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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product $abababab…$ in clifford algebra

Let $a,b$ are vectors in vector space $V \leq \mathcal{Cl}_n(V)$. I would like to know if product $ababab...ab=(ab)^r$ can be written in form $\sum_{\alpha \in A} F_\alpha(a) G_\alpha(b)$. For some ...
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Seeking intuitive explanation of Clifford Algebra

Is there a simple intuitive graphical explanation of Clifford Algebra for the layman? Since Clifford Algebra is a Geometric Algebra, surely the best way to present those concepts is with graphical ...
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Set of 4 anticommutative matrices

How would you go about showing that there cannot be a set of four 2 by 2 matrices that satisfy the anticommutative relation $AB + BA = 0 $ or $2I$ if $A=B$? i.e minimum order has to be 4. I know that ...
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troubles showing existence of Clifford-algebra

We had the following definition in class: Let $V$ be a vector space, $K$ a field and $Q$ be a quadratic form. We call $(C(V,Q),j)=C$ a Clifford-algebra if: $C$ is an assoziative algebra with 1, ...
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About the definition of norm in Clifford algebra

I have seen two definitions for the norm in the Clifford algebra $\mathrm{Cℓ}_{p, q, r}$. According to one of them $\Vert x\Vert = ⟨x. x^\dagger⟩_0$, where the dagger stands for the reversal of the ...
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Ideal of $TV$ which trivially intersects $V$

Let $V$ be a vector space over a fied $\mathbb{K}$ and let $TV = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ be its tensor algebra. Let $b \colon V \times V \rightarrow \mathbb{K}$ be a bilinear form ...
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Space time algebra isomorphic to matrix algebra

i have the following problem: I already know that there exists representation of the Clifford Algebra of the Minkowski space $\mathcal{C}l(M,\eta)$. Here $M$ denotes the Minkowski space and $\eta$ ...
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Bivector as a sum of exterior products of basis vectors.

Prove that for any 2-vector $\alpha=\Lambda^2(V)$ there is a basis $\{e_1,\ldots, e_k\}$ of $V$ such that $\alpha= e_1\wedge e_2 + e_3\wedge e_4 + \ldots +e_{k-1}\wedge e_k$, where $\wedge$ denotes ...
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Hermiticity of a Matrix representing a Clifford alg. element

I'm interested in representations of the elements of a Clifford algebra, $\gamma^a$ with $a\in \{1,...,n\}$, such that $(\gamma^a)^2 =\pm 1$. If these were numbers, for those $\gamma^a=1$ one have ...
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Does Clifford algebra depend on the topology of manifold?

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or ...
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An expression for $e_1e_2e_3e_4$

In a real Clifford algebra over ${\mathbb R}^4$ with the quadratic form defined on the orthogonal basis $e_1,e_2,e_3,e_4$ by $e_1^2=1, e_2^2=1, e_3^2=-1$, and $e_4^2=-1$, find an even number of ...
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Difference between $P(V,q)$ and $\tilde P(V,q)$ (about Lawson, Michelson “Spin Geometry”, page 19)

I'm confused by discussion on page 19 of the "Spin Geometry" book by Blaine Lawson and Marie-Louise Michelson. It claims, that when field $k$ is a spin field, $\widetilde P(V,q)/P(V,q)$ is either $0$ ...
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Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
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Interpretation of a line integral in complex analysis

$\newcommand{\C}{\mathbb{C}}$ Suppose $f\colon \Omega\subset \C\to\C$ is a holomorphic function and $\gamma:[0,1]\to\Omega$ is a continuous path. If $\Omega=\C\setminus\{0\}$, $\gamma(t):= e^{2\pi i ...
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Solving equation in Clifford algebra

Let $g_i$ denote the $i$'th orthonormal basis of a Clifford algebra $G_{p,q}$, then any number in Clifford domain can be represented as follows: $$a=\sum_{i=1}^{2^{p+q}}(a_ig_{i})$$ where $a_i$ is ...
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The pushforward under the left action in the group of units of a Clifford algebra

The following I know to be true: let $A$ and $B$ be elements of $GL(m,\mathbb{R})$ and let $X \in T_BGl(m, \mathbb{R})$ and let $L_A:Gl(m, \mathbb{R}) \to GL(m, \mathbb{R})$ be the left multiplication ...
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Clifford algebra and Spin group of 4-dimensional Euclidean space

I’m seeking for a straightforward construction of well-known $\mathrm{Spin}(4) = \mathrm{Spin}(3)\times\mathrm{Spin}(3)$ isomorphism using geometric algebra-based definition of “Spin”, without ...
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Geometric algebra and quantum field theory

How does the reformulation of QFT with GA look like? I read that GA can be applied to almost every kind of physics, but QFT is rarely mentioned. Is there a lot of research going on in this direction ...
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What are the “irreducible factors” in an algebra?

What are the irreducible factors in an algebra? In the book "Spin Geometry" by Lawson and Michelsohn, page 35, line 1, there occurs "take irreducible factors of...", but I don't know what irreducible ...
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An $SU(3)$ isomorph in Clifford $G(5,0)$?

I am a computer scientist using the geometric (Clifford) algebras $G(n,0)$ over $\mathbb{Z}_3 = \{0,1,-1\}$ to describe distributed computations. My question concerns $G(5,0)$ with generators ...
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Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
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Hermitian conjugation and representations of the Lorentzian Clifford algebras

The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma ...
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Clifford algebra over non-Archimedean field

Usually the Clifford algebra is defined over the Reals $\mathbb{R}$ or the Complex $\mathbb{C}$ numbers. Can the definition be extended over non-Archimedean fields, such as the hyperreal numbers ...
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Suggestions for Clifford Analysis

I'm trying to learn Clifford Calculus and I can't find introductory level books. I found some nice lecture notes of José Figueroa-O'Farrill when I was studying Clifford Algebra: ...
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Element of the spin group

I've got the following question: why it true, that if I have a unitary element $u$ in the (real) Clifford algebra $Cl(V,g)$ which is even and the operator $\varphi(u)$ defined via ...
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Definition of the Spin^c group

Reading some notes about spin geometry I met two definitions of $Spin^c(n)$ group: one as $Spin(n) \times_{\mathbb{Z}_2} \mathbb{S}^1$ (product $Spin(n) \times \mathbb{S}^1$ divided by ...
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A question on orthogonal groups and Clifford Algebras

Does someone know if the following orthogonal groups have some special or distinctive property and their associated Clifford algebra: O(3), O(6), O(91) ?