# Questions tagged [clifford-algebras]

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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### What is the proof of the inner product formula between a vector and an r-blade?

I would like to prove the formula $a\cdot A_r = \frac{1}{2}(aA_r - (-1)^r A_r a)$ for a grade-1 vector $a$ and a simple r-blade $A_r$. How? In Geometric Algebra for Physicists (Doran & Lasenby), ...
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### How to we square unit vectors to 1,-1, or 0? What does ℜ( p, q, r) means?

My Geometric Algebra book starts with the following: Which I did not understand a bit, and went on with the book hoping for it to be further explained. It turns out that doesn't happen! I'm reading ...
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### The bases of Clifford algebra

In order to get the bases of Clifford algebra, I want to understand the proof of the following proposition. Proposition: Let $Cl(V)$ be the Clifford algebra of a $\mathbb{R}$-vector space $V$ with ...
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### algebraic structure of $\otimes_{\Bbb R}$ in $\text{Spin}^c(V)\subset Cl(V)\otimes_{\Bbb R} \Bbb C$

I hope to know a clear explanation on the tensor product notation: $\otimes_{\Bbb R}$: Suppose we write $A \otimes_{\Bbb R} B$, what is the required algebraic structure of $A$ and $B$? What is the ...
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### Rewriting the Geometric Product in 3D Space?

I'm reading a passage from Geometric Algebra for Physicists, and in section 2.4.1 there's a train of logic I'm not quite understanding. I'll copy it word for word, with italics being my comments/...
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### Why are Clifford Algebras Superalgebras?

In Mathematical Gauge Theory With Applications to the Standard Model of Particle Physics Hamilton on page 332 proves that Clifford algebras are superalgebras by the following: First he defines two ...
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### Spin group for the complexified Clifford algebras

The $Spin(p,q)$ group is usually defined as $$Spin(p,q) = \{s \in C\ell^{+}_{p,q}: s \tilde{s}=1\}$$ where $s$ belongs to the Lipschitz group and $C\ell^{+}_{p,q}$ denotes the even element subset of ...
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### Generalizing norms: "$\lambda$-pseudonorms"

The study of norms is generally limited to those that are most interesting, most notably the Euclidean $2$-norm (and rarely the Manhattan $1$-norm, and supremum $\infty$-norm). However, I can ...
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### Clifford map is injective for $B=0$.

Here's my definition of a Clifford algebra: Definition: Let $B(\cdot,\cdot)$ be a symmetric bilinear form on a vector space $V$ over $\mathbb{K}$ and $Q$ its associated quadratic form. The Clifford ...
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### Triality with even and odd spinors

The spin group with mixed metric signs ($p$ for positive and $q$ for negative) is usually defined as $$Spin(p,q) = Pin(p,q) \cap C\ell^{+}_{p,q}\,,$$ and in the special case of $Spin(8,0)$ one has ...