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Article

Algebras of general type: rational parametrization and normal forms

For every algebraically closed field k of characteristic different from 2, we prove
the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type
of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple
of algebraically independent (over k) rational functions of 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∈I and {b_j}_j∈J ,
such that the ideal generated by the set {f_i}_i∈I is prime and, for every tuple c of structure
constants satisfying the property b_j(c) = 0 for all j ∈ 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 ∈ I.