
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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
