3
votes
1answer
53 views

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 ...
4
votes
2answers
94 views

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 ...
0
votes
1answer
87 views

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 ...
1
vote
3answers
189 views

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, ...
2
votes
1answer
50 views

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 $\mathcal{Cl}(V,q)$?
1
vote
0answers
54 views

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 ...
5
votes
2answers
139 views

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 ...
1
vote
0answers
59 views

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 ...
2
votes
0answers
71 views

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 ...
3
votes
1answer
44 views

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 ...
1
vote
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 ...
2
votes
3answers
313 views

What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining $$ ab = a\cdot b + ...
3
votes
2answers
203 views

What is the “grade” in geometric algebra

I'm reading a book (Linear and Geometric Algebra, by Alan Macdonald) where the author uses the term grade without ever defining it. I have a murky sense of what the grade of a blade may be (a ...
2
votes
1answer
57 views

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 ...
-2
votes
1answer
153 views

Clifford Algebra is driving me crazy

How do we determine the Clifford Algebra of a quadratic form $q$? Thanks!
1
vote
1answer
101 views

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 $M_2(K)$?
3
votes
1answer
339 views

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 ...
1
vote
1answer
129 views

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 ...
12
votes
3answers
2k views

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 ...
9
votes
1answer
240 views

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 ...
1
vote
1answer
91 views

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 ...
0
votes
2answers
69 views

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 $C'_n$ ...
4
votes
1answer
168 views

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 ...
5
votes
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 ...
4
votes
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 ...