Distance Theory Algebraically Supplemented Note 1 Ring

Distance Theory Algebraically Supplemented

Note 1

Ring

TANAKA Akio

1

<Ring>     To be meant commutative ring containing 1 and defining the operation addition and multiplication

<Ring of rational integers>     Set of all the integers containing the operation addition and multiplication

<Ring of polynomials in n variables over k>     Ring that is what k is field x₁, …, x_n is variables and k-coefficient n-variables all the polynomials set has addition and multiplication.

<Zero divisor>     When commutative ring R that has element a , there exists element b ≠ 0 in the condition ab = 0. a is zero divisor.

<Integral domain>     Ring that has not zero divisor except 0

<Ideal of R>    Subset I that satisfies next conditions in R  

(1) a, b ∈ I ⇒ = -a +b ∈ I

(2) a ∈ I, r ∈ R ⇒ ra ∈ I

Ideal defines addition, multiplication and quotient.

<Prime ideal of R>     Ring R’s ideal p≠R     a, b∈R, ab∈p ⇒ a∈p or b∈p

<Maximal ideal of R>     Ring R’s ideal p≠R     m ⊂_≠ a ⊂_≠ R     When there does not exist ideal a, m is maximal ideal.

<Principal ideal>     Ideal generated by ring R’s one element a   (a) = { xa | x=R }

<Principal ideal ring>     R’s all the ideal are principal ideals

<Principal ideal domain>     Principal ideal ring when R is integral domain

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<Radical> of a     Commutative ring R’s ideal a is in the condition √a = { a ∈ R | Natural number n has aⁿ ∈ a }.

<Nilradical>       √(0)

<Primary ideal>     R is ring. q ( ≠R ) is ideal of R.     When a, b ∈ R, ab ∈ q and a ∉ q, there exists natural number n and bⁿ ∈ q.

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<Multiplicatively closed set>

R is ring. S is subset of R. When S has next conditions, S is multiplicatively closed set.

(1) x, y∈S   xy∈S

(2) 1∈S

(3) 0 ∉ S

<Quotient ring>

Class containing elements (r, s) is expressed by r/s.

When S^-1R (set of class r/s) has next conditions, S^-1R is quotient ring. Quotient ring has unit 1/1 and zero element is 0/1.

(1) (a₁/s₁) + (a₂/s₂) = (a₁s₂ + a₂s₁)/s₁s₂

(2) (a₁/s₁)(a₂/s₂) = a₁a₂/s₁s₂

<Ring of total quotients>

R is ring. S is all the non-zero divisors. q( R ) = S^-1R is ring of total quotient.

<Quotient field>

q( R ) that has inverse elements except element 0

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<Local ring>

Commutative ring that has only one maximal ring

<Residue field>

R is local ring. m is R’s maximal ideal. R/m is residue field of R.

<Localization>

R’s prime ideal is p. S = R∖p is multiplicatively closed set.

S = R∖p    Rp = S^-1R    Rp is localization of R at p.

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<Noetherian ring>

Noetherian ring satisfies next conditions.

(1) Commutative ring R has maximal one in arbitrary set that ideals of R make.

(2) Infinite sequence of R has number N that is a_N = a _N+1 = … .

(3) Arbitrary ideal of R is finitely generated.

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<Hilbert basis theorem>

When R is Noetherian ring, ring of polynomials in n-variables over R is also Noetherian ring.

[Reference]

Algebraic Linguistics / Linguistic Result / Deep Fissure between Word and Sentence / Tokyo September 10, 2007

Tokyo October 4, 2007

Sekinan Research Field of Language

Noncommutative Distance Theory Kontsevich Invariant

Noncommutative Distance Theory

Note 5

Kontsevich Invariant

TANAKA Akio

R³ : = C × R

Knot     K

Parameter of height     t

Two points on K at t     z ( t )  z’ ( t )

Selected point of z and z’     P

z and z’ Code figure on S¹     D_P

Iteration integral     Z’ ( K ) : = Σ_m=0^∞  ×(-1)^#P1D_P

Quotient vector space that is quoted by 3 relations ( AS, IHX and STU )* over C on which Jacobi figure is described A ( S¹ )

Kontsevich invariant    Z ( K ) ∈ A ( S¹ )

[Note]

*3 relations ( AS, IHX and STU ) are seemed to be related with characters’ descriptive system.

Tokyo January 3, 2008

Sekinan Research Field of Language

Noncommutative Distance Theory Atiyah’s Axiomatic System

Noncommutative Distance Theory

Note 4

Atiyah’s Axiomatic System

TANAKA Akio

1

ATIYAH Michael’s axiomatic system of topological quantum field theory, abbreviated TQFT, is shown below.

Oriented smooth compact d-dimensional manifold     Σ

Finite dimensional complex vector space     Z ( Σ )

d + 1 dimensional manifold that has boundary     Y

Functor     Z    

Z ( Y ) ∈ Z ( Σ )

Axiom 1   Z ( Σ* ) = Z ( Σ )*   Σ* is reverse orientation of Σ. Z ( Σ )* is dual space of Z ( Σ ).

Axiom 2   Z ( Σ₁⋃Σ₂) = Z ( Σ₁ ) ⊗ Z ( Σ₂ )

Axiom 3   Z ( Y ) = Z ( Y₂ ) ◌ Z ( Y₁ )

Axiom 4   Z ( 0 ) = ℂ

Axiom 5   Z ( Σ×I ) = id_{Z( Σ )}

2

In TQFT, when Z ( Y₁ ) = Z ( Y₂ ) and the both are connected, Y becomes a compact manifold.

3

The generated manifold has meridian α and longitude β.

α is oriented by Σ.

β is seemed to be time development by axiom 5.

4

On algebra, α is corresponded to monodromy and β is corresponded to Frobenius automorphism.

5

From algebraic number field K’s Galois extension L/K and K’s prime ideal p, Frobenius automorphism is defined.

6

From prime ideal, prime number is considered for the roots of space that has orientation which shows the distance.

Tokyo December 22, 2007

Sekinan Research Field of Language

Noncommutative Distance Theory Point Space

Noncommutative Distance Theory

Note 3

Point Space

TANAKA Akio

1

Set     X

Points over X     x, y

Distance d over X satisfies next conditions.

d ( x, y ) is nonnegative real number value.

Axiom 1   d ( x, y ) = 0 ⇔ x = y

Axiom 2   d ( x, y ) = d ( y, x )

Axiom 3   d ( x, y ) + d ( y, z) ≥ d ( x, z )

Distance space is ( X, d ).

Distance space is commutative.

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Manifold    ε, M

Differentiable map   π : ε → M

Manifold     E

Open set of M     U_i

Diffeomorphism     φ_i : π ^-1 (U_i) → U_i ×E

E     Fiber

(ε, π)     Fiber bundle

Ε     Total space

M     Base space

x ∈ U_i ∩ U_j

Linear isomorphism over E    φ_i ○φ_j : {x} ×E → {x} ×E

Diffeomorphism     φ_i : π ^-1 (U_i) → U_i ×E

Vector bundle    π : ε → M

Diffeomorphism     s : M → ε

π ( s ( x ) ) = x

s is cross section of vector bundle π.

Set of infinite differentiable cross section     Γ ( M, ε )

Lie group     G that is structure group

Fiber    G

Fiber bundle     π : P → M

(p ·g) ·h = p·(gh),   p ∈ P,   g, h ∈ G

π(p ·g) = π(p)   p ∈ P, g ∈ G

P     Principle bundle

Manifold    E

Group that is all the diffeomorphic of E     Diff ( E )

ρ : G → Diff ( E )

Direct product    P × E

Equivalence relation     ( p ·g, f ) ~ ( p, ρ(g)f )

Quotient space P ×_G E becomes associated bundle.

Group that is all the endomorphic of E     End ( E )

Representation     ρ : G → End ( E )

Principle bundle     P

Associated bundle     ε = P ×_G E

Dual vector space     E*

Dual representation     ρ* : G → End ( E *)

Associated bundle     ε* = P ×_G E*

Tangent vector bundle of M      TM   

Cross section of TM     X ∈ Γ ( M, TM )

Differential map     φ : M₁ →M₂

φ_* ( v ) ∈ T_φ(x) M₂   v ∈ T_xM₁

φ_* : TM₁ →TM₂

Vector bundle over M that fiber is R^N   ε

Fiber bundle     GL (ε)   Fiber over x ∈ M  is all the linear isomorphism from R^N to fiber ε_x .

GL (ε) is frame bundle of ε.     

Frame bundle of TM     GL ( TM )

Representation space of arbitrary representation ρ over GL ( n )     E

Tensor bundle of M     Associated bundle ε = GL ( TM ) ×_ρ E

Representation E has exterior algebra Γ(M, ΛT*M) that is called differential form of space Ω ( M ).

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Square matrix     A = ( a_ij )

Diagonal element     a_ii ( i = 1,2, …, n )

Here a_ii is expressed by a_i.

Now there gives i = 1,2, diagonal matrix is  A =(^a1₀ ⁰_a2)

Here A and a_i are seemed to be functions that expressed by f.

f = (^f1₀ ⁰_f2)

f₁ and f₂ is commutative.

Next there gives matrix D = (⁰_μ ^μ₀).

Linear differential form is defined by the next.

Df := [ D, f ] = (⁰_μ(f1-f2) ^μ(f2-f1)₀).

[Reference]

Distance Theory Algebraically Supplemented / 3 Point / Tokyo October 12, 2007

Tokyo December 9, 2007

Sekinan Research Field of Language

Noncommutative Distance Theory C*-Algebra

Noncommutative Distance Theory

Note 2

C*-Algebra

TANAKA Akio

1

<C*-algebra>

Complex field     C

Algebra over C     A

A algebra satisfies next conditions, it is called *-algebra.

Arbitrary x,y ∈ A

(xy)* = y*x*

(x*)* = x

A ∋ x ↦ x* ∈ A

Norm ||・||　of *-algebra A satisfies next conditions, it is called C*-norm.

Arbitrary x,y ∈ A

||xy|| ≤ ||x|| ||y||

||x*x|| = ||x||²

When algebra A is complete on C*-norm, it is called C*-algebra.

2

<Gel’fand-Naĭmark theorem>

Compact Hausdorf space     X

Universal continuous function over X     C ( X )

C ( X ) has identity element.

C ( X ) is called commutative C*-algebra.

When C ( X ) and C ( Y ) are equal as C*-algebra, X and Y are homeomorphism as space.

3

<Noncommutative 2 dimensional torus>

2-dimensional torus     T²

Function over T² is identified with double periodic function f(x,y) = f(x+2π, y) = f(x, y+2π).

Measurable function that has inner product makes Hilbert space L²(T²).

Operators that product function exp(ix) and exp(iy)     U and V

Sequence space     l²(Z) = { a = (a_n) : |a_n|² <  }

Operator U_θ     U (a)_n = a_n-1

Operator V_θ       V(a)_n = λⁿa_n-1  λ= exp (2πiθ)

V_θU_θ = λU_θV_θ

A_θ = C*( U_θ , V_θ ) is called noncommutative 2-dimensional torus.

When θ = 0, VU = UV , C(T²) is made again.

Tokyo December 4, 2007

Sekinan Research Field of Language

Noncommutative Distance Theory Groupoid

Noncommutative Distance Theory

Note 1

Groupoid

TANAKA Akio

1

<Open covering>

Topological space     X

Family of open sets     U_i ⊂ X ( i ∈ I )

Open covering is sum of the family　     X = ∪_i ∈ I U_i

On topological space, refer to the next.

Distance Theory Algebraically Supplemented / 1 Distance Preparatory Consideration / Tokyo October 8, 2007

On open covering, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 1-3 / Tokyo September 17, 2007

2

<Spectrum>

Set of prime ideals (≠1) over commutative ring A that has unit element 1 is spectrum of A. The expression is Spec A.

On prime ideal, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 2-1 / Tokyo September 20, 2007

3

<Affine open covering>

X = { U_i = Spec A_i }

On affine space, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 3-4 / Tokyo September 24, 2007

4

<Noetherian ring>

All the ideals of commutative ring R are generated from finite elements. R is Noetherian ring.

On Noetherian ring, refer to the next.

Algebraic Linguistics / Linguistic Premise / Premise of Algebraic Linguistics 3-1 / Tokyo September 23, 2007

5

<Locally Noetherian scheme>

Each A_i of affine open covering at X is Noetherian ring. X is locally Noetherian scheme.

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<Noetherian scheme>

When locally Noetherian scheme X is compact as topological space, X is Noetherian scheme.

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<Groupoid>

Noetherian scheme     S

Locally finite scheme over S     U, R

Arrow over S     s : R → U     t : R → U     μ : R ╳ _{t, U, s} R →R

Groupoid    ( U, R, s, t, μ)

Tokyo November 30, 2007

Sekinan Research Field of Language