非交换多项式的值、李斜理想和图零点定理
Values of Noncommutative Polynomials, Lie Skew-Ideals and the Tracial Nullstellensatz
摘要：
具有对合的代数的子空间称为李斜理想，如果它在具有斜对称元的李积下是闭的。对具有对合的中心单代数中的李斜理想进行了分类（第一类对合有8个，第二类对合有4个），并利用这一分类结果通过非交换多项式在这些代数中的值来刻画它们.作为应用，我们证明了一个多项式是$d\times d$矩阵的多项式恒等式和当且仅当它在$d\times d$矩阵代数中的所有值都有零迹.
英文摘要：
A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\times d$ matrices if and only if all of its values in the algebra of $d\times d$ matrices have zero trace.
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