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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How to prove the tensor product of two copies of $\mathbb{H}$ is isomorphic to $M_4 (\mathbb{R})$?

How to prove the tensor product over $\mathbb{R}$ of two copies of the quaternions is isomorphic to the matrix algebra $M_4 (\mathbb{R})$ as algebras over $\mathbb{R}$? More precisely, the problem ...
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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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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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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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condition for a bivector to be simple

If $B$ is a bivector and $x$ is a vector, I have to prove that Bivector $B$ is simple with $x$ (i.e. there's some vector $y$ such that $B=x\wedge y$) iff $B\wedge x = 0$ Now, for necessity part: ...
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Clifford Algebra with Degenerate Bilinear form

This is a follow up question to this question on Clifford Algebras. As I understand it, if the associated bilinear form $\beta$ of a Clifford Algebra $C_{\ell}$ is non-degenerate, then $ker\beta$ is ...
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109 views

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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324 views

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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123 views

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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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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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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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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430 views

In Geometric Algebra, how would one express the result of a tensor product in the language of GA?

Thanks for your time and effort. I appreciate your help. I'm new to geometric algebra and I get that it supersedes linear algebra. I was wondering though how I could learn to express a tensor ...
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383 views

Within Geometric Algebra, is there a way to express a linear operator within the language of geometric algebra?

Thanks for your time and effort. I appreciate your help. I'm new to geometric algebra and I get that it supersedes linear algebra. I was wondering though how I could learn to take an operator in ...
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196 views

In Geometric Algebra, is there a geometric product between matrices?

Thanks for your help in advance. I literally just started to self-study about geometric algebra. I have some coursework background in linear algebra and was trying to make an educational bridge ...
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183 views

Clifford Algebra for understanding Atiyah Singer Index Theorem Reference Request

I am interested in studying Atiyah Singer Index Theorem and Spin Geometry and would like to study Clifford Algebras and their representations for this purpose. I have a book 'Clifford Algebras : An ...
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440 views

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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134 views

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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607 views

Looking for a clear definition of the geometric product

In brief: I'm looking for a clearly-worded definition1 of the geometric product of two arbitrary multivectors in $\mathbb{G}^n$. I'm having a hard time getting my bearings in the world of ...
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3answers
424 views

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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108 views

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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254 views

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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517 views

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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Composition of two axis-angle rotations

Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar ...
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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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200 views

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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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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676 views

Mathematical significance of the “Dirac conjugate”

Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ ...
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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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214 views

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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384 views

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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156 views

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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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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1answer
375 views

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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268 views

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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163 views

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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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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Are there simple examples of Banach spaces with no non-trivial Clifford Isometries?

By a Banach space $X$ I mean, a complete normed vector space and by a Clifford isometry I mean a surjective isometry $\gamma$ of $X$ such that the distance $d(\gamma x, x)$ is constant on $X$. ...
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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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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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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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222 views

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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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 ...