Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables

Abstract

We recall known and establish new properties of the Dieudonné and Moore determinants of quaternionic matrices.Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. Then we introduce and briefly discuss quaternionic Monge-Ampére equations.

42 pages. A mistake in the statement of Theorem 1.2.4(iii) corrected

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• Classification of invariant valuations on the quaternionic plane
• Potential theory in several quaternionic variables
• Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemannian Manifolds
• Dirichlet Problem of Quaternionic Monge-Ampère Equations
• Viscosity solutions to quaternionic Monge-Ampère equations
• Weak solutions to the quaternionic Monge-Ampère equation
• Plurisubharmonic Functions in Calibrated Geometries
• Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds
• An Introduction to Potential Theory in Calibrated Geometry
• Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry
• Cegrell classes and a variational approach for the quaternionic Monge-Ampere equation
• Quaternionic plurisubharmonic functions and their applications to convexity
• Subsolution theorem and the Dirichlet problem for the quaternionic Monge-Ampere equation
• Solvability of the quaternionic Monge-Ampere equation on compact manifolds with a flat hyperKaehler metric
• The Dirichlet Problem with Prescribed Asymptotic Singularities
• The Neumann problem for the $k$-Cauchy-Fueter complexes over $k$-pseudoconvex domains in ${\mathbb{R}}^4$ and the $L^2$ estimate
• On a uniform estimate for the quaternionic Calabi problem
• The Linear algebra in the quaternionic pluripotential theory
• Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets