Quantum-Nerve Theory Part 0 Cicerone to QNT and Part 3 Basic Paper added Text partly revised 2019

Quantum-Nerve Theory Part 0 Cicerone to QNT and Part 3 Basic Paper added 2019

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Quantum-Nerve Theory

Abbreviation  :  QNT

TANAKA Akio

Part 0

Cicerone to QNT

Macro Time and Micro Time  2013

For WITTGENSTEIN Ludwig  Position of Language  2005 - 2014

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Part  1

Making nerve's mathematical model

What is signal? The existence that generates language

Preface, Preparation, Preparation 2, References, Main Issue 1-5,  Anticipation - FAQ

and Comment
Contents Edition

21 November 2018 - 24 April 2019

TANAKA  Akio

Sekinan Library

Tokyo

Original Title 

What is signal? A mathematical model of nerve  ​

For father and mother

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Quantum-Nerve Theory

Part 2

Physical mathematics from quantum group

0

Inspiration The Time of Quantum 2008

1

Quantum Theory for Language 2004

Place where Quantum of language Exists 2004
Frame-Quantum Theory 2005

Frame-Quantum-Time Theory 2005

Quantum Warp Theory Warp 2005

Quantum Linguistics Solidity and Flexibility 2006

Quantum Language Substantiality of Language 2006

2

1.Clifford Algebra Note 1 From Super Space to Quantization 2008

2. Kac-Moody Lie Algebra Note 2 Quantum Group 2008

3. Symplectic Language Theory Note 6 Homological Mirror Symmetry Conjecture by KONTSEVICH​ 2009

4. Floer Homology Language Note 7 Quantization of Language​ 2009

5. Floer Homology Language Note 8 Discreteness of Language​ 2009

6. Simplicial Space Language  Composition of Word 2013

3

1. Floer Homology Language 2009

2. Homology Structure of Word 2009

3. Quantization of Language 2009

4. Discreteness of Language 2009

5. Proto Arithmetic Geometry Language 2009

6.Language, Word, Distance, Meaning and Meaning Minimum 2009

4

1. Manuscript of Quantum Theory for Language Note added 2003

2. Quantum Theory for Language 2004

3. Quantum Theory for Language  Theoretical Summarization and Problem in Future 2004

4. Noncommutative Distance Theory  Note 2  C*-Algebra 2007

5. Kac-Moody Lie Algebra 2008

Read more: https://srfl-paper.webnode.com/news/quantum-nerve-theory-2019/

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Part 3 
Basic Paper

Kac-Moody Lie Algebra

Note 2

Quantum Group

TANAKA Akio

1

Base field     K

Finite index set     I

Square matrix that has elements by integer     A = ( a_ij )_{i, j ∈ I}

Matrix that satisfies the next is called Cartan matrix.

i, j ∈ I

(1) a_ii = 2

(2) a_ij ≤ 0  ( i ≠j )

(3) a_ij = 0 ⇔ a_ji = 0

2

Cartan matrix     A = (a_ij)_i, j ∈I

Family of positive rational number    {d_i}_i∈I

Arbitrary i, j∈I    d_ia_ij = d_ja_ji

A is called symmetrizable.

3

Finite dimension vector space     h

Linearly independent subset of h     {h_i}_i∈I

Dual space of h     h*= Hom_K (h, K )

Linearly independent subset of h*     {α_i} _i∈I

Φ = {h, {h_i}_i∈I, {α_i} _i∈I }

Cartan matrix A = {α_i(h_i)} _{I, j∈I}

Φis called fundamental root data of A that is Cartan matrix.

4

Symmetrizable Cartan matrix    A = (a_ij)_i, j ∈I

Fundamental root data     {h, {h_i}_i∈I, {α_i} _i∈I }

E = α_i ⊂h*

Family of positive rational number     {d_i}_i∈I

d_ia_ij = d_ja_ji

Symmetry bilinear form over E     ( , ) : E×E → K     ( (α_i ,α_j ) = d_ia_ij )

The form is called standard form.

5

n-dimensional Euclid space    Rⁿ

Linear independent vector     v₁, …, v_n

Lattice of Rⁿ     m₁v₁+ … +m_nv_n     ( m₁, …, m_n ∈ Z )

Lattice of h     h_Z

6

From the upper 3, 4 and 5, the next three components are defined.

(Φ, ( , ), h_Z )

When the components satisfy the next, they are called integer fundamental root data.

 i ∈ I

(1)  ∈ Z

(2) α_i ( h_z ) ⊂ Z

(3) t_i :=  h_i ∈ h_z

7

Vector space over K     A

Bilinear product over K     A×A → A

When A is ring, it is called associative algebra.

8

Integer     m

t similarity of m    [m]_t

[m]_t = t^m-t^-m / t- t^-1

Integer   m, n   m≧n≧0

Binomial coefficient     (^m_n)

t similarity of m!     [m]_t! = [m]_t! [m-1]_t!...[1]_t

t similarity of (^m_n)    [^m_n]_t = [m]_t! / [n]_t! [m-n]_t!

[^m₀] = [^m_m]_t = 1

8

Integer fundamental root data that has Cartan matrix A = ( a_ij )_{i, j ∈ I}

      Ψ = ((h, {h_i}_i∈I, {α_i} _i∈I ), ( , ), h_z )

Generating set     {K^h}_h∈hz ∪{E_i, F_i}_i∈I

Associative algebra U over K (q), that is defined the next relations, is called quantum group associated with Ψ.

(1) k^hk^h’ = k^h+h’     ( h, h’∈h_Z )

(2) k⁰ = 1

(3) K^hE_iK^-h = q^αi(h)E_i    ( h∈h_Z , i∈I )

(4) K^hF_iK^-h = q^αi(h)F_i   ( h∈h_Z , i∈I )

(5) E_i F_j – F_jE_i = _ij  K_i - K_i^-1 / q_i – q_i^-1     ( i , j∈I )

(6) ^p [^1-aij_p]_qiE_i^1-aij-pE_jE_i^p = 0     ( i , j∈I , i ≠j )

(7) ^p [^1-aij_p]_qiF_i^1-aij-pF_jF_i^p = 0     ( i , j∈I , i ≠j )

[Note]

Parameter q in K is thinkable in connection with the concept of 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

www.sekinan.org

Read more: https://srfl-lab.webnode.com/products/kac-moody-lie-algebra-note-2-quantum-group/

Note 

13 October 2019

Text Partly revised

SRFL Paper 

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Tokyo
11 October 2019
SRFL Paper

Read more: https://srfl-paper.webnode.com/news/quantum-nerve-theory-part-0-cicerone-to-qnt-and-part-3-basic-paper-added-2019/