The natural Clifford algebra of a vector space

[Manfred]

I don't get it.  The GA community seems to be ignoring the most beautiful GA out there – one that gets close to meeting Hestenes's overblown claims for GA.

Various authors in GA have bemoaned the absence of a known GA that contains the projective transformations, expresses quadrics, etc. [Chris Doran & Anthony Lasenby (2003) Geometric Algebra for Physicists. p. 413; Leo Dorst, Daniel Fontijne, Stephen Mann (2007) Geometric Algebra for Computer Science – An Object-oriented Approach to Geometry, pp. 369, 499; Christian Perwass (2009) Geometric Algebra with Applications in Engineering]  Yet such a beast is known, and has been for a while – before their lamentations [e.g. José María Pozo & Garret Sobczyk (2002) Geometric Algebra in Linear Algebra and Geometry, p. 4].

Okay, so let me back up.  Consider what I will call the natural Clifford algebra of (the direct sum of) a vector space V and its dual V*: Cl(V⊕V*,q).  V and V* come with a natural pairing, namely ⟨a,x⟩ = a(x), a ∈ V*, x ∈ V, and the natural quadratic form is q(a+x) = ⟨a,x⟩.  This algebra has all sorts of magic when it comes to a natural interpretation.  You might object that this algebra has dimension 2^2n (n = dim V) in place of a GA of dimension 2^n, but ... look at the list of magic below.

• This construction works for any free module of finite dimension over a commutative ring (which includes characteristic 2): it is very general.

• We can choose any basis of V and its dual basis in V*.  Together these constitute a null basis, which makes manipulation particularly convenient and efficient (many products of terms vanish).

• We do not need a multitude of awkward constructs that we usually tack onto GAs, since these objects and operations are now within the algebra (i.e. we need only addition, multiplication and inverse):
• The geometrically natural ΛV and ΛV* are subalgebras of the algebra.  No need for an additional product ∧.
• Any invertible linear map on V or V* is expressible within the algebra: T Y T^−1.  The stress–energy tensor is an example.
• The main involution has a simple algebraic form: J Y J^−1, where J is the top element, which can always be normalized so that J^2 = 1.
• We can represent the contraction of a ∈ V* onto ΛV within the algebra.  Apparently no need for the additional contractive products ⌋ and ⌊.
• Every linear map on the entire algebra (not only V or V*) is expressible within the algebra.  Hence, non-algebraic operations like grade projection are unneeded.  (Consider that the main involution is not algebraically expressible in Cl(R^{1,0}) and Cl(R^{0,1}))

• Any quadric is expressible as a single element D: the solutions y ∈ V such that y D y = 0.

• We can simultaneously represent as elements of the algebra multiple nondegenerate bilinear forms (e.g. metrics) on V.

• It is highly intuitive.  Geometric objects are k-vectors and k-covectors, and invertible transformations are a elements of a particular group, all separate sets.  Having separate sets allows us to identify geometrically nonsensical operations like contracting a covector onto a k-covector or onto a transformation, and to prohibit this say at compilation in C++.

• The exterior derivative and divergence operators are straightforward (with an explicit metric element G needed in the latter).

• Projective geometry with its transforms in any dimension is easily expressible (this is obvious, given the general linear transforms).

• Counter-intuitively, due to the null basis, the algebra seems to be very efficient computationally, despite its high dimension (2^2n).  For example, a general linear transform in 2D has 6 coefficients, and in 3D has 20 coefficients.  Because of the null basis, linear transformations T y T^−1 simplify (e.g. to exactly n^2 multiplications on 1-vectors with pre-calculation on T).

I'm hoping someone will tell me this is all well-known – just to go and look in this or that text or paper.

Manfred

[garret sobczyk]

Dear Manfred,

This is all well-known - (for just one example) just go and look up the article by Z. Oziewicz, "From Grassmann to Clifford", pp. 245-255, in Clifford Algebras and Their Applications  in Mathematical Physics, edited by J.S.R. Chisholm and A.K. Common, NATO ASI Series, Vol. 183, Advanced Science Institutes Series, (1986). The simplest way of introducing Clifford geometric algebra to students, or anybody who knows how to multiply matrices, I have set down in my new book, "Matrix Gateway to Geometric Algebra, Spacetime and Spinors", November 7, 2019. See https://www.garretstar.com for publication details. The first 3 Chapters of this book are elementary in nature tying a geometric algebra directly to its coordinate matrix of components. The method is also closely related to the vector-dual space approach which you are advocating.

Garret Sobczyk

https://www.garretstar.com

--
You received this message because you are subscribed to the Google Groups "Geometric_Algebra" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geometric_alge...@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/geometric_algebra/bf9bb683-52f6-4e7e-afd0-442e0541f2a0%40googlegroups.com.

[Manfred]

Hi Garret - (also thanks to Bill Page for the PDF).

It seems clear (from the Oziewicz PDF, your 2002 paper and Doran & Lazenby in 2003) that the theory of a "natural geometric algebra" is well-enough understood by some, including its general linear transforms, but I have not seen it being developed into an applied algebra, or pursued for its efficiency.  That could be an artefact of my poor research skills, lack of access to academia, and failure to investigate what software is out there.  I would expect considerable interest in the applied GA community, given its efficiency versus expressive power.  I expect it to be amenable to implementation with small C++ templates.

I was too harsh on some of the references that I mentioned: two (yours and Doran & Lazenby) acknowledge the object and what it can do, but do not elaborate.

Out of interest, I have ordered your book, though the matrix approach might not hold much appeal for me.  So far I have used it only for exploration of ideas, but little else.

Manfred

To unsubscribe from this group and stop receiving emails from it, send an email to geometri...@googlegroups.com.

[garret sobczyk]

Hi Manfred - nice to hear from you. I am attaching my author's copy of my paper "Notes on Plucker's relations in geometric algebra". It discusses in greater detail the material I have in Chapter 6 in my new book. It also discusses the question of the proper place/representation of a  dual space in Clifford's geometric algebra. It certainly has been a nagging question for many years. I introduce the idea of a "geometric matrix" as simply a real or complex coordinate matrix of a geometric number in a geometric algebra, which also happens to provide an isomorphism between the algebras. I have had many discussions with my friend and colleague Zbigniew Oziewicz over many years, which is why I recognized the familiar questions that you posed.

Best,

Garret

To unsubscribe from this group and stop receiving emails from it, send an email to geometric_alge...@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/geometric_algebra/922b2f41-e4c4-4ffd-8f6d-bb7afd6660b5%40googlegroups.com.