Kac-Moody Lie Algebra
Conjecture 1
Finiteness in Infinity on Language
TANAKA Akio
1 Total of words is finite at a certain point of time.
2 Total of finite words combination, phrase, is probably finite.
3 Sentence is free combination of words. So total of sentences is seemed to be infinite.
4 Now cognition of sentence is probably seemed to be based on finite condition of language.
5 If sentences be really infinite, cognition of them is at last seemed to be impossible.
6 Sentence’s appearance of infinity is possibly finite, or finite in infinite.
7 From freeness on generation of sentence, sentence is radically infinite and from cognition of that, sentence is seemed to be a certain type of finite in infinity.
[References]
Subdivision / Tokyo April 10, 2005
Algebraic Linguistics /Linguistic Result /Deep Fissure between Word and Sentence /Tokyo September 10, 2007
Tokyo February 10, 2008
Sekinan Research Field of Language
[Basis February 26, 2008]
Operator Algebra / Note 1 / Differential Operator and Symbol / Tokyo February 24, 2008
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
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 = ( 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 <Fundamental root data>
Finite dimension vector space h
Linearly independent subset of h {hi}i∈I
Dual space of 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.
3 <Lie algebra>
Cartan matrix A = {αi(hi)} I, j∈I
Fundamental root data Φ what A is Cartan matrix Φ = {h, {hi}i∈I, {αi} i∈I }
Lie algebra that is generated by {ah}h∈h ∪{i ,i }i∈I (Φ)
(Φ) satisfies the next.
h, h’ ∈ h c ∈ K i, j ∈ I
ah + ah’ = ah+h’
cah = ach
[ah, ah’] = 0
[ah, i] = αi(h)i
[ah,i] = -αi(h)i
[i ,i] = ijahi
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
Kac-Moody Lie Algebra
Assistant Site: sekinanlogos
Note
1 Kac-Moody Lie Algebra
2 Quantum Group
Conjecture
1 Finiteness in Infinity on Language
Tokyo
11 July 2015
Sekinan Research Field of Language
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