I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance.

Question: I am seeking further exposition of the material in Chapter 4 or 5 of the text. I have prolonged some tableaus, but, I have doubts and would like some additional examples. Can anyone point me to some such exposition on explicit PDEs and Tableaus ?

Let me sketch the goal of the project as it might give insight into which resources we should seek. We have spent some time studying calculus over an associative real algebra. In short, when a function is $\mathcal{A}$-differentiable it must satisfy the $n^2-n$ generalized Cauchy Riemann equations, but, on the level of the algebra variables it's just calculus. On the other hand, we wonder, when can we take a given PDE or system of PDEs and gain insight into the solutions of the system by studying it in terms of the $\mathcal{A}$-calculus. For a basic example, obviously $\mathbb{C}$-variables gives insight into the solutions of Laplace's equation on $\mathbb{R}^2$.

My naive hope was that the machine in Cartan for Beginners could be put to work in ciphering whether a given system of PDEs was compatible with an $\mathcal{A}$-calculus substitution. I hoped there was some natural way to use the Tableaus to judge compatibility. Maybe the hope is misguided.

Added: $\mathcal{A} = \mathbb{R}^n$ paired with an associative multiplication over $\mathbb{R}$. To say $f: \mathcal{A} \rightarrow \mathcal{A}$ is $\mathcal{A}$-differentiable is to say $J_f$ is in the left-regular (matrix) representation. This is equivalent to insisting the algebra multiplication factors out of the differential; $df(v \star w) = df \star w$. For example, $\mathcal{A} = \mathbb{R} \oplus j \mathbb{R}$ with $j^2=1$ gives $$ J_f = \left[ \begin{array}{cc} a & b \\ b & a \end{array}\right]$$ I have a list of examples in: http://link.springer.com/chapter/10.1007/978-1-4614-9332-7_8 and to get a better idea of the PDE idea, see: is this a known method for solving PDEs which illustrates the idea.

Thanks in advance for your help!

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    $\begingroup$ You need to define what you mean by $\mathcal{A}$-calculus. Without knowing this, we can't help you. $\endgroup$ – Robert Bryant Dec 19 '14 at 10:00
  • $\begingroup$ @RobertBryant thanks for the note, I have added some detail. I suppose $\mathcal{A}$-calculus has been known by various names, I'm not sure what the best name is. $\endgroup$ – James S. Cook Dec 19 '14 at 14:31
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    $\begingroup$ Your question is still too vague. A tableau is essentially a way to represent a constant coefficient first order system of PDE's. So any arbitrary system of first order PDE's gives you an example of a tableau. You can then test whether the tableau is in involution or not and prolong if necessary. You need to say more about what you're hoping to learn about in the examples. $\endgroup$ – Deane Yang Dec 20 '14 at 5:24
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    $\begingroup$ There are other approaches to this theory. A modern cohomological approach was developed by Spencer, Quillen, Guillemin, Sternberg, and Goldschmidt. A nice description of an involutive first order differential operator was given by Guillemin and is now known as Guillemin normal form. Using this description, it is easy to construct examples of involutive systems of PDE's. Still another approach, using only basic linear algebra, is described in the first chapter of my thesis, "Involutive hyperbolic differential systems", Memoirs of the AMS. $\endgroup$ – Deane Yang Dec 20 '14 at 5:29
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    $\begingroup$ @Sintrastes: It has been a while since I thought about this, but, if I remember correctly, the answer is that every $\mathcal{A}$-differentiable function in a neighborhood of $a\in\mathcal{A}$ has a Taylor series expansion of the form $$ f(x) = f_0 + f_1(x-a) + f_2(x-a)^2 + \cdots$$ where the $f_i$ for $i>1$ lie in the left-annihilator of the commutator ideal. Conversely, if $f_i$ are chosen to lie in this ideal and the above series converges, this defines an $\mathcal{A}$-differentiable function in a neighborhood of $a$. $\endgroup$ – Robert Bryant Jun 8 '16 at 8:33

The following may help: Fueter tried to describe quaternionic analysis in the sense of $\mathcal A$-Analysis in the 20's or 30's, but only first order quaternionic polynomials were possible. The working generalization is hypercomplex Analysis or Clifford Analysis, where one generalizes the Laplacian from complex Analysis to more general algebras. I enclose references to two papers below. (I am not a specialist in this, I just listened to some talks during the years.)

  • MR0822855 (87f:30102) Reviewed Bureš, J.(CS-CHRL); Souček, V. Generalized hypercomplex analysis and its integral formulas. Complex Variables Theory Appl. 5 (1985), no. 1, 53–70.

  • MR3277680 Prelim Brackx, F.; Schepper, H. De; Eelbode, D.; Lávička, R.; Souček, V.; Fundaments of Quaternionic Clifford Analysis I: Quaternionic Structure. Adv. Appl. Clifford Algebr. 24 (2014), no. 4, 955–980.

  • $\begingroup$ Thank you for the references. However, I'm mostly focused on the commutative case and I'm currently trying to collect results which apply for all associative algebras over $\mathbb{R}$. Clifford Analysis would be a specialization (granted an interesting one). Sorry my question was not more focused, perhaps I will write a better one in a few weeks. $\endgroup$ – James S. Cook Dec 20 '14 at 3:25

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