
TL;DR
This paper investigates the computational complexity of matrix Lie algebra conjugacy, revealing connections to graph isomorphism and providing polynomial-time algorithms for specific cases, with implications for problems like affine polynomial equivalence.
Contribution
It establishes equivalences between Lie algebra conjugacy problems and graph isomorphism, and develops efficient algorithms for certain restricted cases.
Findings
Abelian Lie algebra conjugacy is as hard as graph isomorphism.
Polynomial-time algorithms exist for Abelian Lie algebra conjugacy with constant dimension.
Semisimple Lie algebra conjugacy reduces to graph isomorphism and is solvable in polynomial time under certain conditions.
Abstract
We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley--Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that implies . Two matrix Lie algebras are conjugate if there is an invertible matrix such that . We show that certain cases of Lie algebra conjugacy are equivalent to graph isomorphism. On the other hand, we give polynomial-time algorithms for other cases of Lie algebra conjugacy, which allow us to essentially derandomize a recent result of Kayal on affine equivalence of polynomials. Affine equivalence is related to many complexity problems such as factoring integers, graph isomorphism, matrix multiplication, and permanent versus determinant. Specifically, we…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Polynomial and algebraic computation
