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Cayley-Dickson construction defines general forms of complex multiplication and conjugate:

$$ (a,b)^* = (a^*, -b) \\ (a,b)(c,d) = (ac-d^*b,da+bc^*) $$

By applying these recursively, progressively higher-dimensional hypercomplex number systems can be constructed.

Is there an equivalent construction for dual numbers? Does $$ (a,b)(c,d) = (ac,ad+bc) $$ only work over reals, or is it sufficiently general to apply recursively?

yuri kilochek
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  • You can generalize dual numbers to multiple dimensions in various ways, from straight multi-dimensional dual numbers (commutative) to Grassman numbers (non-commutative) to nilpotents of higher order (commutative). – Anixx May 31 '25 at 00:57

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I do not know what do you mean by "equivalent", but you can generalize dual numbers to multiple dimensions in various ways, either commutative or not.

Here are some links to lists of low-dimensional associative algebras:

https://arxiv.org/pdf/0910.0932

https://arxiv.org/pdf/1606.03119

Classification of 4 dimensional real associative unital algebra

For instance,

$\mathbb{R}[x]/(x^4) \simeq \begin{bmatrix}a&b&c&d\\0&a&b&c\\0&0&a&b\\0&0&0&a\end{bmatrix}$

$\mathbb{R}[x, y]/(x^2, y^2)\simeq \begin{bmatrix}a&b&c&d\\0&a&0&c\\0&0&a&b\\0&0&0&a\end{bmatrix}$,

$\mathbb{R}[x]/x^2 \times \mathbb{R}[y]/y^2\simeq \begin{bmatrix}a&b&0&0\\0&a&0&0\\0&0&c&d\\0&0&0&c\end{bmatrix}$ (direct product of dual numbers)

$\mathbb{R}\langle x,y\rangle /(x^2,y^2,xy+yx)\simeq \begin{bmatrix}a&b&c&d\\0&a&0&-c\\0&0&a&b\\0&0&0&a\end{bmatrix}$ (Grassmann numbers)

$\mathbb{R}[x, y, z]/(x^2, xy, y^2, yz, z^2, zx)\simeq \begin{bmatrix}a&0&b&d\\0&a&c&b\\0&0&a&0\\0&0&0&a\end{bmatrix}$,

etc. All of these can be considered generalizations of dual numbers.

Anixx
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