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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Geometric explanation of Fueter-Sce-Qian theorem and similar situations

In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
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is the Contraction as Adjoint of Wedging? [closed]

The implicit definition of contraction is $$\DeclareMathOperator{\lc}{\rfloor} (X \wedge A) *B= X*(A \lc B)$$ and the outermorphism has a term called adjoint transformations, which are also implicitly ...
Takayama_Maria's user avatar
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First few cases of the Clifford algebra $\text{Cl}_{r,s}$

I have been reading Lawson & Michelsohn's Spin Geometry, and in the beginning of Chapter $1$ $\S 4$ on classification of the Clifford algebra $\text{Cl}_{r,s}$, they say that With little ...
Dasheng Wang's user avatar
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Polynomial Functions on Matrix Representations of a Clifford Algebra

Hi everyone, this is my first StackExchange post, so all tips on how to improve the question are very welcome! The question I would like to ask comes from mathematical physics, so please also tell me ...
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Associated graded algebra of Clifford algebra $\text{Cl}(V,q)$

I have been reading Lawson & Michelsohn's Spin Geometry, there is a sentence in the proof of Proposition $1.2$ that I don't understand. The proposition is the following: For any quadratic form $q$...
Dasheng Wang's user avatar
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Uniqueness of the spinor representation

Let $V$ be an even-dimensional euclidean vector space and $\mathbb{C}\mathrm{l}(V)$ the complexification of the Clifford algebra, then there exists a spinor module, i.e. a pair $(S,c)$ consisting of a ...
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Vector-like algebraic structure for multivalues?

When evaluating a non-integer exponent, like $ x^{a/b} $ (where a and b are coprime), it's common to only consider the principal root. However, in my use case, I need to be able to evaluate ...
stackshifter's user avatar
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Table of Clifford Algebras

Thanks to Hans Lundmark, I just found the table of Clifford algebras $C_k=C_{\Bbb R}(0,k)$ and $C'_k=C_{\Bbb R}(k,0)$, $0\leq k\leq 7$. But, I couldn't understand some of the slots in the table. $\...
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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....
physicsrev's user avatar
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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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Why are Dirac spinor representations defined as a projection onto the first factor?

Let $\mathbb{C}l(n)$ denote the Clifford algebra over $\mathbb{C}^n$ with the standard bilinear form. Then $$\mathbb{C}l(n) \cong \begin{cases} \text{End}(\mathbb{C}^N) \quad n \text{ is even}\\ \text{...
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Explicit triality representation in Spin(8)

First of all, I'm from a physics background, so pardon my probable lack of mathematical rigor in my question. Let's say I have an 8-dimensional Clifford algebra $C\ell_8$ with generators $e_i$, for ...
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Viewing vectors in a Clifford algebra as reflections

Let $Cl(s,t)$ be the Clifford algebra over $\mathbb{R}^{s,t}$ where $(s,t)$ is the signature of the bilinear form $\eta$. Let $Pin(s,t)$ be the associated pin group and define $$R: Pin(s,t) \times \...
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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 ...
Dasheng Wang's user avatar
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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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Seeking Correct 2x2 Matrix Representation for CL(1,1) in Clifford Algebra

I'm working with the Clifford algebra $ \text{CL}(1,1) $ and attempting to find an appropriate $ 2 \times 2 $ matrix representation. This algebra corresponds to a space with the metric signature $ (1, ...
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What is a Geometric Interpretation of the Product of two Blades?

I am hoping for a meaningful interpretation for the geometric product of two blades of not-necessarily the same grade. I understand that blades can be expressed as a sum of basis k-vectors, and then I ...
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Seeking 4x4 Real Matrix Representation of Generators for Clifford Algebra Cl(3,1)

I am currently delving into the study of Clifford algebras, particularly $ \text{Cl}(3,1) $, in the context of theoretical physics and am seeking clarity on a specific representation issue. As I ...
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How to derive the Log and Exp functions of a Geometric Algebra Rotor for Spherical Interpolation (Slerp) in 4D

I am trying to create a Slerp function between two 4D Rotors Rotor Slerp(Rotor a, Rotor b, t) { ... } I have been told you can do an interpolation using Log and ...
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Express reflection w.r.t. combination of vectors as combinations of reflection w.r.t. vectors

Let $\alpha\in\mathbb{R}^m$, then for every $x\in\mathbb{R}^m$ we call reflection of $x$ w.r.t. $\alpha$ the reflection of $x$ w.r.t. the hyperplane $\alpha^\perp=\{y\in\mathbb{R}^m\mid <\alpha,y&...
Giulio Binosi's user avatar
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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 ...
Michael Edwards's user avatar
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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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Do the Clifford Algebra products $e_1e_0$, $e_1e_{-1}$ and $e_0e_{-1}$ anticommute?

In a Clifford Algebra $\mathbb{CL}_{(1,1,1)}$ we have the following relations: $e_1^2=1$ $e_{-1}^2=-1$ $e_0^2=0$ Question: Do all the products $e_ie_j=-e_je_i$ anticommute? If so, why? $\\ % blank ...
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The algebra $\Gamma^\infty(\mathbb{C}l(M))$ is generated by $\Omega^1(M)$

I'm reading "Elements of Noncommutative Geometry" by Garcia-Bondía. There it was mentioned that the algebra $\Gamma^\infty(\mathbb{C}l(M))$ is generated by $\Omega^1(M)$. Here $\Omega^1(M)$ ...
Schrödinger's cat's user avatar
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Lorentz algebra problem. Relation of $\mathfrak{s}\mathfrak{o}(3,1)$, $\mathfrak{s}{u}(2)$, $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$

In the chapter 7 problems of Hall's Lie Groups, Lie Algerbas and Representations we have the following one: Show that the real Lie algebra $\mathfrak{s}\mathfrak{o}(3,1)$ is isomorphic to $\mathfrak{...
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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 ...
Kyle Taljan's user avatar
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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 ...
Hyeongmuk LIM's user avatar
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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 ...
TrentKent6's user avatar
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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? ...
mathgoblins's user avatar
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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 ...
LolloBoldo's user avatar
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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 ...
Andrius Kulikauskas's user avatar
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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 ...
Rehno Lindeque's user avatar
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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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Question about Clifford modules

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 ...
Schrödinger's cat's user avatar
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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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Solving for the rotor in quaternion rotation

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 $\hat{n}$ ...
George Chiporikov's user avatar
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Possible error in Doran’s “Geometric Algebra for Physicists”

On page 40 of Geometric Algebra for Physicists, Doran states in equation 2.98 that for a unit vector $n$ and an arbitrary vector $a$ with $a_\perp = n n\wedge a$, that $$n\cdot a_\perp=\langle n n \...
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Geometric/Clifford algebra - intuition behind this derivation of a formula that recovers a rotor from a vector basis and its transformation

This short manuscript by Francisco G. Montoya presents and proves a general formula that gives the rotor (up to a sign change) that effects a certain rotation transformation, from a given vector basis ...
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Are Clifford connections even?

I am currently reading the book Heat Kernels and Dirac Operators. I think that all Clifford connection should be even such that the associated Dirac operators are odd. (Dirac operators are defined to ...
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How to prove this identity in geometric/Clifford algebra in a series of reasonable steps [duplicate]

On page 76 of Doran's Geometric Algebra for Physicists, the following identity is presented (Eq. (3.129)): $$(x \cdot \Omega_B)^2 = \langle x \cdot \Omega_B x \Omega_B \rangle = - \Omega_B \cdot (x \...
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