Algebraic constructions of cubic minimal cones

TL;DR

This paper introduces new algebraic tools, quasicomposition and tripling, to study Hsiang algebras, advancing the classification of these nonassociative algebras related to minimal cones and elliptic PDE solutions.

Contribution

It presents two novel methods for analyzing Hsiang algebras, including a tripling construction and the concept of quasicomposition, linking algebraic structures to geometric and PDE applications.

Findings

Tripling construction relates to Cayley-Dickson process.

Exceptional Hsiang algebras correspond to quasicomposition algebras.

New tools facilitate classification of complex algebraic structures.

Abstract

Hsiang algebras are a class of nonassociative algebra defined in terms of a relation quartic in elements of the algebra. This class arises naturally in relation to the construction of real algebraic minimal cones. Additionally, Hsiang algebras were crucial in the construction of singular (trulsy viscosity) solutions of nonlinear uniformly elliptic partial differential equations in a series of papers by Nikolai Nadirashvili and Serge Vl\u{a}du\c{t}. The classification of Hsiang algebras is a challenging problems, based on a Peirce decomposition like that used to study Jordan algebras, although more complicated. This paper introduces two new tools for studying Hsiang algebras: a distinguished class of algebras called \textit{quasicomposition} that generalize the Hurwitz algebras (the reals, complexes, quaternions, and octonions) and cross-product algebras and a tripling construction associating with a given algebra one of three times the dimension of the original algebra, that can be thought of as a kind of analogue of the Cayley-Dickson doubling process. One main result states that the triple is an exceptional Hsiang algebra if and only if the original algebra is quasicomposition. The quasicomposition and tripling notions are interesting in their own right and will be considered in a more general form elsewhere.