The catastrophes of algebras

TL;DR

This paper explores how finite-dimensional associative algebras relate to pseudo-Finsler norms and trace forms, revealing complex invariants and phenomena like catastrophes through a novel transform approach.

Contribution

It introduces a new transform linking algebraic structures to geometric norms, capturing algebra invariants and explaining complex behaviors like catastrophes.

Findings

Associations between algebras and pseudo-Finsler norms are established.

Transform procedures reveal algebra invariants reflecting isomorphism complexity.

Application of Hamiltonian formalism shows potential for catastrophes in algebraic contexts.

Abstract

We show that a real finite-dimensional unital associative algebra is naturally associated with a vector space of pseudo-Finsler norms whose members are linked to the algebra's space of normalized trace forms through an integral transform. Since components of the space of trace forms act as parameters controlling the implied pseudo-Finsler indicatrices, successive application of the usual Hamiltonian formalism can lead to caustics and bifurcations of caustics (i.e., catastrophes and catastrophes of catastrophes) by continuous variation of these parameters. In order to capture influence from the entirety of the Jacobson radical, the transform procedure can be applied at all appropriate levels of a Cuntz-Quillen tower defining the algebra's formal neighborhood. The latter procedure leads to a trove of algebra invariants, whose intricacy reflects the wildness of the algebra isomorphism problem that appears when dimensions of the algebras in question are not small. In that respect, description of algebra structure using this methodology compares favorably with programs whose content is primarily homological.