We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for $\mathbb{Z}^{4 \times 4}$ and decidable for $\mathbb{Z}^{2 \times 2}$. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time.

翻译：我们研究单位元问题，即判定有限生成矩阵半群是否包含单位矩阵的问题；参见Blondel与Megretski（2004）所著《数学系统与控制理论中的未解问题》（第10.3章，问题3）。这一基本问题已知在$\mathbb{Z}^{4 \times 4}$上不可判定，但在$\mathbb{Z}^{2 \times 2}$上可判定。最近，Dong利用李代数与Baker-Campbell-Hausdorff公式，证明复系数任意维海森堡群上的单位元问题可在多项式时间内求解。我们为该问题发展了替代性证明技术，从而向更一般的问题（如成员问题）迈出一步。进一步扩展我们的技术，证明判定给定海森堡矩阵集合是否生成群的这一基本问题同样可在多项式时间内解决。