma^0\gamma^1\gamma^2\gamma^3.$ Then if we define $\psi \to \psi' := \...
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Gamma matrices and special relativity

I understand how gamma matrices generate a Clifford algebra that corresponds to the Minkowski metric. So the next step for me is to understand how gamma matrices are used in the context of special ...
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For two unit non-oriented bivectors $A,B\in \mathbb{R}P^2\subset \Lambda^2\mathbb{R}^3$ is the mapping $\phi:(A,B)\rightarrow AB$ bijective?

For two non-oriented unit bivectors $A,B\in \mathbb{R}P^2\subset \Lambda^2\mathbb{R}^3$ is the mapping $\phi:\mathbb{R}P^2\times \mathbb{R}P^2/\mathbf{D} \rightarrow S^3$, where $\mathbf{D}$ is the ...
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Minkowski metric. Scalar or tensor?

The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation $\displaystyle\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu =...
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Clifford algebra complex representation

A canonical basis for the geometric algebra $\mathcal{G}(3,0)$ has $2^3$ elements: $1, e_1, e_2, e_3, e_1e_2, e_1e_3, e_2e_3, e_1e_2e_3$ That's easy to understand. There are 3 dimensions and ...