Abstract:We classify and count the real-algebra involutions of the multicomplex algebra $\mathbb{M}\mathbb{C} (n)$ that map each element of its canonical monomial basis to a signed monomial, using a matrix model over $\mathbb F_2$.
An \emph{elliptic-admissible pair} consists of such an involution $\sigma$ and a monomial unit $\mathbf{i}\in\mathbb{I}(n)$ satisfying $\mathbf{i^2}=-1$ and $\sigma(\mathbf{i})=-\mathbf{i}$. The fixed algebra of $\sigma$ is a real form for the complex structure $\mathcal{L}_{\mathbf{i}}$ defined by multiplication by $\mathbf{i}$, and the pair yields a Cauchy--Riemann system. We prove that its solution class depends only on $\mathbf{i}$, not on $\sigma$, and is exactly the class of mappings holomorphic with respect to $\mathcal{L}_{\mathbf{i}}$. Multicomplex holomorphy is recovered as the intersection of the classes associated with the elementary generators. We obtain the analogous characterization for anti-holomorphic classes, describe twisted systems intertwining two such complex structures, and prove that every $\mathbf{i}{}$-holomorphic or $\mathbf{i}$-anti-holomorphic mapping is componentwise harmonic (solutions to Laplace's equation).
From: Pierre-Olivier Parise Pop [view email]
[v1]
Fri, 25 Nov 2022 03:34:17 UTC (25 KB)
[v2]
Sun, 2 Aug 2026 03:01:30 UTC (42 KB)
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