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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Clifford Algebras

What would be the best source to learn Clifford Algebras from? Anything online would suffice or any textual sources for that matter.. I'm interested in doing a project in the subject, but I'm not ...
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Proof: Clifford-Algebra representations are semisimple / completely reducible

There is a theorem: Every finite-dimensional Clifford-Algebra representation $V$ is semisimple / completely reducible, which means that it's a direct sum of irreducible subrepresentations. How this ...
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What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining $$ ab = a\cdot b + ...
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index free proof of dot product of two wedge products

I am learning geometric algebra, and meet an identy of (edited according to Andrey's comments below) $$ (a\wedge b)\cdot(c\wedge d) = (a \cdot d)(b\cdot c) - (a \cdot c)(b \cdot d)$$ as in wiki ...
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Formal Definition of Yang Mills Lagrangian

I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as $$ \mathcal ...
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What is the “grade” in geometric algebra

I'm reading a book (Linear and Geometric Algebra, by Alan Macdonald) where the author uses the term grade without ever defining it. I have a murky sense of what the grade of a blade may be (a ...
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The product of a simple k-vector with its reverse in Clifford algebra

Is there a simple proof that the product of a simple k-vector with its reverse always yields a scalar? Assume real Clifford algebra of arbitrary but finite dimension with unspecified metric. By simple ...
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Clifford Algebra Multiplication Intuition

I discovered Clifford Algebra recently and I am deeply impressed with its explanatory power and geometrical intuitiveness. I've been playing with the GAViewer ...
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Geometric algebra approach to Lorentz group representations

Background: Let $\Lambda$ be the Lorentz transformation parameterized by the asymmetric real matrix $w_{\mu \nu}$. That is, let $\Lambda = \exp(\frac{w_{\mu \nu}}{2}J^{\mu \nu})$, where $(J^{\mu ...
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Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based ...
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Odd valued dimensional number impossible to build?

Using numbers of the form $$\alpha_1+\alpha_2e_1+\alpha_3e_2+...+\alpha_ne_{n-1}$$ where $\alpha_n\in\Bbb R$ and for all $a≠b, \alpha_a≠\alpha_b$ with $e_n^2=-1$, can these numbers exist for an odd n? ...
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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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What's the Clifford algebra?

I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
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Good introductory book on geometric algebra

The title of the question already says it all but I would like to add that I would really like the book to be more about geometric algebra than its applications : it should contain theorems' proofs. ...
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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 of diagonal quadratic form

Just curious as I am studying quadratic forms. Is there a special way of viewing the Clifford algebra $C(q)$, given the diagonal quadratic form $q = \langle a_1, a_2, \ldots, a_n\rangle$, where $a_i ...
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When does the next complex split occur?

So I was thinking about complex numbers and how they came about and someting interesting occured to me: the formation of complex numbers occurs because there exists a function (namely $f(x)=x^2$) ...
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What is an even Clifford algebra?

I'm reading a paper and he defines $C_0(f)$ to be the "even Clifford algebra over $R$ associated to $f$", where $R$ is a principal ideal domain and $f$ is a non-degenerate ternary quadratic form. What ...
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Clifford Algebra is driving me crazy

How do we determine the Clifford Algebra of a quadratic form $q$? Thanks!
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Quadratic forms and Clifford Algebra Part 2

So just to ask, if $q(x, y) = ax^2 + by^2$ is a quadratic form in two variables over a field $K$ ($a, b \in K$) with char $K \neq 2$, how is $C(q)$ isomorphic to $M_2(K)$?
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Properties of different matrix representations of clifford algebras

I am looking for some theorems about matrix representations of Clifford algebras. Let $a \in G_{p,q,r}$, where $p$ elements square to $1$, $q$ to $-1$, and $r$ to $0$, that is ...
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Distinguishing the two irreducible representations of odd-dimensional complex Clifford-Algebras

The complex Clifford algebra $A$ of a complex, non-degenerate quadratic space $(V,q)$ of odd dimension $2k+1$ admits up to isomorphism exactly two non-trivial, irreducible and finite-dimensional ...
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An operator in clifford algebra

In my research I found it useful to define a following operator ($^{*}$) in Clifford algebra: If $a \in G_{p,q}$, (where $p$ is number of elements that square to $1$, and $q$ is the number of elements ...
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Solving a geometric algebra equation

I'm reading Hestenes' book "New Foundations of Classical Mechanics" as an introduction to Geometric (Clifford) Algebra. Don't worry, no physics mentioned here :) An exercise asks to solve a vector ...
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Showing: If $w\in C\ell^1(V,Q)$ anticommutes with all $v\in V$, then $w=0$

Show that if an element of the odd part of the Clifford Algebra anticommutes with everything in the vector space, then it is 0. Been having a really hard time making any progress with this one.
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Opposite Clifford-Algebra

for a symmetric bilinearform $\beta$ on a $\mathbb{K}$-vectorspace $V$ the associated Clifford Algebra $Cl(\beta)$ is the associative algebra with unit subject to the relations $$v\cdot ...
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Automorphism groups of real clifford algebras

I'm sure someone has already worked-out what all the relevant groups really are; my question is about how signature duality interacts with these groups. So, by an awful calculation, and choosing a ...
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Vectors in Clifford Algebra

I'm studying Clifford Algebra $\mathcal{Cl}_2$ and got stuck in an exercise: Let $\mathbf{a}=e_2+e_{12},\quad \mathbf{b}=(1/2)(1+e_1).$ Compute $\mathbf{ab}$. The answer is zero, but I can't get to ...
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Claim in Atiyah and Bott and Shapiro's paper on Clifford modules.

In Atiyah, Shapiro, and Bott's paper on Clifford modules, they prove Proposition 4.2 on page 11 that there are isomorphisms $C_k\otimes_\mathbb{R} C_2^\prime\cong C_{k+2}^\prime$ and ...
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Why does $C_{n+8}\approx C_n\otimes_\mathbb{R}M_{16}(\mathbb{R})$?

In an earlier answer, rschwieb kindly pointed me in the direction of Bott periodicity. Just out of curiosity I was reading through a paper on periodicity of Clifford algebras. There was a list of ...
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Curious about extending some Clifford algebra isomorphisms?

I've been considering the Clifford algebra over $\mathbb{R}$. For notation, I denote the clifford algebras $C_n$ associated with the vector space $\mathbb{R}^n$ with negative definite form, and $C'_n$ ...
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Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form? [duplicate]

I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question. Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...
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Quotient of a Clifford algebra by its radical is a Clifford algebra?

I'm fumbling a bit in my reading on Clifford algebras. I'm hoping someone can shed some light on the following isomorphism. Suppose you have a symmetric bilinear form $G$ over a vector space $V$, and ...
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Center of Clifford Algebra depending on the parity of $\dim V$?

While reading about the structure of Clifford algebra, there were two facts listed as bullet points about the center of Clifford algebra based on the parity of the dimension of the underlying vector ...
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On Chevalley's linear identification of the Clifford algebra $C(\mathbf p)$ with the exterior algebra $\wedge \mathbf p$

In reading Sternberg's notes on Clifford algebras and spin representations (page 148) I encountered the following: "...Consider the linear map $$C(\mathbf p)\rightarrow \wedge \mathbf p, x\mapsto ...
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Associativity of Moyal-like products

The Moyal product of two smooth functions $f,g$ on $\mathbb R^{2n}$ can be defined as $$ f\star g = \exp\left(-\omega^{ij} \frac{\partial}{\partial y^i} \frac{\partial}{\partial z^j}\right) f(y)g(z) ...