The Premise for Quantum Group Language

Contents

Derived Category Language, 23 July 2016 Edition

1. Bridge between mathematics and physics / revised

2. Kontsevich's conjecture Category theoretic mirror symmetry conjecture

3. Derived Category Language 1

4. Karcevskij conjecture 1928 and Kawamata conjecture 2002

5. Additional meaning and embedding

6. Derived Category Language 2

7. Language stability and triangulated category

8. Derived Category Language 3

9. Stability of Language

10. Duality of Language

Text

Bridge across mathematics and physics / Revised

TODA Yukinobu. Several problems on derived category of coherent sheaf. Tokyo, 2016

TODA Yukinobu's Several problems on derived category of coherent sheaf has built across mathematics and physics. For my part, further more, physics and language seem to be expected to build over from the book.

Chapter 5. Page 116. Conjecture 5.16 shows us the connection between symplectic geometry and algebraic geometry.

I ever wrote several notes on language related with string theory. Now TODA's book newly lights up the relation between physics and language. This relation is really fantastic for me from now on.

[References]

Distance Theory Algebraically Supplemented

[References 2]

Isomorphism of map Sequence / Symplectic Language Theory

Kontsevich's conjecture
Category theoretic mirror symmetry conjecture

When there exists mirror relation between X1 and X2, derived category of X1's coherent sheaf and derived Fukaya category defined from X2's symplectic structure become equivalence.

M. Kontsevich. Homological algebra of mirror symmetry, Vol. 1 of Proceedings of ICM. 1995.

-----------------------------------------------------------------------------------------------------------------------

[Note by TANAKA Akio]
In the near future, symplectic geometry may be written by derived category. If so, complexed image of symplectic geometry's some theorems will become clearer.

References
Mirror Symmetry Conjecture on Rational Curve / Symplectic Language Theory / 27 February 2009

Homological Mirror Symmetry Conjecture by KONTSEVICH
/ Symplectic Language Theory / 26 April 2009

Quantization of Language / Floer Homology Language / 24 June 2009

Tokyo
10 May 2016
SRFL Theory

Tokyo
19 May 2016
SRFL Theory

Derived Category Language 1

Category theoretic mirror symmetry conjecture

1.
Fukaya category is
Fuk(X, ω).

2.
General symplectic manifold is

.
3.

Derived category of A_∞

4.
Conjecture

When

and

have physical mirror relation,

there exists the next triangle category's equivalence

[Reference 1]

Symplectic Language Theory Note 3 Mirror Symmetry Conjecture on Rational Curve

[References 2]

Tokyo

18 May 2016

SRFL Theory

Karcevskij conjecture 1928 and Kawamata conjecture 2002

TANAKA Akio

Sergej Karcevskij declared a conjecture for language's asymmetric structure on the TCLP of the Linguistic Circle of Prague in 1928. I briefly wrote about the conjecture as the following.

.....................................................................................................................................

Prague in 1920s, The Linguistic Circle of Prague and Sergej Karcevskij's paper "Du dualisme asymetrique du signe linguistique"

From Print 2012, Chapter 18

Non-symmetry. It was the very theme that I repeatedly talked on with C. Prague in 1920s. Karcevskij's paper "Du dualisme asymetrique du signe linguistique" that appeared in the magazine TCLP. Absolutely contradicted coexistence between flexibility and solidity, which language keeps on maintaining, by which language continues existing as language. Still now there will exist the everlasting dual contradiction in language. Why can language stay in such solid and such flexible condition like that. Karcevskij proposed the duality that is seemed to be almost absolute contradiction. Sergej Karcevskij's best of papers, for whom C called as the only genius in his last years' book Janua Linguisticae reserata 1994.

Source:

Tale / Print by LI Koh / 27 January 2012

Reference:

Fortuitous Meeting, What CHINO Eiichi Taught Me in the Class of Linguistics / 5 December 2004

Reference 2:

Linguistic Circle of Prague / 13 July 2012 - 19 July 2012

References 3:

References 4:

Reference 5:

Reversion Conjecture Revised / 1 May 2014 - 20 May 20

[Note, 2 October 2014]
In this Tale, Print 2012, C is CHINO Eiichi who was the very teacher in my life, taught me almost all the heritage of modern linguistics. I first met him in 1969 at university's his Russian class as a student knowing nothing on language study.

From Distance to Pseudo-Kobayashi Distance / 5 February 2012

Tokyo
23 February 2015
SIL
....................................................................................................................................
This asymmetric duality of linguistic sign presented by Karcevskij has become the prime mover for my study from the latter half of the 20th century being led by my teacher CHINO Eiichi.
But the theme was very hard even to find a clue. The turning point visited after I again learnt mathematics especially algebraic geometry in 1980s.
In 2009 I successively wrote the trial papers of the theme assisted by several results of contemporary mathematics. The papers are the following.
......................................................................................................................................

Notes for KARCEVSKIJ Sergej

TANAKA Akio

The papers on this site have been published by
Sekinan Research Field of Language

....................................................................................................................................

Entering in this year 2016, I read TODA Yukinobu's book, Several Problems on Derived Category of Coherent sheaf, Tokyo, 2016. The book shows me the update overview on derived category of coherent sheaf. The essence of my notable points are noted at the following.

....................................................................................................................................

Homological Mirror Symmetry Conjecture by KONTSEVICH
/ Symplectic Language Theory / 26 April 2009

Quantization of Language / Floer Homology Language / 24 June 2009

Tokyo
10 May 2016
SRFL Theory

............................................................................................................................................

In the TODA's book, I received the great hint on Karcevskij's conjecture for language's hard problem.
The hint exists at Kawamata conjecture presented in 2002. The details are the following.

............................................................................................................................................

Derived Category Language 2

Kawamata Conjecture

Conjecture

is birational map between smooth objective algebraic manifolds.

And

At This condition,

there exists next fully faithful embedding.

[Reference]

TODA Yukinobu. Several problems on derived category of coherent sheaf. Tokyo, 2016.Chapter 6, Derived category of coherent sheaf and birational geometry, page 148, Conjecture 6.43.

[References 2]

[Reference 3]

Stability of Language / Language and Spacetime

Tokyo

19 May 2016

SRFL Theory

............................................................................................................................

Kawamata conjecture will hint me the new meaning's entrance in the old meaning at a word.

Notes for KARCEVSKIJ Sergej that I ever wrote will be newly revised through TODA's fine work over viewing the recent 20 year development on derived category that began by Grothendieck.
For TODA's book, refer to the next my short essay.

...........................................................................................................................................

Bridge across mathematics and physics / Revised

TODA Yukinobu. Several problems on derived category of coherent sheaf. Tokyo, 2016

Chapter 5. Page 116. Conjecture 5.16 shows us the connection between symplectic geometry and algebraic geometry.

I ever wrote several notes on language related with string theory. Now TODA's book newly lights up the relation between physics and language. This relation is really fantastic for me from now on.

[References]

Distance Theory Algebraically Supplemented

[References 2]

Isomorphism of map Sequence / Symplectic Language Theory

Tokyo
19 May 2016
SRFL Theory

...............................................................................................................................................

This paper is unfinished.

Tokyo

20 May 2016

SRFL Theory

Additional meaning and embedding

1.Derived category

Category theoretic mirror symmetry conjecture

Fukaya category is
Fuk(X, ω).

2.
General symplectic manifold is

.
3.

Derived category of A_∞

4.
Conjecture

When

and

have physical mirror relation,

there exists the next triangle category's equivalence

2.Embedding

Derived Category Language 2

Kawamata Conjecture

Conjecture

is birational map between smooth objective algebraic manifolds.

And

At This condition,

there exists next fully faithful embedding.

3.Stable and mobile

Notes for KARCEVSKIJ Sergej

Condition of Meaning
TANAKA Akio
September 11, 2011

[Preparation]

Graded differential algebra

Minimal model of graded differential algebra

Degree of homogenious element x of graded differential algabra |x|

Basis of linear space is given by homogenious and elements x₁, ....., x_n

Λ (V) = Λ(V)_k =Λ (x₁, ....., x_n )

Operation of minimal model

Spherical surface Sⁿ, n≥2

de Rham complex *(Sⁿ)

When n is even number,

Volume element of S_n

M_n = Λ (x), |x| = n, dx = 0,

M_2n-1gives minimal model S_n to de Rham complex .

When n is odd number,

M_ngives minimal model S_n to de Rham complex .

[Interpretation]

Word is given by spherical surface.

Meaning of word is given by elements x₁, ....., x_n.

Word has minimal model.

Word becomes formal.

Fundamental group of word contains free group of rank b₁(M).

Here KARCEVSKIJ's "stable part" is identified to fundamental group and " mobile part" is identified to free group.

Refer to the paper Notes for KARCEVSKIJ Sergej, "Du dualisme asymétrique du signe linguistique" .

This paper has been published by Sekinan Research Field of Language.
All rights reserved.
© 2011 by The Sekinan Research Field of Language

4.Additional meaning

Stable and Unstable of Language

For the Supposition of KARCEVSKIJ Sergej

Completion of Language

TANAKA Akio

September 23, 2011

[Preparation]

n dimensional complex space Cⁿ

Open set

Whole holomorphic function over U

Ring sheaf for

U →O^an(U)

Complex analytic manifold C_anⁿ

Algebraic manifold Aⁿ multinomial of C_anⁿ

Ideal of multinomial ring a

[x₁, x₂, ..., x_n]

V(a) = {(a₁, a₂, ..., a_n)

Cⁿ f (a₁, a₂, ..., a_n) = 0,

a }

Whole closed set of V(a)

Fundamental open set D(f) = {(a₁, a₂, ..., a_n)

Cⁿ | f (a₁, a₂, ..., a_n) ≠ 0}

Arbitrary family of open set {Ui}

Easy sheaf F

Zariski topological space

Ring sheaf O

Affine space Aⁿ = (

, O)

Ring R

Set of whole maximum ideal Spm R1

Spm R Spectrum of R

Spm R is Noether- like.

◊

R is integral domain.

Whole of open sets without null set U_x

Quotient field K

Mapping from U_xto whole partial set of K O

O(V(a)^c) =

R_f

^c expresses complementary set.

O is easy sheaf of ring over Spm R that is whole set K.

◊

R is finite generative integral domain over k.

Triple (i) (ii) (iii) is called affine algebraic variety.

(i) Set Spm R

(ii) Zariski topology

(iii) Ring's sheaf O

O is called structure sheaf of affine algebraic variety.

◊

Ring homomorphism between definite generative integral domains

Upper is expressed by

Ring holomorphism O_X(U) → O_Y((^t

)^-1U)

Morphism from affine algebraic variety Y to X ( O_X(U) → O_Y((^t )^-1U), X→Y )

When

is surjection, ^t

is isomorphism overclosed partial set defined by p= Ker

Upper is called to closed immersion.

Ring holomorphism

Morphism between affine algebraic varieties

Kernel of

Image of

It is called that when

is injection

is dominant.

◊

R is medium ring between S and its quotient field K.

When

that is given by natural injection

is isomorphism over open set,

is called open immersion.

◊

When X is algebraic variety, longitude of maximum chain is equal to transcendental dimension of function field k(X).

It is called dimension of algebraic variety X, expressed by dim X.

◊

Defined generative field over k K

Space ( X, O_x )added ring that is whole sets of K that has open covers {U_i}

satisfies next conditions is called algebraic variety.

(i) Each U_iis affine algebraic variety that has quotient K .

(ii) For each i, j

I, intersection

is open partial set of

◊

Tensor product between ring and itself becomes ring by each elements products.

Elements

that defines surjective homomorphism is expressed by

Image

of closed embedding defined by

is called diagonal.

◊

Field K

Ringed space that have common whole set K (A, O_A) (B, O_B)

Topological space C

Open embedding

A and B have common partial set C.

Topological space glued A and B by C

Easy sheaf over W O_W

ahere, arbitrary open set Ø ≠

Ringed space

is called glue of A and B by C.

◊

Intedgral domains that have common quotient field K R, S

Element R a_m ≠ 0

Element S b_n ≠ 0

Spm T

Spm R, Spm T

Spm S

Glue defined by the upper is called simple.

◊

Affine algebraic varieties U¹, U²

Common open set of U¹, U²U^C

Diagonal embedding

When the upper is closed set, glue is called separated.

◊

For simple glue

, next is equivalent.

(*) It is separated.

(**) Ring

is generated by R and S.

◊

R and S are integral domains that have common quotient field K.

For partial ring T=RS generated by R and S, when <Definition> simple is satisfied, it is called "Spm R ad Spm S are simple glue."

◊

Projective space Pⁿ is simple glue.

◊

Algebraic Variety's morphism is glue of affine algebraic variety's ring homomorphism image.

Algebraic direct product is direct product of affine algebraic variety.

◊

Affine algebraic variety X

Ring over k R

is called R value point of X.

Whole

is called set of R value point of X, expressed by X(R).

Ring homomorphism over k

X(f) := X(R)

X(S)

Ring homomorphism

is function from ring category over k to category of set.

◊

Functors from ring category to set category F, G

Ring R

Family of

over ring R {

}

{

}

has functional morphism.

Functors F,G have isomorphism ( or natural transformation).

Functor from ring's category to set's category that is isomorphic to algebraic variety, is called representable or represent by X, or fine moduli.

◊

Functor from ring's category to set's category F

When

satisfies the next conditions, X is called coarse moduli.

(i) There is natural transformation

(ii) Natural transformation

Morphism that satisfies

is existent uniquely.

(iii) For algebraic close field k'

k, (k') is always bijection.

◊

Algebraic variety G that

is functor to group's category is called algebraic group.

◊

Finite generative ring over k A

When G = Spm A satisfies 3 conditions on the next triad is called affine algebraic group.

Triad

Conditions

(i)

are commutative for

(ii)There is identity map for A.

(iii) There is coincident with

for A.

◊

Projective space over C Pⁿ

(2n+1) dimensional spherical surface

{

}

Pⁿ has continuous surjection from

Pⁿis compact.

Map

is called closed map when

is closed set image

becomes closed set.

◊

Algebraic variety X is called complete when projection

is closed map for arbitrary manifold Y.

◊

Morphism from complete algebraic manifold X to separated algebraic manifold Y,

is closed map.

◊

Projective space Pⁿ is complete.

◊

Algebraic manifold that has closed embedding at Pⁿ is complete.

This algebraic manifold is called projective algebraic manifold.

[Interpretation]

Here language is expressed by P^n.

Word is expressed by projective algebraic manifold.

Meaning of word is expressed by closed embedding.

5.
Note

Provisional philosophic conjecture on additional meaning is the following.

From symplectic geometry to Fukaya category.
From Fukaya category to derived category.
Kawamata conjecture at derived category.
Smooth objective algebraic manifolds satisfies birational map.
Fully faithful embedding exists between two manifolds.

[References]
Extract of Zoho papers at Sekinan Library.
More details refer to Sekinan Zoho.

[2012]

Birational Language

[2011]

Notes for KARCEVSKIJ Sergej
Stable and Unstable of Language

[2009]

Symplectic Language Theory
Floer Homology Language

Tokyo

23 May 2016

SRFL Theory

Derived Category Language 2

Kawamata Conjecture

Conjecture

is birational map between smooth objective algebraic manifolds.

And

At This condition,

there exists next fully faithful embedding.

[Reference]

TODA Yukinobu. Several problems on derived category of coherent sheaf. Tokyo, 2016.Chapter 6, derived category of coherent sheaf and birational geometry page 148 Conjecture 6.43.

[References 2]

[Reference 3]

Stability of Language / Language and Spacetime

Tokyo

19 May 2016

SRFL Theory

Language stability and triangulated category

T. Bridgeland defined stability conditions on triangulated categories in 2007.
Language's stability presented by Sergej Karcevskij in 1928.
Bridgeland's stability seems to hint for Karcevskij's stability.

This paper is unfinished.

Tokyo
23 May 2016
SRFL Theory

Language stability and triangulated category

[Epitome]

Stability conditions on triangulated categories Data ( Z, A ) satisfies the next condition.

is bounded kernel of t-structure.

gives stability condition on A .

------------------------------------------------------------------------------------------------------

2.
[Paper on Karcevskij conjecture by TANAKA Akio 2011]

Notes for KARCEVSKIJ Sergej

Condition of Meaning
TANAKA Akio
September 11, 2011

[Preparation]

Graded differential algebra

Minimal model of graded differential algebra

Degree of homogenious element x of graded differential algabra |x|

Basis of linear space is given by homogenious and elements x₁, ....., x_n

N (V) = L (V) _k = N ( x ₁ , ....., x _n )

Operation of minimal model

Spherical surface Sⁿ, n≥2

de Rham complex *(Sⁿ)

When n iseaven number,

Volume element of S_n

M_n = Λ (x), |x| = n, dx = 0,

M_2n-1gives minimal model S_n to de Rham complex .

When n is odd number,

M_ngives minimal model S_n to de Rham complex .

[Interpretation]

Word is given by spherical surface.

Meaning of word is given by elements x₁, ....., x_n.

Word has minimal model.

Word becomes formal.

Fundamental group of word contains free group of rank b₁(M).

Here KARCEVSKIJ's "stable part" is identified to fundamental group and " mobile part" is identified to free group.

Refer to the paper Notes for KARVESKIJ Sergej "asymmetric Dusualisme the linguistic sign" .

--------------------------------------------------------------------------------------------

3.
[Paper on description of language by TANAKA Akio 2011]

Notes for KARCEVSKIJ Sergej

Description of Language
TANAKA Akio
September 9, 2011

Manifold M

Cup product map of M

Dual map of

Free Lie algebra that generates £ ( )

is identified as the partial space of £(

) that quadrastic Lie bracket of

generates.

Ideal of £(

) that Im η generates a

Holonomy Lie algebra of M

Completion of holonomy Lie algebra

If M has quadrastic homology connection, Malcev completion

becomes isomorphic with holonoly Lie algebra completion

Primary minimum model M(1)of differential manifold M is isomorphic with Malcev completion of

of M's fundamental group.

For the description of a language model there is a need primary minimum model M(1) of differential manifold M.

-----------------------------------------------------------------------------------------------------------

[Note]

If word satisfies Bridgeland 2002's Data{Z, A] , word has a stability in language.

For the problem of additional meaning refer to Karcevskij conjecture 1928 and Kawamata conjecture 2002.

-----------------------------------------------------------------------------------------------------

#This paper is unfinished

Tokyo
25 May 2016
SRFL Theory

Perspective for Derived Category Language. Stability of Language.

TANAKA Akio

1. Model
Derived Category Language.

2. Base
Manifold.

3.Main method
Stability conditions.

4. Theorem
R. Pandharipande and R.P. Thomas. Pandharipande-Thomas Theorem (1998).
K. Behrend. Behrent contructable function (2009).
Y. Toda. Curve-counting theories via stable objects (2010).
T. Bridgeland. Hall algebras and curve-counting invariants (2011).

5. Definition
Mu limit semi-stable object.

6. Space
Mu limit semi-stable object's moduli space.

7. Stability of Language
In the space(6.) language keeps stability as the set of derived duality.

8. Essence of Language
Duality is the essence of language.

Refer to the next.
Derived Category Language, 25 May 2016 Edition.
Actual Language and Imaginary Language. To LÉVI-STRAUSS Claude.
This paper is unfinished.

Tokyo
19 July 2016
Sekinan Library

Duality of Language

TANAKA Akio1. Model
Calabi-Yau 3-folds.

2. Theorem
R. Pandharipande and R.P. Thomas.
Pandharipande-Thomas Theorem (1998).
K. Behrend.
Behrent contructable function (2009).

Y. Toda.
Generating functions of stable pair invariants via wall-crossing in derived categories in algebraic geometry. (2008)
Curve-counting theories via stable objects (2010).
T. Bridgeland.
Hall algebras and curve-counting invariants (2011).
D. Joyce and Y.Song.
A theory of generalized Donaldson-Thomas invariants. (2012)

3. Definition
Generalized DT invariants. ( Joyce and Song. 2012)

4. Isomorphism
Isomorphism is induced in the defined moduli spaces.

5. Duality and mirror
If language is identified as Calabi-Yau 3 folds, language inevitably induce dual or mirror element.

Reference
Rensetsuso no Doraiken ni kakawaru shomondai. TODA Yukinobu. 2016. Sugaku shobo. Tokyo.
Reference 2
Derived Category Language, 25 May 2016 Edition.

Reference 3.
Actual Language and Imaginary Language. To LÉVI-STRAUSS Claude.

Reference 4.
Symplectic Language Theory Note 3 Mirror Symmetry Conjecture on Rational Curve.
Symplectic Language Theory Note 6 Homological Mirror Symmetry Conjecture by KONTSEVICH.

Reference 5.
Main papers of Sekinan Library.
SRFL Paper.

Paper ends.
------------------------------------------------------------------------------------------------------

This paper is unfinished.

Tokyo
22 July 2016
Sekinan Library

Saturday, 20 April 2024

The Premise for Quantum Group Language ー Main issue 2 of What is signal?

The Premise for Quantum Group Language

Bridge across mathematics and physics / Revised

Derived Category Language 1

Bridge across mathematics and physics / Revised

Additional meaning and embedding

1.Derived category

2.Embedding

3.Stable and mobile

4.Additional meaning

5.
Note

Language stability and triangulated category

Language stability and triangulated category

Perspective for Derived Category Language. Stability of Language.

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Post a Comment

Saturday, 20 April 2024

The Premise for Quantum Group Language ー Main issue 2 of What is signal?

The Premise for Quantum Group Language

Bridge across mathematics and physics / Revised

Derived Category Language 1

Bridge across mathematics and physics / Revised

Additional meaning and embedding

1.Derived category

2.Embedding

3.Stable and mobile

4.Additional meaning

5.Note

Language stability and triangulated category

Language stability and triangulated category

Perspective for Derived Category Language. Stability of Language.

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Post a Comment

5.
Note