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By "algebraic structure" I mean a finite set of linear operators between tensor products of copies of one (or more) finite-dimensional (complex or real) vector spaces, fulfilling a set of axioms with are equations between different ways of composing the operators. Examples are (unital/commutative/associative) algebras, their (projective) representations, (special/symmetric) Frobenius algebras, (weak/quasi-) Hopf algebras, and so on. With a bit of tweaking, fusion categories and similar things are also structures of this type.

Some of those algebraic structures are "classified", while for most of them no real classification is known. For practical purposes (in particular thinking about physics applications such as in TQFT), the relevant solutions of the axioms are usually the low-dimensional ones. The linear operators (written out in a fixed basis) are nothing but a bunch of numbers, and the axioms are sets of polynomial equations for those numbers. It should be possible to find the roots to those equations by a numerical method like conjugate-gradient or Gauss-Newton. Of course, such an algorithm cannot be efficient for high dimensions, but in low dimensions it should be feasible.

Is anyone aware of numeric attempts like that? I'd be interested in whether there are implementations of such algorithms for specific algebraic structures, as well as implementations where you can put in the combinatorial data of arbitrary structures. If yes, what are the limitations of those methods? To my surprise, I couldn't find anything on google.

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