Derived Category Language 3

Derived Category language 3

Language stability and triangulated category

[Epitome]

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.

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

1.
[Overview of Bridgeland 2002]

Stability conditions on triangulated categories 

A is Abelian category.

K group K(A) is defined as the next.

Here R is generated at complete exact sequence as the next.  
And 2  ~  E 1 + E 3.

Group homomorphism .

D is triangulated category.
Stability condition on D is defined as by the next.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 

<Example>

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-1 gives minimal model S_n to de Rham complex  .

When n is odd number,

M_n gives 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

<Preparation>

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 

<Theorem>
If M has quadrastic homology connection, Malcev completion  becomes isomorphic with holonoly Lie algebra completion  .

<Therorem>
Primary minimum model M(1)of differential manifold M is isomorphic with Malcev completion of  of M's fundamental group.

<Inrterpretation>

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

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

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

4.
[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