Rank-Stability of Polynomial Equations

Abstract Extending the thoroughly studied theory of group stability, we study Ulam stability type problems for associative and Lie algebras. Namely, we investigate obstacles to rank-approximation of “almost” solutions by exact solutions to systems of polynomial equations. This leads to a rich theory of stable associative and Lie algebras, with connections to linear soficity, amenability, growth, and group stability. We develop rank-stability and instability criteria, examine the effect of algebraic constructions on rank-stability, and prove that while finite-dimensional associative algebras are rank-stable, “most” finite-dimensional Lie algebras are not.