# Tagged Questions

1answer
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### Clifford Algebras in Characteristic Two

A Clifford algebra $Cl(V)$ for a vector space $V$ is defined to be the quotient of the tensor algebra $T(V)$ with the ideals generated by elements of the form $v \otimes v - (v,v)1$, where $(,)$ is ...
2answers
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### Identifying the algebra

In order to solve an obscure (physics) problem I have been considering whose details are not important, I am looking for elements (I am thinking in terms of matrices and their products but this may ...
1answer
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### A question on the dual relationship between the regressive product and the exterior product

I am trying to understand the following sentence, which I came across in a book: The underlying beauty of the Ausdehnungslehre is due to this symmetry [the duality between the regressive and ...
3answers
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### Is Geometric Algebra isomorphic to Tensor Algebra?

Is geometric algebra (Clifford algebra) isomorphic to tensor algebra? If so, how then would one relate a unique 2-vector (this is what I'm going to call a multivector that is the sum of a scalar, ...
1answer
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### Number of Involutive Automorphisms on a Clifford Algebra

Let $V$ be a vector space with dimension $n$ and $q$ a quadratic form on $V$. How many involutive automorphisms are there in $\mathcal{Cl}(V,q)$?
0answers
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### What are the “irreducible factors” in an algebra?

What are the irreducible factors in an algebra? In the book "Spin Geometry" by Lawson and Michelsohn, page 35, line 1, there occurs "take irreducible factors of...", but I don't know what irreducible ...
2answers
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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 ...
0answers
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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 ...
0answers
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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 ...
1answer
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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 ...
4answers
373 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 ...
3answers
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1answer
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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 ...
2answers
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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 $C'_n$ ...
1answer
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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 ...
0answers
119 views

### 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 ...
1answer
382 views

### 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 ...