Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables

By Semyon Alesker. 2001

102 citations recorded by the source corpus.

We recall known and establish new properties of the Dieudonne 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-Ampere equations.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/63cdd8425347cc795a0f0ebdf53e6e6ccce3d7a6

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In Algebra and Number Theory

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry (2005, methodology): “dered as quaternionic hyperhermitian matrix is equal to its usual determinant. (ii) For any hyperhermitian matrix A and any quaternionic matrix C det(C∗AC) = detA ·det(C∗C). For the proof we refer to [A1] though this result was known earlier and is probably a folklore. 2.6 Example. (a) Let A = diag(”
Valuations on convex functions and convex sets and Monge–Ampère operators (2017, background): “1, it was shown by the author [3] that this expression can be defined, as a signed measure, for any convex (more generally, continuous quaternionic plurisubharmonic) function f and this measure satisfies all the properties stated in Theorem 2.”
Plurisubharmonicity in a General Geometric Context (2008, background): “ves the set of standard plurisubharmonic functions in complex analysis. 8 5. G= Pn−1(H) = G H(1,Hn) ⊂ G(4,R4n) gives the set of quaternionic plurisubharmonic functions on quaternionic n-space Hn (cf. [Al], [AV]). 6. G= GC(p,Cn) for 1 <p<ngives complex p-plurisubharmonic functions on Cn. 7. G= ”
Quaternionic Monge-Ampère equations (2002, methodology): “Note also that all such affine transformations preserving the domain f2 must preserve the class of psh functions (see [ 2 ], Section 2.1).”
Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemannian Manifolds (2009, methodology): “ deﬁned, and when the hyperKahler structure is integrable, they coincide with the ones used by Alesker and Verbitsky to study the Quaternionic Monge-Ampere equation in the principal-branch case. (See [A1,2], [AV]) However a manifold with a topological Spn-structure carries other almost calibrations ”