Grant: Discovery Grants Program - Individual

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.