Christophersen’s problem

Christophersen’s problem asks whether every finite-dimensional local algebra $A$ of dimension $n$ satisfies $\dim\operatorname{Aut}(A)^\circ\ge n-1$, with equality only for $A\cong\mathbb{C}[t]/(t^n)$. The retrieved literature does not claim a resolution for arbitrary local algebras.

Known results

• Roman Stasenko records $\dim\operatorname{Der}(A)\ge n-\dim\operatorname{Soc}(A)$ for non-negatively graded local algebras.
• For arbitrary local algebras, he records $\dim\operatorname{Der}(A)\ge\dim(\mathfrak m/\mathfrak m^2)\dim\operatorname{Soc}(A)$.
• Full-null-index Gorenstein algebras have solvable $\operatorname{Der}(A)$, while an example containing $\mathfrak{so}_3$ shows broader solvability fails.

September 2026 monomial-algebra result

Roberto Díaz and Alvaro Liendo claim the bound and equality characterization throughout the monomial-algebra class, with stronger bounds and additional extremal classifications in reducible and irreducible cases. This is substantial progress, but it does not cover all finite-dimensional local algebras and remains unverified in the retrieved material.

Current status (as of September 2026): The monomial-algebra case is claimed solved but unverified; Christophersen’s problem for arbitrary finite-dimensional local algebras remains open.