Chapter 7 - Cross-dimensional Lie algebra and Lie group

Abstract

Consider the quotient space of square matrices, Σ1, which is a vector space. First, we generalize the Lie algebraic structure of general linear algebra gl(n,R) to this dimension-free quotient space. With natural Lie-bracket, Σ1 becomes an Lie algebra. It is obvious that Σ1 is an infinite dimensional Lie algebra. But it is interesting that Σ1 has many interesting properties of finite dimensional Lie algebra, those are in general not true for other infinite dimensional Lie algebra.