AI 中文总结
本文针对满足特定条件的非负分次代数，证明其非分次同构必导出分次同构，推广了已有连通代数及路代数版本的相关定理，并讨论了分次代数的若干开放问题。
AI 中文摘要
设$A_\ullet$和$B_\ullet$为非负分次代数，在小于2的次数上有限生成，且具有半单基环$A_0$和$B_0$。我们证明，若$A$作为非分次代数同构于$B$，则$A_\ullet$作为分次代数同构于$B_\ullet$。这推广了Bell与Zhang（arXiv:1509.08812）在连通情形（$A_0 = k = B_0$）下的定理，以及Gaddis（arXiv:1712.01650）在$A_0$和$B_0$为$k$-初等的路代数版本。我们还讨论了分次代数的若干相关开放问题。
英文摘要
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.
Comments12 pages. Comments very welcome!
