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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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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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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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 $$v\cdot ...
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Are Clifford algebras and differential forms equivalent frameworks for differential geometry?
I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based ...
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When does the next complex split occur?
So I was thinking about complex numbers and how they came about and someting interesting occured to me:
the formation of complex numbers occurs because there exists a function (namely $f(x)=x^2$) ...
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Automorphism groups of real clifford algebras
I'm sure someone has already worked-out what all the relevant groups really are; my question is about how signature duality interacts with these groups.
So, by an awful calculation, and choosing a ...
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Geometric algebra approach to Lorentz group representations
Background:
Let $\Lambda$ be the Lorentz transformation parameterized by the asymmetric real matrix $w_{\mu \nu}$. That is, let $\Lambda = \exp(\frac{w_{\mu \nu}}{2}J^{\mu \nu})$, where $(J^{\mu ...
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On Chevalley's linear identification of the Clifford algebra $C(\mathbf p)$ with the exterior algebra $\wedge \mathbf p$
In reading Sternberg's notes on Clifford algebras and spin representations (page 148) I encountered the following:
"...Consider the linear map $$C(\mathbf p)\rightarrow \wedge \mathbf p, x\mapsto ...
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Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form?
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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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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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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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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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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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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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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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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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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Associativity of Moyal-like products
The Moyal product of two smooth functions $f,g$ on $\mathbb R^{2n}$ can be defined as
$$
f\star g = \exp\left(-\omega^{ij} \frac{\partial}{\partial y^i} \frac{\partial}{\partial z^j}\right) f(y)g(z) ...
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Clifford Algebra of diagonal quadratic form
Just curious as I am studying quadratic forms. Is there a special way of viewing the Clifford algebra $C(q)$, given the diagonal quadratic form $q = \langle a_1, a_2, \ldots, a_n\rangle$, where $a_i ...
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Mathematical significance of the “Dirac conjugate”
Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ ...
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Why does $C_{n+8}\approx C_n\otimes_\mathbb{R}M_{16}(\mathbb{R})$?
In an earlier answer, rschwieb kindly pointed me in the direction of Bott periodicity. Just out of curiosity I was reading through a paper on periodicity of Clifford algebras. There was a list of ...
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Showing: If $w\in C\ell^1(V,Q)$ anticommutes with all $v\in V$, then $w=0$
Show that if an element of the odd part of the Clifford Algebra anticommutes with everything in the vector space, then it is 0.
Been having a really hard time making any progress with this one.
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Odd valued dimensional number impossible to build?
Using numbers of the form $$\alpha_1+\alpha_2e_1+\alpha_3e_2+...+\alpha_ne_{n-1}$$
where $\alpha_n\in\Bbb R$ and for all $a≠b, \alpha_a≠\alpha_b$ with $e_n^2=-1$, can these numbers exist for an odd n? ...
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An operator in clifford algebra
In my research I found it useful to define a following operator ($^{*}$) in Clifford algebra:
If $a \in G_{p,q}$, (where $p$ is number of elements that square to $1$, and $q$ is the number of elements ...
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Quadratic forms and Clifford Algebra Part 2
So just to ask, if $q(x, y) = ax^2 + by^2$ is a quadratic form in two variables over a field $K$ ($a, b \in K$) with char $K \neq 2$, how is $C(q)$ isomorphic to $M_2(K)$?
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The product of a simple k-vector with its reverse in Clifford algebra
Is there a simple proof that the product of a simple k-vector with its reverse always yields a scalar? Assume real Clifford algebra of arbitrary but finite dimension with unspecified metric. By simple ...
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Matrices defining Clifford Algebra
Given the two matrices:
$\sigma_i$ and $\sigma_j$
we can construct a Clifford algebra based on the anti commutator rule:
$$\{\sigma_i,\sigma_j\}=\delta_{ij}1$$
where $\delta_{ij}$ is the Kronecker ...
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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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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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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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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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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 ...
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Clifford Algebra is driving me crazy
How do we determine the Clifford Algebra of a quadratic form $q$? Thanks!
