# Tagged Questions

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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 finite-...
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### What are the “correct” conventions for defining Clifford algebras?

I have three related questions about conventions for defining Clifford algebras. 1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...
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### Why do Clifford algebras determine $KO$ (and $K$-)-theory?

In the paper "Clifford modules" by Atiyah-Bott-Shapiro, they construct a family of Clifford algebras $C_k$ over the real numbers, so that $C_k$ is the algebra associated to a negative definite form on ...
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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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### Examples of Clifford Modules

For a Riemannian manifold $M$, a Clifford module is a bundle over $M$ that is, fiberwise, a representation of the Clifford algebra bundle $Cl(TM)$ and has a connection compatible with this action. ...
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### Which real Pin groups agree?

In the Lie theory notes on my website it is claimed (Example 7.3.3.5) that $\mathrm{Pin}(4,0)$ and $\mathrm{Pin}(0,4)$ are not isomorphic. As Nigel Ray pointed out to me, this claim is not quite ...
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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 ...
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### What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero. If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...
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### Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over $\mathbb{R}$)....
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### Spin and SO groups associated to a degenerate symmetric bilinear form

In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But the rest of the book ...
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### Is there a way to embed Clifford algebras into the corresponding tensor algebra?

$\newcommand{\talg}{\mathcal{T}(V)}$$\newcommand{\clalg}{\mathcal{Cl}_q(V)}$$\newcommand{\qalg}{\mathcal{I}_q(V)}$Is there a way to embed Clifford algebras into the corresponding tensor algebra? ...
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### Finitely generated projective modules over the algebra of sections of the Clifford bundle

Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...
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### Is an associative division algebra required for this phenomenon?

For which integers $d \geq 1$ can we find real matrices $R_1, \dotsc, R_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R_1 v, \dotsc, R_d v$$ is an orthonormal basis? ...
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### Clifford Lie Algebras

I'm studying the "Clifford Lie Algebra" (see http://arxiv.org/pdf/1007.2481.pdf page 30). It's basically a way to look at Clifford algebras and their properties in a Lie algebraic setting (which I ...
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### Mathematica package for supergravity and string theory

I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
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### Exceptional symmetric spaces embedded in exceptional Lie group

In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective ...
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### Element in spin group

I've got the following question: why is it true (if it really is?), that if I have a unitary element $u$ in the (real) Clifford algebra $Cl(V,g)$ which is even and the operator $\varphi(u)$ defined ...
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### Discriminants of Clifford algebras

I have a Clifford algebra defined over a field of characteristic not equal to $2$. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...
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### 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
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### How can the Cayley table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$, $e_1=i$, $e_2=j$ and so on. I'm looking for an ...
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### Automorphisms of Clifford Algebras

What are the automorphisms of real Clifford algebras $Cl_{n,0}$? Of course, I'm interested in the case where they are not central simple.
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### What is the relation (if any) between Clifford algebras and Azumaya algebras?

Suppose the base field is $\mathbb{C}$ and the Clifford algebra is the classical one (i.e. associated to a quadratic form in $n$ variables). It seems that there are relations between Clifford algebras ...
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### $Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here. Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi$ is an automorphism of $O(2n)$, and ...
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### Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$. Q1) Is it ...
In order to ask the question in the title more precisely, let me recall some standard stuff introduced in [1; Atiyah, Bott, Shapiro]. Suppose $X$ is a compact CW complex and $V \to X$ is an oriented ...