Octonionic Hessian - Knowlpedia

Definition

Write an octonion as $q=\sum_{r=0}^7x_re_r$ , with $e_0=1$ . For an octonion-valued function $F$ , set

\frac{\partial F}{\partial\bar q} =\sum_{r=0}^7e_r\frac{\partial F}{\partial x_r}, \qquad \frac{\partial F}{\partial q} =\sum_{r=0}^7\frac{\partial F}{\partial x_r}\bar e_r.

For a real-valued $C^2$ function $u$ on $\mathbb O^2$ , its octonionic Hessian is

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

It is an octonionic Hermitian $2\times2$ matrix.

Positivity criterion

A $C^2$ function is octonionic plurisubharmonic exactly when this Hessian is positive semidefinite at every point.

Dimension restriction

Mixed octonionic derivatives can be written for more coordinates, but the determinant and covariance used in this theory are specific to the $2\times2$ Hermitian Hessian. The established octonionic pluripotential theory here is therefore a theory on $\mathbb O^2$ , not a formal replacement of $\mathbb H^n$ by arbitrary $\mathbb O^n$ .

Convention warning

Left and right placement of octonionic units matters even more than in the quaternionic case because multiplication is nonassociative. The displayed operators and derivative order are part of the definition.

References

Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$ -invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record. Relevant: §§0.1, 1.2, and 3.1.