# 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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### Why is it that we can represent vectors using the even part of the Clifford algebra?

You can represent a vector by a quaternion with no scalar part, and you can also represent the rotation itself as a quaternion. Then the rotation is applied to the vector by conjugation. The ...
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### Point at infinity in geometric algebra

In Homogeneous Coordinates, the point at infinity is represented by a vector pointing to "the horizon". Is there an equivalent representation of the Point at infinity in geometric algebra (...
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### Is the definition $a\cdot b= 0.5(ab+ba)$ in geometric algebra justified, or is it mostly arbitrary?

According to one of the basic axioms of geometric algebra, the square of a vector with itself is a scalar. For two vectors $a$ and $b$, this results in $ab+ba = (a+b)^2-a^2-b^2$. Therefore $ab+ba$ ...
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### Introductory articles on Spin group and Clifford Algebra

I am searching for some introductory articles on Spin Group and Clifford Algebra. Please help me if you know about such articles or any other material. Thank you.
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### What would an infinite dimensional Clifford algebra look like in terms of its generators?

In Infinite dimensional Clifford algebras?, the answers spoke on how to construct a Clifford algebra in infinite dimensions. What I want to know is: what does $Cl^{p,q,r}(\Bbb R)$ look like when any ...
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### Dimension of filtered algebras and their associated graded algebras

Let $F$ be a finite dimensional filtered algebra and let $G$ be the associated graded algebra. Will the dimension of $G$ and $F$ coincide or differ in general? If they differ in general then what is ...
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### How much novelty is there in spectral theory over Clifford algebras?

Asking here seems like the most efficient way of getting an answer. Maybe I'll get an opinion from a relevant expert. A demonstration of my idea is to take the Takagi decomposition: Given a complex-...
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### What is the center of a Clifford algebra

I have for short interess at Clifford algebras, and I've read on the german wikipedia that the center, that the set $$\{x\in CL| x\cdot y= y\cdot x \;\forall y\in CL\}$$ was $$\mathbb{R}\cdot 1_{CL}$$...
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### Unique irreducible complex representation of Clifford algebra implies isomorphism with matrix algebra

Consider the Clifford algebra $\mathrm{Cl}(n)$ over Euclidean space $\mathbb{R}^n$ (with the standard inner product). Now, in the case that $n$ is even, it is known (cf. ), then $\mathrm{Cl}(n)$ ...
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### a question about clifford-algebras

in n -imensional linear space,e_i are the standard Orthogonal basic, we use the axiom of clifford algebra : $x\cdot y = \frac{1}{2}(xy+yx) ,x\wedge y = \frac{1}{2}(xy-yx), xy=x\cdot y + x\wedge y$ to ...
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### Outer product of 3 vectors in geometric algebra

I am currently self-studying axiomatic geometric algebra from a couple of books(Doran-Lasenby(DL), Hestenes(H), notes on the internet), and I got stuck while checking some calculations. When defining ...
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### Classifying space of Clifford Algebra

I am a physics student, and I am reading a paper in topological condensed matter employing the classifying space of Clifford Algebra.This paper In particular, I feel that I am a bit confused by the ...
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### Mechanics of Clifford Algebra valued Möbius Maps

The Möbius maps $$(ax+b)(cx+d)^{-1}$$ Where $a,b,c,d,x \in \mathbb{C}$ are well-understood, but how do the ones where the variable and coefficients are in Clifford Algebras act? Are there any good ...
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### Construction of dual numbers

Out of curiosity, I was seeing about hypercomplex numbers. In that article, the definition says that, "Where possible, it is conventional to choose the basis so that $i_k^2 \in \{ -1, 0, +1 \}$. ...
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### Quotient algebra of tensor algebra, waiting with the quotients?

I was just wondering if we in a quotient algebra of a tensor algebra can "wait" with the quotients until we have computed the expression in the tensor algebra? This would make it simpler for ...
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### Clifford multiplication with specific 2-forms in 6 dimensions

I am new to spin geometry and I am trying to understand spinor bundles in dimension 6 for something else I am reading. $\mathcal{Cl}(6,0)$ has a unique irreducible real 8-dimensional representation $S$...
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### Proof that contraction $𝒖 \rfloor$ is an anti-derivation

In geometric algebra, contraction by a vector $𝒖$ is an anti-derivation, meaning for all (possibly inhomogeneous) multivectors $A, B$, we have  \DeclareMathOperator{\lc}{\rfloor} 𝒖 \lc (AB) = (𝒖 \...
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### To the Clifford algebra, what is Clifford multiplication?

What I know about Clifford algebras has come from my friend who is studying them right now, who explained some basics to me over the last fifteen minutes. I have a small conceptual question. My ...
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### Non-trivial squares of bases in Clifford algebras?

I've been learning about Clifford algebras recently, and I know that bases can square to +1, -1, or 0 (but are not real numbers themselves). Is there any value in considering bases that square to ...
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### Notes on Spinors by Deligne

I was reading section 2 (Clifford Modules) of the Notes on Spinors by Deligne, and am a little puzzled by Proposition 2.2. In this proposition, it is assumed $V$ is a complex vector space and $Q$ a ...
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### Connections between Weyl, Clifford, Exterior, and Symmetric algebras

The Clifford algebras are analogous to the Weyl algebras in the same way that the exterior algebras are analogous to the symmetric algebras. How are the Clifford and Weyl algebras analogous? The ...
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### What changes for a complexified Clifford algebra?

If you complexify a Clifford algebra, does it mean just using complex scalars instead of reals? Or do any of the algebraic rules change? I suppose $\langle M\rangle$ will extract the full complex ...
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### How do you prove this left contraction identity for the dual in the homogeneous model, $X\rfloor (I_n^{-1}e_0^{-1})=X^\star e_0^{-1}$?

I would like to prove that $X\rfloor (I_n^{-1}e_0^{-1})=X^\star e_0^{-1}$ as claimed in Geometric Algebra for Computer Science (Dorst et al). I don't see where this pattern matches to any of the ...
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### Hopf fibration from $SO(3)$ Lie algebra generators?
One can use the Pauli matrices $\sigma_i$ to generate $Cl_3(\mathbb{R})$ and taking commutators of these matrices gives the $SU(2)$ Lie algebra \$\mathfrak{su}(2)=\biggl(\begin{matrix} ia&-z\\ z&...