Student Seminar: Classical and Quantum Integrable Systems

Note that both ad tz and ad y are diagonal. Thus, if x = at z + by then ad x is also
diagonal. Explicitly we find
⎛
⎞
a 0 0 0 0 0 0 0
0 −a 0 0 0 0 0 0
0 0 0 0 0 0 0 0
ad x =
0 0 0 − 1 2 a + b 0 0 0 0
1
0 0 0 0
2 a − b 0 0 0
.
1
0 0 0 0 0
2
⎜
a + b 0 0
⎝ 0 0 0 0 0 0 − 1a − b 0 ⎟
2
⎠
0 0 0 0 0 0 0 0
In other words, the basis elements (t + , t − , t z , u + , u − , v + , v − , y) are all eigenvectors
of ad x with eigenvalues a, −a, 0, − 1a + b, 1a − b, − 1 a − b and 0 respectively. The
2 2 2
procedure we followed in crucial for analysis of other (larger) Lie algebras. We found
a two-dimensional subalgebra generated by t z and y which is abelian. Further, we
have chosen a basis for the rest of the Lie algebra such that each element of the basis
is an eigenvector of ad x if x is from this abelian subalgebra. This abelian subalgebra
is called the Cartan subalgebra.
In general the Cartan subalgebra H is determined in the following way. An
element h ∈ H is called regular if ad h has as simple as possible number of zero eigenvalues
(i.e. multiplicity of zero eigenvalue is minimal). For instance, for su(3) the
element ad tz has two zero eigenvalues, while ad y has for zero eigenvalues. Thus, the
element ad tz is regular, while ad y is not. A Cartan subalgebra is a maximal commutative
subalgebra which contains a regular element. In our example the subalgebra
generated by t z and y is commutative and its maximal since there is no other element
we can add to it which would not destroy the commutativity.
Roots. It is very important fact proved in the theory of Lie algebras that any simple
Lie algebra has a Cartan subalgebra and it admits a basis where each basis vector
is an eigenstate of all Cartan generators; the corresponding eigenvalues depend of
course on a Cartan generator. In our example of su(3) for an element x = at z + by
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