9 problems

Valenti–Zaicev conjecture. Every grading on $U$ is graded isomorphic to

Let $L$ be a simple complex Lie algebra. A grading of $L$ is called fine if it cannot be properly refined by another grading, and a fine group grading is a fine grading whose index…

Let $L$ be a Lie algebra with a group grading $\Gamma$ that is a coarsening of an abelian grading of $L$. Abelianness conjecture. Then $\Gamma$ is abelian. The paper states that no…

Let $X$ be a finite poset, let $\mathbb{F}$ be a field of characteristic zero, and let $G$ be a group. Consider a $G$-grading on the incidence algebra $I(X)$. For each minimal idem…

Let $p>2$ be a prime, and let $h(p)$ be the minimal integer such that every Lie ring $L$ with a grading … by $(\mathbb{Z}_p,+)$ satisfying $L_{\overline{0}}=\\{0\\}$ is nilpotent o…

Classification conjecture. The algebra $A$ is $*$-graded simple if and only if it is isomorphic as a graded $*$-algebra either to the $G$-graded simple algebra…

Let $W$ be an algebra over an algebraically closed field $F$ of characteristic zero satisfying a polynomial identity of degree $n$. Suppose that $W$ is nondegenerately graded by a…

Bahturin–Regev's conjecture. If $A$ admits a minimal regular grading by a finite abelian group $G$, then the order $\lvert G\rvert$ is uniquely determined by $A$.

Aljadeff's graded-exponent conjecture. The exponent of $A$ as a $G$-graded algebra equals the maximum of these sums.