On the Identity and Group Problems for Complex Heisenberg Matrices

TL;DR

This paper presents new proof techniques for the Identity Problem in complex Heisenberg matrices, showing it is decidable in polynomial time and advancing understanding of related group membership problems.

Contribution

The paper introduces alternative methods for solving the Identity Problem for complex Heisenberg matrices, extending polynomial-time decidability results and addressing group generation.

Findings

Identity Problem for complex Heisenberg matrices is decidable in polynomial time.

New proof techniques applicable to broader problems like Membership Problem.

Decidability of group generation by Heisenberg matrices in polynomial time.

Abstract

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.