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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Algebra generators of Clifford Algebra

A Clifford Algebra $C_k$ is a Real Algebra of dimension $2^k$ with its algebra generators being $\{e_1,\ldots,e_k\}$, satisfying the following relations: $$ e_i^2 = -1 ~~\& ~~e_je_i = -e_ie_j ~~\...
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Why Clifford algebras do not have basis vectors squaring to i?

Clifford's algebras have basis elements squaring either to $1$, $0$ or $-1$. Why not $i$? Is that already covered by $1$, $0$ or $-1$? If there is an algebra squaring to $i$, what is his name? I have ...
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Bivector Wedge Bivector

I'm struggling with bivector wedge bivector. If i do via the formula $$A\wedge B=\frac12(AB-BA)$$ I get the correct answer but i can't do it directly. For Example, $$A=\hat i\hat j+\hat j\hat k+\hat k\...
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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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Am I interpreting the wedge product correctly?

On "The Inner Products of Geometric Algebra" (Leo Dorst), page 39, equation 2.3, the outer product between multivector $A_r$ and $B_s$, of grades $r$, $s$, is defined as: $$A_r \wedge B_s = \...
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Irreducible representations and linear independence: a matrix irrep of dimension $d$ contains $d^2$ linearly independent matrices

I'm interested in the following statement: an irreducible representation of dimension $d$ contains $d^2$ linearly independent matrices that can be found in H. C. Lee (1948), On Clifford Algebras and ...
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Lie algebra of Spin Group

In Peskin's QFT textbook, eq.(3.23): \begin{equation} S^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}] \end{equation} gives a Lie algebra representations of Lorentz group or more generally $SO(p,q)$, ...
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Clifford algebra dimension is $2^n$, problem with part of proof that it is $\leq 2^n$

I have a problem with part of the proof about dimensionality of Clifford algebra over finite dimensional vector space, but let me just state for reference what that is (Lang, Algebra XIX.4.): Let $E$ ...
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Clifford Algebra for 2D spaces

I am studying Clifford and geometric algebra. Specifically, decomposing the line element by finding the matrices that satisfy the necessary algebraic relations. I know that for 4D spacetime, the ...
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How do I take the divergence of a multi-vector?

Say I have a multi-vector of $\mathbb{G}(2,\mathbb{R})$: $$ \mathbf{u}=a+xe_0+ye_1+be_0e_1 $$ How do I take the divergence? How do I even define it for multi-vector in general?
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What is the difference between Projective Geometric, Clifford Algebra, Grassman Algebra, Geometric Algebra, Quaternion Algebra and Exterior Algebra?

Since a few years now, Special Interest Group on Computer Graphics have been shilling this new type of algebra that they advertise fixes all the problem with Linear Algebra like no Gimbal locks, error ...
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Divergence of a vector field in geometric calculus

I have been studying geometric algebra from a couple of books, and have just moved into geometric calculus in the book by Doran-Lasenby(DL). DL shows that the vector derivative can be written as $\...
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Complete the pattern to 6D: Norm of a multi-vector.

In this article https://people.kth.se/~dogge/clifford/files/clifford.pdf, the norm of a multi-vector from 0D to 5D is given on page 91 (section 6.7). They are $$ \begin{align} N_0(x)&=x\\ N_1(x)&...
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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. [1]), 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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What is the meaning of $e_{n+i}$ in Reduced Geometric Algebra papers, when the vector part of the basis of $G_n$ only includes $e_1$ through $e_n$?

https://ieeexplore.ieee.org/document/8878099 defines some basis objects $\{\varepsilon_i \}$ for Reduced Geometric Algebra based on the basis vectors $\{e_i\}$ of geometric algebra: $$\varepsilon_i = ...
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Isomorphism between Clifford basis and Pauli matrices

Suppose If I have a Clifford multivector $MV = e_{0}+ae_{1}+be_{2}+ce_{3} $ I was seeing in some papers that Clifford basis are isomorphic to Pauli matrices $\sigma_{0},...\sigma_{3}$ so can i replace ...
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Is there a unified description of the geometric derivative?

The exterior derivative ($d$ or $\nabla\wedge$) is a very unifying concept, in that it subsumes the gradient of a scalar, the curl of a vector, and the "divergence" of a bivector, thus also ...
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Defining free Dirac operator on $\big(L^2(\mathbb{R}^n)\big)^{n+1}$

So I have been considering free Dirac operator $D_0:(L^2(\mathbb{R}^3))^4 \supset (W^{2,1}(\mathbb{R}^3))^4 \to (L^2(\mathbb{R}^3))^4$ given by formula \begin{equation} D_0 = \left[\begin{matrix} ...
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How do you calculate the magnitude of a directly represented flat in conformal geometric algebra?

A flat connected to the origin point $o$ is represented by $X=o \wedge \textbf{A}_k \wedge \infty$, where $\textbf{A}_k$ is a k-blade of directions in the Euclidean subspace, and $\infty$ is the point ...
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How to calculate $\alpha=(x \wedge b)(a \wedge b)^{-1}$ in the projection $P[x]=P[\alpha a + \beta b] = \alpha a$ using the reciprocal frame?

In Dorst et al pg. 104, there is a projection function defined such that $P[a]=a$ and $P[b]=0$ for some specific vectors $a$ and $b$. There is a plane associated with the 2-blade $a \wedge b$. A ...
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Spin$^c$-structure necessary for K-orientation

In Atiyah, Bott, and Shapiro's Clifford Modules, Theorem 12.3, they prove that a Spin$(k)$-structure (resp. Spin$^c(2k)$)-structure gives a KO(K)-orientation on the associated vector bundle of rank $k$...
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Clifford algebra of a free bimodule over a noncommutative ring

I have an apparent definition (and construction) of the Clifford algebra of a free $R$-$R$-bimodule $M$ with a quadratic form $q: M \rightarrow R$ with noncommutative $R$. I am not aware of any ...
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Differentiation in Geometric Calculus, Computing Vector Derivatives of Multivector-Valued Functions

I haven't found an explicit formula and way to compute vector derivatives in geometric calculus. For instance, let $V \simeq \mathbb{R}^3$ with the usual orthonormal basis $\{\textbf{e}_i\}_{i=1}^3$ ...
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Decide the dimension of a Clifford algebra

k is a field. The Clifford algebra $C_2(k)$ is defined as $$k\left<e_1,e_2\right> / (e_1^2+1,e_2^2+1,e_1 e_2 + e_2 e_1 = 0).$$ I know that $\{1, e_1, e_2, e_1 e_2\}$ spans $C_2(k)$. But how can ...
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Basic multivector derivatives $∂_X X = n$ and $∂_X X^2 = 2X$, etc…

Using geometric algebra, one may define the multivector derivative $∂_X$ with respect to a general multivector $X$ as $$ ∂_X ≔ \sum_J 𝒆^J (𝒆_J * ∂_X) $$ where each “component” $𝒆_J * ∂_X$ is ...
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Derivation of hodge dual of different bivectors

So, I read that the Hodge dual operator acts in the following way (you can find the following in https://en.wikipedia.org/wiki/Hodge_star_operator#Geometric_explanation): $$\alpha \ \wedge \ \star \...
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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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Invertibility conditions on multivectors

Given a split number $n=a+bj$, its inverse is $(a-bj)/(a^2-b^2)=n^*/nn^*$ A similar thing holds for complex and duel numbers. However, it fails for split numbers if $a^2=b^2$ and for dual numbers if $...
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How to recover a rotor in higher dimensions from the vectors?

I'd like to recover a rotor in geometric algebra where I know which (multi)vectors transform to given rotated (multi)vectors. I've found many references for 3D/4D dimensions (something like $\sum ...
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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&...
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