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

  

Kac-Moody Lie Algebra

Note 1

Kac-Moody Lie Algebra

 

TANAKA Akio

 

 

1 <Cartan matrix>

Base field     K

Finite index set     I

Square matrix that has elements by integer     A = ( a_ij )_{i, j ∈ I}

Matrix that satisfies the next is called Cartan matrix.

i, j ∈ I

(1) a_ii = 2

(2) a_ij ≤ 0  ( i ≠j )

(3) a_ij = 0 ⇔ a_ji = 0

2 <Fundamental root data>

Finite dimension vector space     h

Linearly independent subset of h     {h_i}_i∈I

Dual space of      h*= Hom_K (h, K )

Linearly independent subset of h*     {α_i} _i∈I

Φ = {h, {h_i}_i∈I, {α_i} _i∈I }

Cartan matrix A = {α_i(h_i)} _{I, j∈I}

Φis called fundamental root data of A that is Cartan matrix.

3 <Lie algebra>

Cartan matrix A = {α_i(h_i)} _{I, j∈I}

Fundamental root data Φ what A is Cartan matrix     Φ = {h, {h_i}_i∈I, {α_i} _i∈I }

Lie algebra that is generated by {a_h}_h∈h ∪{_i ,_i }^i∈I     (Φ)

(Φ) satisfies the next.

h, h’ ∈ h   c ∈ K   i, j ∈ I

a_h + a_h’ = a_h+h’

ca_h = a_ch

[a_h, a_h’] = 0

[a_h, _i] = α_i(h)_i

[a_h,_i] = -α_i(h)_i

[_i ,_i] = _ija_hi

4 <Kac-Moody Lie algebra>

Subset of (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).

 

 

Tokyo February 7, 2008

 

Sekinan Research Field of Language

 

www.sekinan.org