Computing the Zariski closure of a finitely generated matrix group
Willem A. de Graaf
TL;DR
The paper addresses computing the Zariski closure $\overline{\mathcal{H}}$ of a finitely generated subgroup of $GL(n,\mathbb{C})$ by representing the output as a Lie algebra basis for the identity component and a finite set of component representatives, avoiding Gröbner bases. The approach combines torus diagonalization, lattice computations, and multiplicative-relations in eigenvalues to determine $Lie(G(g))$ for generators and iteratively enlarges the candidate Lie algebra until it matches $Lie(G)$; the algorithm guarantees termination and yields the full Zariski closure. A central contribution is an explicit, Gröbner-basis-free method for constructing the identity component and component structure, suitable for nontrivial examples, implemented in OSCAR with reliance on number-field arithmetic and interoperability with GAP and Julia. The work demonstrates practical performance on representative Lie types and provides a workable path for computing algebraic envelopes of matrix groups in characteristic zero, with potential applications in quantum automata and related areas.
Abstract
We describe an algorithm for determining the algebraic subgroup of GL(n,C) that is defined as the closure of the group generated by a finite number of elements of GL(n,C). The algorithm avoids the use of Groebner bases and can be used on non-trivial examples. In the last section we report on an implementation of the algorithm in the computer algebra system {\tt OSCAR}.