Note 2
Quantum Group
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 <Symmetrizable>
Cartan matrix A = (aij)i, j ∈I
Family of positive rational number {di}i∈I
Arbitrary i, j∈I diaij = djaji
A is called symmetrizable.
3 <Fundamental root data>
Finite dimension vector space h
Linearly independent subset of h {hi}i∈I
Dual space of h 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.
4 <Standard form>
Symmetrizable Cartan matrix A = (aij)i, j ∈I
Fundamental root data {h, {hi}i∈I, {αi} i∈I }
E = 
αi ⊂h*
αi ⊂h*
Family of positive rational number {di}i∈I
diaij = djaji
Symmetry bilinear form over E ( , ) : E×E → K ( (αi ,αj ) = 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, ( , ), hZ )
When the components satisfy the next, they are called integer fundamental root data.
i ∈ I
(1) 
∈ Z
∈ Z
(2) αi ( hz ) ⊂ Z
(3) ti := hi ∈ hz
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]t = tm-t-m / t- t-1
Integer m, n m≧n≧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 A = ( aij )i, j ∈ I
Ψ = ((h, {hi}i∈I, {αi} i∈I ), ( , ), hz )
Generating set {Kh}h∈hz ∪{Ei, Fi}i∈I
Associative algebra U over K (q), that is defined the next relations, is called quantum group associated with Ψ.
(1) khkh’ = kh+h’ ( h, h’∈hZ )
(2) k0 = 1
(3) KhEiK-h = qαi(h)Ei ( h∈hZ , i∈I )
(4) KhFiK-h = qαi(h)Fi ( h∈hZ , i∈I )
(5) Ei Fj – FjEi = 
ij Ki - Ki-1 / qi – qi-1 ( i , j∈I )
ij Ki - Ki-1 / qi – qi-1 ( i , j∈I )
(6) 
p [1-aijp]qiEi1-aij-pEjEip = 0 ( i , j∈I , i ≠j )
p [1-aijp]qiEi1-aij-pEjEip = 0 ( i , j∈I , i ≠j )
(7) p [1-aijp]qiFi1-aij-pFjFip = 0 ( i , j∈I , i ≠j )
p [1-aijp]qiFi1-aij-pFjFip = 0 ( i , j∈I , i ≠j )
[Note]
Parameter q 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.