Student Seminar: Classical and Quantum Integrable Systems

3. The Jacobi identity
[[ξ, η], ζ] + [[η, ζ], ξ] + [[ζ, ξ], η] = 0
Let J be a Lie algebra of dimension n. Choose a basis e 1 , · · · , e n ∈ J . We have
[e i , e j ] = C k ije k
The numbers C k ij are called structure constants of the Lie algebra. Upon changing
the basis these structure constants change as the tensor quantity. Let e ′ i = A j i e i and
[e ′ i, e ′ j] = C ′k
ij e ′ k then C ′k
ij A m k e m = A r i A s j[e r , e s ] = A r i A s jC m rse m
Thus, the structure constants in the new basis are related to the constants in the
original basis as
C ′k
ij = A r i A s jC m rs(A −1 ) k m . (6.2)
Skew-symmetry and the Jacobi identity for the commutator imply that the tensor
C k ij defines the Lie algebra if and only if
C k ij = −C k ij , C m p[iC p jk] = 0 .
Classify all Lie algebras means in fact to find all solutions of these equations modulo
the equivalence relation (6.2).
Example. The Lie algebra so(3, R) of the Lie group SO(3, R). It consists of 3 × 3
skew-symmetric matrices. We can introduce a basis in the space of these matrices
⎛
0 0 0
⎞
⎛ ⎞
0 0 1
⎛ ⎞
0 −1 0
X 1 = ⎝ 0 0 −1 ⎠ , X 2 = ⎝ 0 0 0 ⎠ , X 3 = ⎝ 1 0 0 ⎠ .
0 1 0
−1 0 0
0 0 0
In this basis the Lie algebra relations take the form
[X 1 , X 2 ] = X 3 , [X 2 , X 3 ] = X 1 , [X 3 , X 1 ] = X 2 .
These three relation can be encoded into one
[X i , X j ] = ɛ ijk X k .
Example. The Lie algebra su(2) of the Lie group SU(2). It consists of 2 × 2 skewsymmetric
matrices. The basis can be constructed with the help of the so-called
Pauli matrices σ i
σ 1 =
( ) 0 1
, σ 2 =
1 0
( )
( )
0 −i
1 0
, σ 3 = .
i 0
0 −1
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