Generic algebras: rational parametrization and normal forms

TL;DR

This paper establishes a rational parametrization and a unique normal form for finite dimensional algebras over algebraically closed fields of characteristic not 2, simplifying classification up to isomorphism.

Contribution

It introduces a universal rational parametrization and an algebraic normal form for finite dimensional algebras, extending classification methods beyond associative cases.

Findings

Finite dimensional algebras are parametrized by algebraically independent rational functions.

Existence of a unique algebraic normal form for algebras via polynomial systems.

Normal forms facilitate algebra classification by basis transformation.

Abstract

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent over $\boldsymbol k$ rational functions in the structure constants. (2) There exists an "algebraic normal form", to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis, namely: there are two finite systems of nonconstant polynomials on the space of structure constants, $\{f_i\}_{i\in I}$ and $\{b_j\}_{j\in J}$, such that the ideal generated by the set $\{f_i\}_{i\in I}$ is prime and, for every tuple $c$ of structure constants satisfying the property $b_j(c)\neq 0$ for all $j\in J$, there exists a unique new basis of this algebra in which the tuple $c'$ of its structure constants satisfies the property $f_i(c')=0$ for all $i\in I$.