Lie Groups and Lie Algebras

Abstract

The theory of differential equations had flourished to such a level by the 1860s that a systematic study of their solutions became possible. Sophus Lie, a Norwegian mathematician, undertook such a study using the same tool that was developed by Galois and others to study algebraic equations: group theory. The groups associated with the study of differential equations, now called Lie groups, unlike their algebraic counterparts, are uncountably infinite, and, as such, are both intricate and full of far-reaching structures. It was beyond the wildest dream of any 19th-century mathematician to imagine that a concept as abstract as Lie groups would someday find application in the study of the heart of matter. Yet, three of the four fundamental interactions are described by Lie groups, and the fourth one, gravity, is described in a language very akin to the other three.