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

Abstract

Communicated by E. ZelmanovWe reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for some class of finite-dimensional 2-step nilpotent Lie algebras over a field and for some 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 (2012). 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.

A fast isomorphism test for groups whose Lie algebra has genus 2

“…The problem itself has different aspects, ranging from practical methods for use in the sciences [CH, ELGO], to questions of computability [R, HL, BCGQ, LW], to the intimate but complex relationship it has with the graph isomorphism problem [M3,p. 132;HL;LV,Theorem 3.1]. The classification of finite simple groups, combined with natural recursive methods based on Sylow subgroups and the lower central series, gives a reduction to the case of p-groups of exponent p-class 2.…”