Student Seminar: Classical and Quantum Integrable Systems

i.e. H is an ideal which contradicts to the assumption that J is the semi-simple Lie
algebra.
Cartan subalgebra. To demonstrate the construction of the adjoint representation
and introduce the notion of the Cartan subalgebra of the Lie algebra we use the
concrete example of su(3). The Lie algebra su(3) comprises the matrices of the form
iM, where M is traceless 3×3 hermitian matrix. The basis consists of eight matrices
which we chose to be the Gell-Mann matrices:
⎛ ⎞
⎛ ⎞
⎛ ⎞
0 1 0
0 −i 0
1 0 0
λ 1 = ⎝ 1 0 0 ⎠ , λ 2 = ⎝ i 0 0 ⎠ , λ 3 = ⎝ 0 −1 0 ⎠
0 0 0
0 0 0
0 0 0
⎛ ⎞
⎛ ⎞
⎛ ⎞
0 0 1
0 0 −i
0 0 0
λ 4 = ⎝ 0 0 0 ⎠ , λ 5 = ⎝ 0 0 0 ⎠ , λ 6 = ⎝
1 0 0
i 0 0
⎛ ⎞
⎛ ⎞
0 0 0
λ 7 = ⎝ 0 0 −i ⎠ , λ 8 = 1 1 0 0
√ ⎝ 0 1 0 ⎠ .
3
0 0 0
0 0 −2
0 0 1
0 1 0
There are two diagonal matrices among these: λ 3 and λ 8 which we replace by T z =
1
λ 2 3 and Y = √ 1
3
λ 8 . We introduce the following linear combinations of the generators
t ± = 1 2 (λ 1 ± iλ 2 ) , v ± = 1 2 (λ 4 ± iλ 5 ) , u ± = 1 2 (λ 6 ± iλ y ) .
One can easily compute, e.g.,
[t + , t + ] = 0 , [t + , t − ] = 2t z , [t + , t z ] = −t + , [t + , u + ] = v + , [t + , u − ] = 0 ,
[t + , v + ] = 0 , [t + , v − ] = −u − , [t + , y] = 0 .
Since the Lie algebra of su(3) is eight-dimensional the adjoint representation is eightdimensional
too. Picking up (t + , t − , t z , u + , u − , v + , v − , y) as the basis we can realize
the adjoint action by 8 × 8 matrices. For instance,
⎛ ⎞ ⎛
⎞ ⎛ ⎞
t + 0 0 0 0 0 0 0 0 t +
t −
0 0 2 0 0 0 0 0
t −
t z
−1 0 0 0 0 0 0 0
t z
ad t+ u +
u −
=
0 0 0 0 0 1 0 0
u +
0 0 0 0 0 0 0 0
u −
v +
0 0 0 0 0 0 0 0
v +
⎜ ⎟ ⎜
⎟ ⎜ ⎟
⎝ v − ⎠ ⎝ 0 0 0 0 −1 0 0 0 ⎠ ⎝ v − ⎠
y 0 0 0 0 0 0 0 0 y
} {{ }
matrix realization of t +
⎠
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