李代数共轭性

摘要：

研究了矩阵李代数的共轭性问题。李代数集中出现在微分方程，粒子物理，群论和Mulmuley-Sohoni几何复杂性理论程序等不同领域。一个矩阵李代数是一个矩阵集合L，使得$A, B\in L$蕴涵$AB - BA \in L$。两个矩阵李代数是共轭的，如果存在可逆矩阵$M$使得$L_1 = M L_2 M^{-1}$ .我们证明了李代数共轭的某些情形等价于图同构。另一方面，我们给多项式时间算法的其他情况下的李代数共轭，这使我们能够基本上derandomize最近的结果Kayal仿射等价的多项式。仿射等价关系到许多复杂性问题，如整数分解、图同构、矩阵乘法、积和行列式。具体来说，我们表明：阿贝尔李代数共轭是等价的代码等价问题，因此是硬图同构。当$n \times n$矩阵的维数为O（1）时，可在poly（n）时间内求出其Abel李代数共轭.半单李代数共轭等价于图同构。一个李代数是半单的，如果它是单李代数的直和。当李代数仅由$O(\log n)$单直和项组成时，$n \times n$矩阵的半单李代数共轭性可以在多项式时间内求解.完全可约李代数的共轭性--即交换和半单李代数的直和--当交换部分的维数为O（1）且半单部分有$O(\log n)$单直和项时，可以在多项式时间内求解。

英文摘要：

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 $A, B\in L$ implies $AB - BA \in L$. Two matrix Lie algebras are conjugate if there is an invertible matrix $M$ such that $L_1 = M L_2 M^{-1}$. 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 show: Abelian Lie algebra conjugacy is equivalent to the code equivalence problem, and hence is as hard as graph isomorphism. Abelian Lie algebra conjugacy of $n \times n$ matrices can be solved in poly(n) time when the Lie algebras have dimension O(1). Semisimple Lie algebra conjugacy is equivalent to graph isomorphism. A Lie algebra is semisimple if it is a direct sum of simple Lie algebras. Semisimple Lie algebra conjugacy of $n \times n$ matrices can be solved in polynomial time when the Lie algebras consist of only $O(\log n)$ simple direct summands. Conjugacy of completely reducible Lie algebras---that is, a direct sum of an abelian and a semisimple Lie algebra---can be solved in polynomial time when the abelian part has dimension O(1) and the semisimple part has $O(\log n)$ simple direct summands.

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