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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multiplication table for conformal geometric algebra?

I'm trying to find the full multiplication table for conformal geometric algebra (should be a 32 by 32 matrix). It does not seem to be available in explicit form anywhere on the web. Can anybody help ...
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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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How do you show that the Laplacian is the square of the (Euclidean) Dirac operator?

If I understand correctly, the Euclidean Dirac operator is given by $$D=\sum_{i=1}^n e_i \frac{\partial}{\partial x_i},$$ where $e_i$ are bases for $Cl_{0,n}(\mathbb{R})$, i.e., the $n$-dimensional ...
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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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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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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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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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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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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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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) ?
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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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Choosing a spinor space

I still have a hard time on trying to fully understand spinors (on real spaces). Let $V$ be a $n$-dimensional real quadratic space $V$ with a non degenerated signature $(p, q)$. From what I had ...
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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 ...