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Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables

Semyon Alesker

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.

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.

Paper Structure

This paper contains 7 sections, 25 theorems, 159 equations.

Table of Contents

  1. Introduction.
  2. Linear algebra.
  3. Hyperhermitian forms and the Moore determinant.
  4. Dieudonné determinant.
  5. Plurisubharmonic functions of quaternionic variables.
  6. Main notions.
  7. The minimum principle.

Key Result

Theorem 1.1.8

There exists a polynomial $P$ defined on the space of all hyperhermitian $n \times n$-matrices such that for any hyperhermitian $n \times n$-matrix $A$ one has $det({}^{\hbox{\rm $~\!\!$R}} A)= P^4(A)$ and $P(Id)=1$. $P$ is defined uniquely by these two properties. Furthermore $P$ is homogeneous of

Theorems & Definitions (41)

  • Definition 1.1.1
  • Example 1.1.2
  • Claim 1.1.3
  • Claim 1.1.4
  • Remark 1.1.5
  • Definition 1.1.6
  • Claim 1.1.7
  • Theorem 1.1.8
  • Theorem 1.1.9
  • Example 1.1.10
  • ...and 31 more