7. Lie algebras

Lie algebras are a fundamental noncommutative algebraic structure that is related to symmetries. You have already met the simplest interesting Lie algebra in secondary school. That is, the vectors in the space with the cross product. But this is just the very first example in one of many families of the so-called simple Lie algebras.

In Lie algebras, one can add and subtract freely and so one has a multiplication × that satisfies

• a × b = - b × a,
• a × (b × c) + b × (c × a) + c × (a × b) = 0.

Lie algebras are of fundamental importance in mathematics and in theoretical physics. In many ways, one can say that Lie theory is the axis about which much of mathematics and theoretical physics turns around. It is not unreasonable to say that in mathematics one finds three god-given structures, they are the natural numbers, the spheres of various dimensions, and the so-called simple Lie algebras. The study of Lie algebras was initiated by Sophus Lie in the 1870s and the classification of the simple Lie algebras was completed in 1895 by W. Killing, in an article that later came to be known as "the greatest mathematical paper of all times".