The twistor space of a real associative algebra: incidence geometry, semisimple classification and slice-regular functions

Abstract

To every finite-dimensional associative real algebra $A$ we associate its twistor space $S(A)$, the real algebraic set of square roots of $-1$. Left multiplication defines an almost complex structure on $S(A)$, integrable precisely because $A$ is associative. The resulting complex manifold embeds biholomorphically into the Grassmannian of $\mathbb{C}\otimes A$ by sending $s$ to the $(-i)$-eigenspace of the complexified operator $L_s$. This realization turns the tautological pairing $\pi :\mathbb{C}\otimes A\times S\to A$ into an incidence correspondence. For every compact complex subvariety $K\subseteq S$, the associated zero variety $Z_K\subset \mathbb{C}\otimes A$ is complex analytic by Remmert's theorem, and the pull-back of the incidence variety along a holomorphic lift describes the corresponding zero set. We also give an intrinsic, section-free formulation of the twistor transform of Gentili, Salamon and Stoppato as a holomorphic map into $\mathbb{P}(\mathcal{V}\oplus \mathcal{V})$ over $S$. After choosing a Euclidean structure on $A$, we study the compact subvariety $S_0\subset S$ formed by those $s$ for which $L_s$ is orthogonal. For semisimple algebras, the Wedderburn decomposition shows that $S_0$ is a finite union of products of compact Hermitian symmetric spaces, including $O(2m)/U(m)$, $Sp(m)/U(m)$ and complex Grassmannians. In the classical simple cases, we compute explicit equations for its Euclidean zero variety, obtaining isotropic or determinantal cones. The construction is motivated by slice-regular function theory. For $A=\mathbb{H}$, it recovers the classical twistor sphere and the Gentili--Salamon--Stoppato transform. Finally, for a compact connected space $S$, we introduce generalized slice-regular functions and extend the maximum modulus principle, the representation formula and a twisted Cauchy--Riemann characterization via holomorphic reparametrizations of $S$.

References in corpus (2)

• Twistor transforms of quaternionic functions and orthogonal complex structures
• Singularities of slice regular functions over real alternative *-algebras

Cited by in corpus (1)

• Slice regular functions as covering maps and global $\star$-roots