Isomorphisms of graded semiconnected algebras

Abstract: Let $A_\bullet$ and $B_\bullet$ be nonnegatively graded algebras, finitely generated in degrees less than $2$ and with semisimple base rings $A_0$ and $B_0$. We prove that if $A \simeq B$ as ungraded algebras, then $A_\bullet \simeq B_\bullet$ as graded algebras. This generalises a theorem of Bell and Zhang arXiv:1509.08812 in the connected case $A_0 = k = B_0$, and the path-algebra version when $A_0$ and $B_0$ are $k$-elementary due to Gaddis arXiv:1712.01650. We also discuss some related open problems for graded algebras.