On Borel complexity of the isomorphism problems for graph-related classes of Lie algebras and finite p-groups

TL;DR

This paper investigates the complexity of isomorphism problems for certain algebraic structures by reducing graph isomorphism to problems for Lie algebras and p-groups, revealing their relative difficulty in Borel reducibility terms.

Contribution

It establishes reductions from graph isomorphism to algebraic isomorphism problems and compares their complexities, introducing a computable Borel reducibility framework.

Findings

Graph isomorphism is harder than Lie algebra and p-group isomorphism problems.

A reduction from graph isomorphism to Lie algebra and p-group isomorphism problems is constructed.

The complexity hierarchy of these isomorphism problems is clarified using Borel reducibility.

Abstract

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J.D. Hamkins, and R. Miller. A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied.