Isomorphism problem for enveloping algebras

李代数自然而然地出现在数学和物理的许多领域。李代数是非交换的、非结合的对象。什么是“非对易”？每天非交换行为的一个例子是穿袜子和鞋子，因为这些行为的顺序确实会影响结果。相反，穿袜子和穿裤子是互换的。想一想，我们把一种化学物质倒入圆底烧瓶中，然后在烧瓶中加入另一种化学物质。在这里，我们首先将哪种产品倒入烧瓶中并不重要。换句话说，行动是可交换的。相比之下，“非关联性”涉及三个对象。例如，考虑将三种不同的化学品A、B和C混合在烧瓶中而得到的化学产品。所以秘诀是先把A和B加在一起，然后再加C。在数学公式中，我们可以把这个过程概括为C+(A+B)。相反，如果我们先将C和A相加，然后再加上B，我们可能会得到不同的产品。换句话说，C+(A+B)可能不等同于(C+A)+B。我们可以解释这一经验，并认为添加化学物质的行为是交换的而不是结合的。由于非结合使对象更难研究，所以人们以通用的方式将另一个仍然是非交换但具有结合的李代数与另一个代数联系在一起。现在，我们有兴趣知道从李代数的万能代数中可以推断出关于李代数的哪些信息。换句话说，如果两个泛代数相同，我们能说它们是相同的吗？这个问题的解决对相关领域也有很大的影响，也可以帮助回答其他问题。

Lie algebras arise naturally in many areas of Mathematics and Physics. Lie algebras are non-commutative and non-associative objects. What is "non-commutativity"? A daily example of a non-commutative action is to wear socks and shoes, as the order of these actions does effect the results. On the contrary to wear socks and to wear pants are commutative. Consider the action where we pour a chemical in a round-bottom flask and then add another chemical in the flask. Here, it doesn't matter which product we pour first into the flask. In other words, the action is commutative. In comparison "non-associativity" involves three objects. For example, consider a chemical product obtained by mixing three different chemicals A, B, and C in a flask. So the recipe is to add together A and B first and then add C. In Mathematics formulation, we can summarize this procedure as C+(A+B). Instead, if we first add C and A and then add B we could possibly get a different product. In other words, C+(A+B) may not be the same as (C+A)+B. We can interpret this experience and say that the action of adding chemicals is commutative bot not associative.Since non-associativity makes the objects harder to study, one associates in a universal way to every Lie algebra another algebra which is still non-commutative but it is associative. Now, it is of fundamental interest to know what information about a Lie algebra can be inferred from its universal algebra. In other words, if two universal algebras are the same, can we say the Lie algebras are the same? A solution to this question has a great impact in the related fields as well and can help to answer other questions.