Kac-Moody Lie Algebra Note 2 Quantum Group

01/10/2019 19:56
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*
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
(2) αi ( hz ) ⊂ Z
(3) ti :=  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 )
(6) 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 )
 
[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.
 
 
 

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