The catastrophes of algebras

Abstract

A parametrized family of pseudo-Finsler norms is derived from application of a particular transform procedure to the space of ``normalized" trace forms of any given real finite-dimensional unital associative algebra. The implied parametrized family of dual norms on a connected conic subbundle of the relevant cotangent space supports wavefront phenomena that lead to a trove of novel algebra isomorphism invariants associated with a cascade of caustics typically arising from algebras that do not admit a direct sum decomposition whose non-simple blocks all have dimension less than four. The algebra's commutator subspace can be accessed by reapplying the above procedure at all relevant orders of the algebra's infinitesimal neighborhoods. The general character of this set of invariants appropriately reflects the wildness of the algebra isomorphism problem.

arXiv admin note: text overlap with arXiv:2207.11358 [author note: arXiv:2207.11358 was withdrawn due to the overlap]. There is necessarily some overlap with the much shorter arXiv:2209.14119, since the latter concerns an alternative geometrical motivation for the program of the present paper