# 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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### Is every element of Spin(4) an element of Spin(8)?

My question is simple: I have an element of Spin(4) written in terms of $16\times 16$ matrices from a Clifford algebra $C\ell_{8,0}$. Does this imply that any such element is also an element of Spin(8)...
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### Conjugate of an operator and its relation with its squaring value

This question is straightforward. Is it possible for a unit length operator (vector, multivector, etc.), excepts for $-1$ itself that squares to $+1$ to have its hermitian conjugate equal to $-1$?, i....
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### Is the inverse of a $k$-vector a $k$-vector?

In geometric (Clifford) algebra, both $k$-vectors and inhomogeneous multivectors may have inverses, which are unique (if they exist). I want to prove the following statement. Let $A = ⟨A⟩_k$ be a $k$-...
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### $R \otimes_k \mathrm{End}_R(W) \cong \mathrm{End}_k(W)$ for a $k$-algebra $R$.

Suppose $k$ be a field of characteristic $0$, and $R$ be an unital $k$-algebra. Then, does it hold $$R \otimes_k \mathrm{End}_R(W) \cong \mathrm{End}_k(W)?$$ for an $R$-module $W$? This is an ...
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### What justifies the statement that a Dirac spinor can be written as two Weyl spinors?

I am cross listing this from physics SE in case it is more appropiate here. That post can be found here: https://physics.stackexchange.com/questions/794843/what-justifies-the-statement-that-a-dirac-...
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### Adjoint representation of $V\subset \text{Cl}(V,q)$

I have been trying to learn Clifford algebra from Lawson & Michelsohn's Spin Geometry. There is a geometric explanation for the adjoint representation formula ($V$ is a vector space, $q$ is a ...
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### What is the intuition behind the regressive product and its axioms?

I'm self-studying Grassmann algebra and I find it hard to visualize the regressive product, specifically the magnitude and orientation of the outcome. My main soure is John Brown's Grassman Algebra ...
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### Computing $x\cdot e_n \cdot x \cdot e_n$ in a Clifford algebra over $\mathbb{C}^d$

I am closely following Hamilton's Mathematical Gauge Theory. Let $V$ be a vector space, $Q$ a bilinear form on $V$, and $CL(V,Q)$ the corresponding Clifford algebra. We can construct the Clifford ...
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### Show that $\tfrac{1}{2}(b B\cdot (c \wedge a) - a B\cdot (c \wedge b)) = (c\cdot b) a \cdot B$ in geometric algebra

I would like to show that $$\frac{1}{2}\left( b B\cdot (c \wedge a) - a B\cdot (c \wedge b)\right) = (c\cdot b) a \cdot B$$ where B is a bivector, and the others are vectors. I've tried to coerce ...
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### Understanding's Wikipedia's definition of a spinor

I am trying to understand spinors from a mathematical view. I've seen similar questions on this website but I'm still unclear on what they are exactly. On Wikipedia they state: Although spinors can ...
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### On triality, $Cl_{0,6}(\mathbb{R})$, and the outer automorphism of $S_6$

I ran across this paper recently, which explores Conway's proof that $L_4(2) = A_8$: https://archive.maths.nuim.ie/staff/jmurray/Preprints/jmurrayalt8.pdf . A central part of the argument involves ...
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### Projective Geometric Algebra - what does an "inheritance property" mean in this context and how does it relate to line representations as bivectors?

On page 15 of this manuscript, the author says the following on why lines can be represented by 2-blades: I'm facing trouble understanding the point the author is trying to make in the second ...
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### How does the following representation of $\mathbb{C}l(3)$ decompose into irreducibles?

Consider the three dimensional complex Clifford algebra $\mathbb{C}l(3)$ and the following representation $$S=\text{Span}\{|0\rangle,c^\dagger|0\rangle,\gamma^3|0\rangle,\gamma^3c^\dagger|0\rangle\},$$...
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### Reducible representations of the Clifford Algebra

I would like to construct reducible representations of the Clifford algebra, that consist of $8\times8$ matrices with purely real or purely imaginary elements. Assume that I am familiar with the Dirac,...
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### Geometric Algebra: show $A_r \cdot B_s = (-1)^{r(s-1)} B_s \cdot A_r$ and $A_r\wedge B_s = (-1)^{rs} B_s \wedge A_r$ from Hestenes and Sobczyk's book

I'm making a go at self-study from Hestenes and Sobczyk's book Clifford Algebra to Geometric Calculus. I'm stuck on the simple formulas in the first section for reversing the order for the inner and ...
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### Inferring classification of Clifford algebras from classification of Clifford modules

Let $Cl_n$ be the Clifford algebra (over reals) $$Cl_n = T^{*}\mathbb{R}^n/\langle v\otimes v - q(v) \rangle.$$ There is a periodic table of $K$-representations of $Cl_n$, i.e. $\mathbb{R}$-linear ...
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### Can a vector have a wedge product with a scalar? And a geometric product?

One of the necessary requirements of a vector space is to be endowed with the multiplication by a scalar. I am now interested in other operations, the wedge and the geometric product in geometric ...
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### Can you show an example of conformal geometric algebra being applied to synthetic geometry?

I have been studying conformal geometric recently where points, lines, etc are described as $n$-blades in geometric algebra. The notes I am studying claim that conformal geometric algebra can be ...
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### Is Clifford product only for scalars already defined?

hi i'm a high school student interested in Clifford algebra. Since it is hard for me to understand several things thus i hope to know as follows: Is Clifford product only for scalars already defined? ...
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### Intersection of *segments* in 3D using projective geometric algebra?

There is a relatively common technique to find the intersection of two segments in 3D, which can be found in page 304 of Graphics Gems Note that a segment is a compact subset of a line. The technique ...
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### Spinors in Spin Geometry

First of all I am a physicist with a decent knowledge of graduate-level geometry. I'm studying Spin Geometry from Bär Lecture Notes and I have some trouble understanding what spinors are from his ...
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### How to find generators of the real Clifford algebra $\text{M}_2(\mathbf{H})$?

The real Clifford algebra $\text{Cl}_{0,4}(\mathbf{R})$ is isomorphic to $\text{M}_2(\mathbf{H})$, the algebra of $2\times 2$ matrices of quaternions. In $\text{M}_2(\mathbf{H})$, how can I find a ...
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### How can one generalize the split-complex algebra to a coordinate space with an arbitrary number of asymptotes?

Given that split-complex numbers generates 2D coordinates with 4 asymptotes (when multiplying) that look as follows: The arrows along the hyperbola indicate a positive direction for boosting points ...
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### Algebra over a field generated by a vector space?

I have just learned about Clifford algebras and am a bit confused by the construction. My understanding is that there is a universal construction, but here is the one I was given for now (Szekeres's A ...
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I'm reading "Elements of Noncommutative Geometry" of Garcia-Bondía. A Clifford module was defined as the following: A Clifford module over a compact Riemannian manifold $(M,g)$ is a finitely ...
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### Why does the wedge product produce a scalar and not a graded element?

Reading the definition of the geometric product we get that it satisfies: $$a_1 \wedge \cdots \wedge a_r = \frac{1}{r!} \sum_{\sigma \in G_r} \mathrm{sgn}(\sigma)a_{\sigma(1)}\cdots a_{\sigma(r)}$$ ...
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If we have 2 vectors $v_1,v_2$ which have been rotated into $v'_1,v'_2$ by the following operations: $v'_1 = e^{θ\hat{n}/2}v_1e^{-θ\hat{n}/2}$ $v'_2 = e^{θ\hat{n}/2}v_2e^{-θ\hat{n}/2}$ Where $\hat{n}$ ...