🤖 AI Summary
This paper addresses two fundamental decision problems for finitely generated Heisenberg matrix semigroups over the complex numbers: (1) the Identity Problem—whether the semigroup contains the identity matrix; and (2) the Group Problem—whether the generating set forms a group. Departing from classical approaches reliant on Lie algebras and the Baker–Campbell–Hausdorff formula, we introduce a novel framework grounded in the structure of nilpotent Lie algebras, precise truncated estimates of exponential and logarithmic maps, and geometric analysis of matrix groups. We present the first deterministic polynomial-time algorithms for both problems; notably, the polynomial decidability of the Group Problem is established for the first time. Our results fully resolve the theoretical decidability of these problems in the Heisenberg setting and provide a new paradigm and key technical methodology for broader matrix semigroup membership problems.
📝 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 $mathbb{Z}^{4 imes 4}$ and decidable for $mathbb{Z}^{2 imes 2}$. 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.
