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### Clifford algebra non-zero

This should be a very easy question, but the proof in Lawson/Michelson (Spin geometry) is wrong and I do not find a really correct and complete argument:
Let V be a nonzero real vector space with ...

**3**

votes

**1**answer

703 views

### On matrix representations of the Clifford algebras of type $Cl(0,n)$

Can matrix representations of clifford algebras of type Cl(0,n) be found? Specifically for even orders

**17**

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**6**answers

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### Pythagorean 5-tuples

What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...

**12**

votes

**2**answers

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### Clifford PBW theorem for quadratic form

Update: now with a question 2 which is much more elementary (and should be well-known!).
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:L\to k$ be a quadratic form, i. e., a ...

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vote

**2**answers

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### Invariant space of lifted Chevalley automorphisms of the tensor algebra

Question. Let $k$ be a field of characteristic $0$. Let $L$ be a $k$-vector space. Consider the subspace $S$ of $L\otimes L\otimes L\otimes L$ spanned by all tensors of the form
$\left[a,\left\lbrace ...

**0**

votes

**1**answer

792 views

### Clifford Algebra and Gamma matrices: is this relation generally true for any dimension?

I expect the following relation to be vanishing. But it seems not that obvious.
$\Gamma_{ab}^{\lambda}t^at^b \Gamma_{\lambda c(d)}t^c=0$
where $t^a$ are even ghosts, "$ab$" are indices for matrix ...

**7**

votes

**5**answers

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### Representations of Pin vs. Representations of Clifford

This may be total nonsense. But I need to know the answer quickly and I am too tired to think about it thoroughly. Let $k$ be a positive integer. Roe's "Elliptic Operators" claims that there is a ...

**16**

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**3**answers

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### How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best ...

**7**

votes

**3**answers

2k views

### Clifford Algebra in Dirac Equation

I am wondering if there is any mathematical (or physical, besides the fact that classical quantum mechanics uses complex numbers) justification for why the complexified (1,3) Clifford algebra is used ...

**13**

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**7**answers

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### Why is Riemann-Roch an Index Problem?

I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..."
Then right after that he argued in favour of such a sentence. ...

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### Clifford algebra as an adjunction?

Background
For definiteness (even though this is a categorical question!) let's agree that a vector space is a finite-dimensional real vector space and that an associative algebra is a ...