Student Seminar: Classical and Quantum Integrable Systems

we have
ad x t + = at +
ad x t − = at −
ad x t z = 0t z
ad x u + = (− 1 2 a + b)u +
ad x u − = ( 1 2 a − b)u −
ad x v + = ( 1 2 a + b)v +
ad x v − = (− 1 2 a − b)v −
ad x y = 0y .
We see that all eigenvalues are linear functions of the Cartan element x, in other
words, if we denote by e α the six elements t ± , v ± , u ± and by h i the two Cartan
elements t z , y we can write all the relations above as
[h i , h j ] = 0
[h i , e α ] = α(h i )e α ,
where α(h i ) is a linear function of h i . The generators e α , which are eigenstates of the
Cartan subalgebra, are called root vectors, while the corresponding linear functions
α(h) are called roots. To every root vector e α we associate the root α which is a
linear function on the Cartan sualgebra H. Linear functions on H, by definition,
form the dual space H ∗ to the Cartan subalgebra H.
The Cartan-Weyl basis. Now we can also investigate what is the commutator of
the root vectors. By using the Jacobi identity we find
[h, [e α , e β ]] = −[e α , [e β , h]] − [e β , [h, e α ]] = (α(h) + β(h))[e α , e β ] .
This clearly means that there are three distinct possibilities
• [e α , e β ] is zero
• [e α , e β ] is a root vector with the root α + β
• α + β = 0 in which case [e α , e β ] commutes with every h ∈ H and, therefore, is
an element of the Cartan subalgebra.
Thus,
[e α , e β ] = N αβ e α+β
– 93 –