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. Using our techniques we also show that the problem of determining if a given set of Heisenberg matrices generates a group can be decided in polynomial time.

翻译：我们研究恒等问题，即判定一个有限生成矩阵半群是否包含单位矩阵的问题；参见Blondel与Megretski所著《数学系统与控制理论中的未解问题》（2004）第10章第3节问题3。这一基本问题在$\mathbb{Z}^{4 \times 4}$矩阵范围内已被证明是不可判定的，而在$\mathbb{Z}^{2 \times 2}$范围内是可判定的。近期，Dong通过运用李代数与Baker-Campbell-Hausdorff公式，证明了在任意固定维度下，复数域上海森堡群的恒等问题具有多项式时间复杂度。我们针对该问题提出了替代性证明技术，为推进解决更一般性问题（如成员判定问题）迈出重要一步。利用我们的技术，我们还证明了判定给定海森堡矩阵集合是否生成群的问题可在多项式时间内判定。