Note 1
Kac-Moody Lie Algebra
1 <Cartan matrix>
Base field K
Finite index set I
Square matrix that has elements by integer A = ( aij )i, j ∈ I
Matrix that satisfies the next is called Cartan matrix.
i, j ∈ I
(1) aii = 2
(2) aij ≤ 0 ( i ≠j )
(3) aij = 0 ⇔ aji = 0
2 <Fundamental root data>
Finite dimension vector space h
Linearly independent subset of h {hi}i∈I
Dual space of 
h*= HomK (h, K )
h*= HomK (h, K )
Linearly independent subset of h* {αi} i∈I
Φ = {h, {hi}i∈I, {αi} i∈I }
Cartan matrix A = {αi(hi)} I, j∈I
Φis called fundamental root data of A that is Cartan matrix.
3 <Lie algebra>
Cartan matrix A = {αi(hi)} I, j∈I
Fundamental root data Φ what A is Cartan matrix Φ = {h, {hi}i∈I, {αi} i∈I }
Lie algebra that is generated by {ah}h∈h ∪{
i ,
i }i∈I 
(Φ)
i ,i }i∈I (Φ)(Φ) satisfies the next.
h, h’ ∈ h c ∈ K i, j ∈ I
ah + ah’ = ah+h’
cah = ach
[ah, ah’] = 0
[ah, i] = αi(h)i
i] = αi(h)i
[ah,
i] = -αi(h)
i
i] = -αi(h)i
[i ,i] = 
ijahi
i ,i] = ijahi
4 <Kac-Moody Lie algebra>
Subset of (Φ) {ad(i)1-aij(j), ad(
i)1-aij(j)|i,j∈I, i ≠j }
(Φ) {ad(i)1-aij(j), ad(i)1-aij(j)|i,j∈I, i ≠j }
Ideal of the subset r0(Φ)
r0(Φ) = r0+(Φ) ⊕ r0-(Φ)
max(Φ) = (Φ)/ r0(Φ)max(Φ) is Lie algebra by definition.max(Φ) is called Kac-Moody Lie algebra attended with fundamental root data max(Φ).