Day 20 (2/12/14)

Time: 8 - 2:30

After a long break, I met up with Professor Dolbin again to continue my internship. We briefly reviewed the basic notions of Lie algebras, and then he let me study independently, with frequent advice and guidance. As I learned previously, Lie algebras form a special family of mathematical structures, with the introduction of a new operator [x,y] for elements x and y. Specifically, these structures are not commutable (i.e. [x,y] is not equal to [y,x]), but rather they have a very similar property: namely, they satisfy the equation

[x,y]=-[y,x].

Also, they are not associative (i.e. [x[y,z]] is not equal to [[x,y],z]), but they satisfy another very similar and cool identity, called the Jacobi identity. It states that for any x, y, and z in the Lie algebra, we have that

[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.

These structures carry a lot of significance and cool properties, many of which are derived using linear algebra. And because I never had an extensive experience with linear algebra, I spent most of today reviewing the necessary knowledge that I needed to continue learning about these Lie algebras.

We concluded today's internship with a brief discussion about a very complex branch of mathematics called Category Theory. Category Theory is an extension of all of these abstract algebraic structures that I learned, including groups, rings, and Lie algebras. In a sense, categories are generalizations of all of these structures, and this explains why in a general perspective all of the major theorems and facts about these structures can be proven in similar fashions. Sadly, we had to stop at this point for today, but I can't wait to continue learning more about these abstract notions even more!