Let $G$ be a unitriangular matrix group of nilpotency class at most ten. We show that the Identity Problem (does a semigroup contain the identity matrix?) and the Group Problem (is a semigroup a group?) are decidable in polynomial time for finitely generated subsemigroups of $G$. Our decidability results also hold when $G$ is an arbitrary finitely generated nilpotent group of class at most ten. This extends earlier work of Babai et al. on commutative matrix groups (SODA'96) and work of Bell et al. on $\mathsf{SL}(2, \mathbb{Z})$ (SODA'17). Furthermore, we formulate a sufficient condition for the generalization of our results to nilpotent groups of class $d > 10$. For every such $d$, we exhibit an effective procedure that verifies this condition in case it is true.

翻译：设 $G$ 为幂零类至多为十的单位三角矩阵群。我们证明：对于 $G$ 的有限生成子半群，恒等问题（半群是否包含单位矩阵？）和群问题（半群是否为群？）可在多项式时间内判定。我们的可判定性结果也适用于任意类至多为十的有限生成幂零群 $G$。这推广了 Babai 等人关于交换矩阵群（SODA'96）以及 Bell 等人关于 $\mathsf{SL}(2, \mathbb{Z})$（SODA'17）的早期工作。此外，我们提出了一个充分条件，用于将结果推广至类 $d > 10$ 的幂零群。对于每个这样的 $d$，我们给出一个有效过程，当条件成立时对其进行验证。