Special Unitary Group - Lie Algebra

Lie Algebra

The Lie algebra corresponding to SU(n) is denoted by su(n). Its standard mathematical representation consists of the traceless antihermitian n × n complex matrices, with the regular commutator as Lie bracket. A factor i is often inserted by particle physicists, so that all matrices become Hermitian. This is simply a different, more convenient, representation of the same real Lie algebra. Note that su(n) is a Lie algebra over R.

If we choose an (arbitrary) particular basis, then the subspace of traceless diagonal n × n matrices with imaginary entries forms an n - 1 dimensional Cartan subalgebra.

Complexify the Lie algebra, so that any traceless n × n matrix is now allowed. The weight eigenvectors are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra h is only n − 1 dimensional, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which should not be thought of as an element of the Lie algebra!) for the purpose of computing weights and that only. So, we have a basis where the i-th basis vector is the matrix with 1 on the i-th diagonal entry and zero elsewhere. Weights would then be given by n coordinates and the sum over all n coordinates has to be zero (because the unit matrix is only auxiliary).

So, SU(n) is of rank n − 1 and its Dynkin diagram is given by A_{n − 1}, a chain of n − 1 vertices. Its root system consists of n(n − 1) roots spanning a n − 1 Euclidean space. Here, we use n redundant coordinates instead of n − 1 to emphasize the symmetries of the root system (the n coordinates have to add up to zero). In other words, we are embedding this n − 1 dimensional vector space in an n-dimensional one. Then, the roots consists of all the n(n − 1) permutations of (1, −1, 0, ..., 0). The construction given two paragraphs ago explains why. A choice of simple roots is

…,

Its Cartan matrix is

Its Weyl group or Coxeter group is the symmetric group S_n, the symmetry group of the (n − 1)-simplex.