Articles of clifford algebras

Real orthogonal Lie algebra isomorphic to Clifford bivectors

I’m studying Clifford algebras on this moment, and I frequently find the statement $$\left(\mathbb{R}_m^{(2)},[\cdot,\cdot]\right) \cong \mathfrak{so}_{\mathbb{R}}(m)$$ stating that the bivectors of a real Clifford algebra are isomorphic to the real special orthogonal Lie algebra. Unfortunately I’m not able to find the original theorem proving this. Is there anyone that can help me with the finding […]

Clifford Algebra Isomorphic to Exterior Algebra

Let $E$ be a vector space over a field $k$ and $Q$ be a quadratic form, that is, $$Q:E\to k$$ such that $$Q(\lambda e)=\lambda^2Q(e)\forall\lambda\in k\,e\in E$$ and such that $P_Q:E^2\to k$ is bilinear, where $$ P_Q(e_1,e_2):=\frac{1}{2}\left[Q(e_1+e_2)-Q(e_1)-Q(e_2)\right]$$ We define the tensor algebra $T(E)$ from $E$ as $$T(E):=\bigoplus_{n=0}^{\infty}E^{\otimes n}$$where $E^{\otimes 0}\equiv k$ and $E^{\otimes n}\equiv E\otimes\dots\otimes E$ […]

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 “geometric algebra”, even though I’m using as my guide an introductory undergraduate-level2 book (Linear and geometric algebra by Macdonald). Among the general problems that I’m […]

Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford’s geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based on differential forms. Is this true? More generally, are these frameworks strictly equivalent? I have heard that geometric algebra is only a suitable approach once a metric tensor has […]

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 is? (Keep in mind that I don’t know abstract algebra, nothing except some group theory.) Does it make sense to write the sum […]

Good introductory book on geometric algebra

The title of the question already says it all but I would like to add that I would really like the book to be more about geometric algebra than its applications : it should contain theorems’ proofs. Just adding that I have never taken a course on geometric algebra. I’m a 2nd year engineering student, […]

Geometric Intuition about the relation between Clifford Algebra and Exterior Algebra

It is common to see a relation being established between the Clifford Algebra and the Exterior Algebra of a vector space. Recently reading some texts written by Physicists I’ve seem applications of Clifford Algebras on which it seems that the author is dealing with the usual exterior algebra somehow. It seems, in that case, that […]

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 let $\mathrm{Cl}_G(V)$ be the corresponding Clifford algebra. I’ll denote it by $C_G$ for short when the vector space is clear. […]