Trace (linear Algebra) - Lie Algebra

Lie Algebra

The trace is a map of Lie algebras from the Lie algebra gl_n of operators on a n-dimensional space ( matrices) to the Lie algebra k of scalars; as k is abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes:

The kernel of this map, a matrix whose trace is zero, is often said to be traceless or tracefree, and these matrices form the simple Lie algebra sl_n, which is the Lie algebra of the special linear group of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the special linear algebra is the matrices which infinitesimally do not change volume.

In fact, there is an internal direct sum decomposition of operators/matrices into traceless operators/matrices and scalars operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as:

Formally, one can compose the trace (the counit map) with the unit map of "inclusion of scalars" to obtain a map mapping onto scalars, and multiplying by n. Dividing by n makes this a projection, yielding the formula above.

In terms of short exact sequences, one has

which is analogous to

for Lie groups. However, the trace splits naturally (via times scalars) so but the splitting of the determinant would be as the nth root times scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose: