Isomorphism problem for enveloping algebras

李代数自然地出现在数学和物理的许多领域。 李代数是非交换和非结合的对象。什么是“非交换性”？一个日常的非交换行为的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，穿袜子和穿裤子是可以互换的。* 考虑这样一个动作，我们将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，我们先倒入哪个产品并不重要。换句话说，这个作用是可交换的。相比之下，“非结合性”涉及三个对象。例如，考虑通过在烧瓶中混合三种不同的化学品A、B和C获得的化学产品。所以配方是先把A和B加在一起，然后再加C。在数学公式中，我们可以将此过程概括为C+（A+B）。相反，如果我们先加上C和A，然后加上B，我们可能会得到不同的产品。换句话说，C+（A+B）可能与（C+A）+B不同。我们可以解释这个经验，说添加化学物质的作用是交换的而不是联想的。由于非结合性使得对象更难研究，人们以一种普遍的方式将每个李代数与另一个仍然是非交换的但它是结合的代数相关联。现在，了解李代数的什么信息可以从它的泛代数中推导出来是一个基本的兴趣。换句话说，如果两个泛代数是相同的，我们能说李代数是相同的吗？这个问题的解决方案在相关领域也有很大的影响，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.**************************