论文标题
组环中的身份，包裹代数和泊松代数
Identities in group rings, enveloping algebras and Poisson algebras
论文作者
论文摘要
这是对组环，代数，泊松对称代数和其他相关代数结构的相同关系作品的简短调查。首先，Passman的经典作品指定的组环满足了非平凡的相同关系。这个结果是近距离研究项目的起源和动力。其次，Latyshev和Bahturin确定了代数，以使它们的普遍包围代数满足非平凡的相同关系。接下来，在限制包围代数的情况下，Passman和Petrogradsky解决了类似的问题。第三，Farkas开始研究泊松代数的相同关系。另一方面，Shestakov证明了任意谎言的对称代数$ s（l）$代数$ l $满足身份$ \ {x，x，\ {y，z \} \} \} \ equiv 0 $，并且只有当时，$ l $是Abelian。我们调查了有关（截断）泊松对称代数（lie代数）中相同关系的进一步结果。特别是，我们报告了有关（强）lies nilpotency和（截断）泊松对称代数和相关nilpotency类别的最新结果。另外，我们讨论实现这些结果的结构和方法。
This is a short survey of works on identical relations in group rings, enveloping algebras, Poisson symmetric algebras and other related algebraic structures. First, the classical work of Passman specified group rings that satisfy nontrivial identical relations. This result was an origin and motivation of close research projects. Second, Latyshev and Bahturin determined Lie algebras such that their universal enveloping algebra satisfies a non-trivial identical relation. Next, Passman and Petrogradsky solved a similar problem in case of restricted enveloping algebras. Third, Farkas started to study identical relations in Poisson algebras. On the other hand, Shestakov proved that the symmetric algebra $S(L)$ of an arbitrary Lie algebra $L$ satisfies the identity $\{x,\{y,z\}\}\equiv 0$ if, and only if, $L$ is abelian. We survey further results on existence of identical relations in (truncated) Poisson symmetric algebras of Lie algebras. In particular, we report on recent results on (strong) Lie nilpotency and (strong) solvability of (truncated) Poisson symmetric algebras and related nilpotency classes. Also, we discuss constructions and methods to achieve these results.
