Kac-Moody Lie Algebra Note 2 Quantum Group

Note 2
Quantum Group
 
 
 
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 <Symmetrizable>
Cartan matrix     = (aij)ij I
Family of positive rational number    {di}iI
Arbitrary i, jI    diaij djaji
A is called symmetrizable.
3 <Fundamental root data>
Finite dimension vector space     h
Linearly independent subset of h     {hi}iI
Dual space of h     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.
4 <Standard form>
Symmetrizable Cartan matrix    = (aij)ij I
Fundamental root data     {h, {hi}iI, {αi} i}
E = αh*
Family of positive rational number     {di}iI
diaij = djaji
Symmetry bilinear form over E     ( , ) : E×E → K     ( (α,α) = diaij )
The form is called standard form.
5 <Lattice>
n-dimensional Euclid space    Rn
Linear independent vector     v1, …, vn
Lattice of Rn     m1v1+ … +mnvn     ( m1, …, mn ∈ Z )
Lattice of h     hZ
6 <Integer fundamental root data>
From the upperv3, 4 and 5, the next three components are defined.
(Φ, ( , ), h)
When the components satisfy the next, they are called integer fundamental root data.
 ∈ I
(1)  ∈ Z
(2) αhz ) ⊂ Z
(3) t:=  hi ∈ hz
7 <Associative algebra>
Vector space over K     A
Bilinear product over K     A×A → A
When A is ring, it is called associative algebra.
8 <Similarity>
Integer     m
t similarity of m    [m]t
[m]= tm-t-m / tt-1
Integer   m  mn≧0
Binomial coefficient     (mn)
t similarity of m!     [m]t! = [m]t! [m-1]t!...[1]t
t similarity of (mn)    [mn]t = [m]t! / [n]t! [m-n]t!
[m0] = [mm]t = 1
8 <Quantum group>
Integer fundamental root data that has Cartan matrix = ( aij )i, j  I
      Ψ = ((h, {hi}iI, {αi} i), ( , ), h)
Generating set     {Kh}hh∪{EiFi}iI
Associative algebra U over K (q), that is defined the next relations, is called quantum group associated with Ψ.
(1) khkh = kh+h     ( hh’∈hZ )
(2) k0 = 1
(3) KhEiK-qαi(h)Ei    hhZ , i)
(4) KhFiK-qαi(h)Fi   ( hhZ , i)
(5) Ei Fj – FjEi ij  Ki - Ki-1 qi – qi-1     ( i , j)
(6) p [1-aijp]qiEi1-aij-pEjEip = 0     ( i , jI , i ≠)
(7) p [1-aijp]qiFi1-aij-pFjFip = 0     ( i , ji ≠)
 
[Note]
Parameter in K is thinkable in connection with the concept of <jump> at the paper Place where Quantum of Language exists / 27 /.
Refer to the next.