Quaternionic Hessian

Definition

For a real-valued $C^2$ function $u$ on an open subset of $\mathbb H^n$ , its quaternionic Hessian is

\operatorname{Hess}_{\mathbb H}u =\left(\frac{\partial^2u} {\partial\bar q_i\,\partial q_j}\right)_{i,j=1}^n,

using the Cauchy–Fueter operator convention fixed there. This matrix is hyperhermitian.

Positivity criterion

A $C^2$ function is quaternionic plurisubharmonic exactly when its quaternionic Hessian is positive semidefinite at every point. Strict plurisubharmonicity corresponds to positive definiteness.

Transformation law

Under a right quaternionic-linear change of variables $q\mapsto Aq$ , the Hessian transforms by hyperhermitian congruence. Consequently its Moore determinant has the covariance needed to define the quaternionic Monge–Ampère operator.

Order convention

The transpose/order in the displayed matrix is essential. Early versions of several foundational papers used the transposed convention and were later corrected. In this knowl, rows are indexed by $\partial_{\bar q_i}$ and columns by $\partial_{q_j}$ .

References

Semyon Alesker, “Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables,” revised 2024. arXiv record. Relevant: §§2–3.
Semyon Alesker, “Quaternionic Monge–Ampère equations,” Journal of Geometric Analysis 13 (2003), 205–238. arXiv record.