Kac-Moody Lie Algebra Note 2 Quantum Group

Kac-Moody Lie Algebra
Note 2

Quantum Group

TANAKA Akio

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.

Place where Quantum of Language exists / Tokyo July 18, 2004

Tokyo February 9, 2008

Sekinan Research Field of Language

http://www.sekinan.org