Thursday 13 September 2018

Kac-Moody Lie Algebra Note 1 Kac-Moody Lie Algebra 2008

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


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