The catastrophes of algebras

Abstract

A parametrized collection of flat (conic pseudo-)Finsler spaces is derived from application of a particular transform procedure to the space of normalized trace forms of a real finite-dimensional unital associative algebra. The associated collection of co-(pseudo-)Finsler symplectic manifolds is thereby indexed by an inherited set of parameters that control the Lagrangian submanifold dynamics. Largrangian submanifolds are defined with respect to the indicatrix H(q,p)=1/2, with the t=0 slice such that q is fixed by the unit Euclidean sphere in configuration space and p is directed inward. The wavefronts resulting from projection of the Lagrangian submanifold time slices to configuration space in the context of variation of the other control parameters leads to a trove of novel algebra isomorphism invariants associated with a cascade of caustics and their bifurcations typically arising from algebras that do not admit a direct sum decomposition whose non-simple blocks all have dimension less than four. The above procedure is replicated at all relevant orders of an algebra's infinitesimal neighborhoods as defined by a Cuntz-Qullen tower. The general character of this set of invariants appropriately reflects the wildness of the algebra isomorphism problem.