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Joint work with Yves Fรฉlix, Aniceto Murillo and Daniel Tanrรฉ
If ๐ป: ๐ฟ โถ ๐ is a continuous map between simply
connected CW-complexes, the following properties are
equivalent:
1. ๐๐ ๐ป โจโ โถ ๐๐ ๐ฟ โจโโถ ๐๐ ๐ โจโ , ๐ โฅ 2.โ
2. ๐ป๐ ๐ป โจโ โถ ๐ป๐ ๐ฟ ;โ โถ ๐ป๐ ๐ ;โ , ๐ โฅ 2.โ
Such a map is called a rational homotopy equivalence.
๐ฟ is rational if its homotopy groups are โ-vector spaces.
A rationalisation of ๐ฟ is a pair (๐ฟโ , ๐), with ๐ฟโ a rational
space and and ๐ โถ ๐ฟ โถ ๐ฟโ a rational homotopy equivalence.
The study of the rational homotopy type of ๐ฟ is the
study of the homotopy type of its rationalisation ๐ฟโ.
The study of the rational homotopy type of ๐ฟ is the
study of the homotopy type of its rationalisation ๐ฟโ.
Then, if ๐ฟ is a finite simply connected CW-complex
๐๐ ๐ฟ = โจ๐โค โจ โค๐1๐1โจโฏโจ โค๐๐๐๐
๐๐ ๐ฟโ โ ๐๐ ๐ฟ โจโ = โจ๐โ
The rational homotopy
type of ๐ฟ is completely
determined in algebraic
terms.
DGL CDGA
A differential graded Lie algebra is a
graded vector space ๐ฟ = โจ๐๐โค ๐ฟ๐ with:
A bilinear operation . , . โถ ๐ฟ ร ๐ฟ โถ ๐ฟ
such that ๐ฟ๐ , ๐ฟ๐ โ ๐ฟ๐+๐ satisfying:
๐, ๐ = โ โ1 ๐๐ ๐, ๐ , ๐ โ ๐ฟ๐ , ๐ โ ๐ฟ๐
๐, [๐, ๐] = ๐, ๐ , ๐ + โ1 ๐๐ ๐, [๐, ๐] ,
๐ โ ๐ฟ๐ , ๐ โ ๐ฟ๐ , ๐ โ ๐ฟ
a)
b)DGL
A linear map ๐ โถ ๐ฟ โถ ๐ฟ such that
๐๐ฟ๐ โ ๐ฟ๐โ1 satisfying:
๐ โ ๐ = 0
๐ ๐, ๐ = ๐๐, ๐ + โ1 ๐ ๐, ๐๐ ,
๐ โ ๐ฟ๐ , ๐ โ ๐ฟ
A differential graded Lie algebra is a
graded vector space ๐ฟ = โจ๐๐โค ๐ฟ๐ with:
a)
b)DGL
DGL
๐๐ข๐ฆ๐ฉ๐ฅ๐ฒ ๐๐จ๐ง๐ง๐๐๐ญ๐๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ DGL +โถ
๐ โถ
< โ >๐
< ๐(๐ฟ) >๐ โ ๐ฟโ
Rational homotopy
equivalencesQuasi-isomorphisms
(๐ฟ, โ,โ , ๐) is a DGL-model of ๐ฟ if
๐(๐ฟ) ๐ฟโฏโถโ โถโ
โถโ โถโ
DGL
๐๐ข๐ฆ๐ฉ๐ฅ๐ฒ ๐๐จ๐ง๐ง๐๐๐ญ๐๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ DGL +โถ
๐ โถ
< โ >๐
< ๐(๐ฟ) >๐ โ ๐ฟโ
Rational homotopy
equivalencesQuasi-isomorphisms
(๐ฟ, โ,โ , ๐) is a DGL-model of ๐ฟ if
(๐ฟ, โ,โ , ๐)โถโ
(๐(๐), ๐)
Minimal Quillenmodel
๐ป๐( )๐ฟ
Rational homotopy groups
โ ๐๐+1(๐ฟ)โจโ
Rational homology groups
๐ฟ(๐ ๐ , ๐) โถโ
๐ (๐ โโ) โ ๐ปโ(๐; โ)DGL
DGL
Spheres
Products
Wedge products
๐ ๐ฃ , 0 , ๐ฃ = ๐ โ 1is a model for the ๐-dimensional sphere ๐๐.
If (๐ฟ, ๐) and (๐ฟโฒ, ๐โฒ) are DGL-models
for ๐ฟ and ๐ respectively, then ๐ฟ ร ๐ฟโฒ, ๐ ร ๐โฒ
is a DGL-model of ๐ฟ ร ๐.
If (๐ ๐ , ๐) and (๐ ๐ , ๐โฒ) are minimal DGL
models for ๐ฟ and ๐ respectively, then
(๐ ๐ โ๐ ,๐ท) is a DGL-model of ๐ฟ โจ ๐.
A based topological space (๐ฟ, โ) ( ๐ ๐ , ๐ )
๐ฟ๐ ๐-skeleton ( ๐(๐<๐), ๐ )
= ๐๐ผ๐+1 ๐ฃ๐ผ โ ๐๐
๐๐ผ โถ ๐๐ โ ๐ฟ๐ ; [๐๐ผ] โ ฮ ๐(๐ฟ๐)
๐๐ฃ๐ผ โ ๐ป๐โ1(๐ ๐<๐ )
โ
ฮ ๐(๐ฟ๐)โจโ
CW-decomposition
A commutative differential graded algebra
is a graded vector space ๐ด = โจ๐๐โค ๐ด๐ with:
A bilinear operation โ โถ ๐ด ร ๐ด โถ ๐ด
such that ๐ด๐ โ ๐ด๐ โ ๐ด๐+๐ satisfying:
๐ โ ๐ = โ1 ๐๐๐ โ ๐, ๐ โ ๐ด๐ , ๐ โ ๐ด๐
๐ โ (๐ โ ๐) = (๐ โ ๐) โ ๐ , ๐, ๐, ๐ โ ๐ดa)
b)CDGA
CDGA
A commutative differential graded algebra
is a graded vector space ๐ด = โจ๐๐โค ๐ด๐ with:
A linear map ๐: ๐ด โถ A such that
๐๐ด๐ โ ๐ด๐+1 satisfying:
๐ โ ๐ = 0
๐(๐ โ ๐) = (๐๐) โ ๐ + โ1 ๐๐ โ (๐๐) ,
๐ โ ๐ฟ๐ , ๐ โ ๐ฟ
a)
b)
CDGA
๐๐ข๐ฅ๐ฉ๐จ๐ญ๐๐ง๐ญ, ๐๐ข๐ง๐ข๐ญ๐ ๐ญ๐ฒ๐ฉ๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ CDGA +โถ
๐ด๐๐ฟ โถ
< โ >๐Rational homotopy
equivalencesQuasi-isomorphisms
< ๐ด๐๐ฟ(๐ฟ) >๐ โ ๐ฟโ
(๐ด, ๐) is a CDGA-model of ๐ฟ if
๐ด๐๐ฟ(๐ฟ) ๐ดโฏโถโ โถโ
โถโ โถโ
CDGA
โถ๐ด๐๐ฟ
๐๐ข๐ฅ๐ฉ๐จ๐ญ๐๐ง๐ญ, ๐๐ข๐ง๐ข๐ญ๐ ๐ญ๐ฒ๐ฉ๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ CDGA +
โถ
< โ >๐Rational homotopy
equivalencesQuasi-isomorphisms
< ๐ด๐๐ฟ(๐ฟ) >๐ โ ๐ฟโ
(๐ด, ๐) is a CDGA-model of ๐ฟ if
(๐ด, ๐)โถโ
(โ๐, ๐)
CDGA
Rational homotopy groups
Rational cohomology groups
๐ปโ ๐ด, ๐ โ ๐ปโ(๐ฟ; โ )
(๐ด, ๐)โถโ
(โ๐, ๐)
Hom(๐๐ , โ) โ ๐๐(๐ฟ)โจ โ
Eilenberg-MacLane spaces
Wedge products
Products
โ๐ฃ, 0 , ๐ฃ = ๐is a model of the Eilenberg-MacLane
space ๐พ(โค, ๐).
If (๐ด, ๐) and (๐ดโฒ, ๐โฒ) are CDGA-models
for ๐ฟ and ๐ respectively, then ๐ด ร ๐ดโฒ, ๐ ร ๐โฒ
is a CDGA-model of ๐ฟ โจ ๐.
If (โ๐, ๐) and (โ๐, ๐โฒ) are minimal
models for ๐ฟ and ๐ respectively, then
(โ ๐ โ๐ ,๐ท) is a CDGA-model of ๐ฟ ร ๐.CDGA
CDGA
Postnikov decomposition
๐๐ข๐ฆ๐ฉ๐ฅ๐ฒ ๐๐จ๐ง๐ง๐๐๐ญ๐๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ DGL +โถ
๐ โถ
< โ >๐
๐๐ข๐ฅ๐ฉ๐จ๐ญ๐๐ง๐ญ, ๐๐ข๐ง๐ข๐ญ๐ ๐ญ๐ฒ๐ฉ๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ CDGA +โถ
๐ด๐๐ฟ โถ
< โ >๐
๐๐ข๐ฆ๐ฉ๐ฅ๐ฒ ๐๐จ๐ง๐ง๐๐๐ญ๐๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ DGL +
โถ๐
โถ
< โ >๐
๐๐ข๐ฅ๐ฉ๐จ๐ญ๐๐ง๐ญ, ๐๐ข๐ง๐ข๐ญ๐ ๐ญ๐ฒ๐ฉ๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ CDGA +โถ
๐ด๐๐ฟ โถ
< โ >๐
โถโถ sLAโถโถsCHAโถโถsGrp
< ๐ด >๐= Hom๐ช๐ซ๐ฎ๐จ(๐ด, ฮฉ )
< ๐ด >๐ is defined for ๐ด a โค-graded CDGA but
< ๐ฟ >๐ has only sense for positively graded DGLโs
๐๐ข๐ฆ๐ฉ๐ฅ๐ฒ ๐๐จ๐ง๐ง๐๐๐ญ๐๐, ๐๐ข๐ง๐ข๐ญ๐ ๐ญ๐ฒ๐ฉ๐๐๐โ ๐๐จ๐ฆ๐ฉ๐ฅ๐๐ฑ๐๐ฌ
โถ
โถโ ๐โ
The goal is to understand the rational behaviour of the spaces:
map ๐ฟ, ๐ , mapโ ๐ฟ, ๐ , map๐ ๐ฟ, ๐ , map๐โ (๐ฟ, ๐)
If ๐ฟ and ๐ are finite type
CW-complexes
map ๐ฟ, ๐ has the homotopy
type of a CW-complex.
If ๐ฟ and ๐ are nilpotent map๐ ๐ฟ, ๐ is nilpotent.
If ๐ฟ is a finite CW-complex map๐ ๐ฟ, ๐ is of finite type.
If ๐ฟ and ๐ are 1-connected map๐ ๐ฟ, ๐ is 1-connected.
Let ๐ฟ be a finite CW-complex and ๐ be a nilpotent,
finite type CW-complex.
(๐ตโ, ๐ฟ)
โ Vโจ๐ตโ , ๐ท
๐ท ๐ฃโจ๐ฝ = ฯ๐(โ1)๐ค |๐ฝ๐
โฒ|(๐ขโจ๐ฝ๐โฒ) (๐คโจ๐ฝ๐
โฒโฒ) + ๐ฃโจ๐ฟ๐ฝ
๐ฟ (๐ต , ๐) CDGA CDGC ๐ตโ๐ = ๐ตโ๐
(โ๐, ๐)๐ Sullivan model
if ๐ฃ โ V such that d๐ฃ = ๐ข๐ค and ๐ฝ โ ๐ตโ with
โ๐ฝ = ฯ๐๐ฝ๐โฒโจ๐ฝ๐
โฒโฒVโจ๐ตโ
The following Sullivan algebra is a model for map ๐ฟ, ๐ .
Let ฯ: (โ๐, ๐) โถ ๐ต be a model of ๐: ๐ฟ โถ ๐ .
Let ๐พฯ be the differential ideal generated by ๐ด1 โช ๐ด2 โช ๐ด3
๐ด1 = (๐ โ ๐ตโ)<0 ๐ด2 = ๐ท(๐ โ ๐ตโ)0
๐ด3 = ๐ผ โ ๐ ๐ผ ๐ผ โ (๐ โ ๐ตโ)0 }
The projection
map๐ ๐ฟ, ๐ โถ map ๐ฟ, ๐
โ Vโจ๐ตโ , ๐ท โถ โ Vโจ๐ตโ , ๐ท /๐พฯ
is a model of the injection
โ Vโจ๐ตโ , ๐ท /๐พฯ โ โ๐ โ๐ตโ1โจ(๐ โ ๐ตโ)โฅ2, ๐ท๐
๐ฟ , ๐ with DGL-models ๐ฟ , ๐ฟโฒ , respectively.
๐: ๐ฟ โถ ๐ .
map๐ ๐ฟ, ๐ ,DGL-model of map๐ ๐ฟ, ๐ ?
(๐ตโ, ๐ฟ)๐ฟ CDGC ๐ตโ, ๐ฟ = ๐โ(๐ฟ) Cartan-Eilenberg
(โ๐, ๐)๐ Sullivan model โ๐, ๐ = ๐โ ๐ฟโฒ cochains on ๐ฟโฒ
map๐ ๐ฟ, ๐ ,DGL-model of map๐ ๐ฟ, ๐ ?
(๐ตโ, ๐ฟ)๐ฟ CDGC ๐ตโ, ๐ฟ = ๐โ(๐ฟ) Cartan-Eilenberg
(โ๐, ๐)๐ Sullivan model โ๐, ๐ = ๐โ ๐ฟโฒ cochains on ๐ฟโฒ
Consider the โค-graded vector space ๐ป๐๐โ(๐โ ๐ฟ โฒ ๐ฟโฒ).
Let f โถ ๐โ ๐ฟ โถ ๐ฟโฒ be a linear map of degree ๐.
๐f = f โ ฮด โ โ1 f ๐โฒ โ f โถ ๐โ ๐ฟ โถ ๐ฟโฒ
(๐ตโ, ๐ฟ)๐ฟ CDGC ๐ตโ, ๐ฟ = ๐โ(๐ฟ) Cartan-Eilenberg
(โ๐, ๐)๐ Sullivan model โ๐, ๐ = ๐โ ๐ฟโฒ cochains on ๐ฟโฒ
Consider the โค-graded vector space ๐ป๐๐โ(๐โ ๐ฟ โฒ ๐ฟโฒ).
Let f , g โถ ๐โ ๐ฟ โถ ๐ฟโฒ be linear maps of degree
๐ and ๐ respectively.
๐โ ๐ฟ
๐โ ๐ฟ โจ๐โ ๐ฟ
โถ
โ
โถf โ g
๐ฟโฒโจ ๐ฟโฒ
โถ[f, g ]
โถ
๐ฟโฒ
[โ,โ]
( ๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒ , ๐, โ,โ ) is a DGL-model of map (๐ฟ, ๐)
๐โ ๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒ , ๐, โ,โ = โ Vโจ๐ตโ , ๐ท
U. Buijs, Y. Fรฉlix and A. Murillo, Trans. Amer. Math.Soc. 2008
๐โ ๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒ , ๐, โ,โ = โ Vโจ๐ตโ , ๐ท
What about the components โ๐ โ ๐ตโ1โจ(๐ โ ๐ตโ)โฅ2, ๐ท๐ ?
ฯ|(๐ ๐ฟโฒ)โโ : ๐โ ๐ฟ โถ ๐ ๐ฟโฒโถ ๐ฟโฒเทจ๐
๐ป: ๐ โถ ๐
๐: ๐โ(๐ฟโฒ) โถ ๐โ ๐ฟโ
๐:โ(๐ ๐ฟโฒ)โโถ ๐โ ๐ฟโ
๐|(๐ ๐ฟโฒ)โ: (๐ ๐ฟโฒ)โโถ ๐โ ๐ฟโ
Continuous function
๐: (โ๐, ๐) โถ ๐ต CDGA-morphism
Linear map of degree 0
Linear map of degree -1
๐โ ๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒ , ๐, โ,โ = โ Vโจ๐ตโ , ๐ท
What about the components โ๐ โ ๐ตโ1โจ(๐ โ ๐ตโ)โฅ2, ๐ท๐ ?
ฯ|(๐ ๐ฟโฒ)โโ : ๐โ ๐ฟ โถ ๐ ๐ฟโฒโถ ๐ฟโฒเทจ๐ Linear map of degree -1
เทฉ๐ โ ๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒโ1 ๐เทช๐ = ๐+ ๐๐เทช๐
Maurer-Cartan equation
is a differential if and only if
๐เทฉ๐ = โ1
2[ เทฉ๐ , เทฉ๐ ]
๐โ ๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒ , ๐, โ,โ = โ Vโจ๐ตโ , ๐ท
What about the components โ๐ โ ๐ตโ1โจ(๐ โ ๐ตโ)โฅ2, ๐ท๐ ?
ฯ|(๐ ๐ฟโฒ)โโ : ๐โ ๐ฟ โถ ๐ ๐ฟโฒโถ ๐ฟโฒเทจ๐ Linear map of degree -1
เทฉ๐ โ ๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒโ1 ๐เทช๐ = ๐+ ๐๐เทช๐
๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒ๐
เทฉ๐=
0 if ๐ < 0
๐ป๐๐โ ๐โ ๐ฟ โฒ ๐ฟโฒ๐ if ๐ > 0
if ๐ = 0Ker ๐เทช๐
Let ๐จ be an associative algebra.
๐จ ร ๐จโถ ๐จ bilinear and associative.
(๐, ๐) โผ ๐ โ ๐
๐จ ๐ก = {
๐=0
โ
๐๐๐ก๐ , ๐๐ โ ๐จ }
A deformation of ๐จ in ๐จ ๐ก is a new bilinear and
associative product โ: ๐จ ๐ก ร ๐จ ๐ก โถ ๐จ ๐ก
(ฯ๐=0โ ๐๐๐ก
๐) โ (ฯ๐=0โ ๐๐๐ก
๐) = ๐0 โ ๐0 + ๐1๐ก1 + ๐2๐ก
2 + ๐3๐ก3 +โฏ ๐๐ โ ๐จ
We can deform any algebraic structure on a vector sace ๐จ.
Instead of ๐จ ๐ก , we can consider a local ring ๐ with a
unique maximal ideal ๐ with ๐ /๐ โ ๐.
A deformation of ๐จ in ๐จโจ๐ is an operation:
โ: ๐จโจ๐ ร ๐จโจ๐ โถ (๐จโจ๐ )
satisfying the same properties of the original such that
we recover the original operation taking quotient by ๐
๐จร ๐จโถ ๐จ.
We denote by Def ๐จ, ๐ the set of equivalence
clases of deformations of ๐จ in ๐จโจ๐
โIn characteristic 0 every deformation problem
is governed by a differential graded Lie algebraโ
There exists a differential graded Lie algebra
๐ฟ = ๐ฟ(๐จ , ๐ ) such that we have a bijection
Def ๐จ, ๐ โ ๐๐ถ(๐ฟ)/๐ข
Let ๐ฟ be a DGL. ๐ โ ๐ฟโ1 is a Maurer-Cartan
element if satisfies the equation
๐๐ = โ1
2[ ๐ , ๐ ]
Two Maurer-Cartan elements ๐, ๐ โ ๐๐ถ(๐ฟ) are
gauge-equivalent ๐ โผ๐ข ๐ if there is an element
๐ฅ โ ๐ฟ0 such that
๐ = ๐ad๐ฅ ๐ โ๐ad๐ฅ ๐ โ id
ad๐ฅ(๐๐ฅ)
For each ๐ โฅ 0 , consider the standard ๐-simplex โ๐
โ๐๐= ๐0, โฆ , ๐๐ 0 โค ๐0 โค โฏ โค ๐๐ โค ๐ }, if ๐ โค ๐ ,
โ๐๐= โ , if ๐ > ๐.
Let ๐ ๐ โ1โ๐ , ๐ be the complete free DGL on the desuspended
rational simplicial chain complex
๐๐๐0,โฆ,๐๐ =
๐=0
๐
(โ1)๐๐๐0,โฆ,๐๐,โฆ,๐๐
where ๐๐0,โฆ,๐๐ denotes the generator of degree ๐ โ 1 represented
by the ๐-simplex ๐0, โฆ , ๐๐ โ โ๐๐.
Sullivanโs question
Problem:
Define a new differential in ๐ ๐ โ1โ๐ such that:
For each ๐ = 0,โฆ , ๐ the generator ๐๐ โ โ0๐ is
a Maurer-Cartan element.
1.
๐๐๐ = โ1
2[๐๐ , ๐๐]
2. The linear part ๐1 of ๐ is precisely ๐
Sullivanโs question
Case ๐ = 0 ๐0
๐ ๐ โ1ฮ0 , ๐ = ๐ ๐0 , ๐
where ๐๐0 = 0.
The first condition implies:
๐ ๐0 , ๐๐0 = โ1
2[๐0, ๐0]
Sullivanโs question
Case ๐ = 1
๐ ๐ โ1ฮ1 , ๐ = ๐ ๐0, ๐1, ๐01 , ๐
where ๐๐0 = ๐๐1 = 0.
๐๐01 = ๐1 โ ๐0.
The first condition implies:
๐๐0 = โ1
2๐0, ๐0 , ๐๐1 = โ
1
2[๐1, ๐1]
The second condition implies:
๐1๐01 = ๐1 โ ๐0.
๐0 ๐01 ๐1
๐2 ๐01 = ๐(๐1 โ ๐0)
= โ1
2๐0, ๐0 +
1
2๐1, ๐1 โ 0
๐2 ๐01 = ๐(๐1 โ ๐0 +1
2๐01, ๐0 +
1
2[๐01, ๐1])
=1
4๐01, ๐0 , ๐0 โ
1
4๐01, ๐1 , ๐1
+1
4[๐01, ๐1, ๐0 ] โ 0
๐๐01 = ๐1 โ ๐0 +๐1 ๐01, ๐0 + ๐1[๐01, ๐1]+1
2+1
2
+๐2 ๐01, [๐01, ๐0] + ๐2[๐01, ๐01, ๐1 ]
โฏ
Sullivanโs question
Case ๐ = 1
๐ ๐ โ1ฮ1 , ๐ = ๐ ๐0, ๐1, ๐01 , ๐
where ๐๐0 = ๐๐1 = 0.
๐๐01 = ๐1 โ ๐0.
The first condition implies:
๐๐0 = โ1
2๐0, ๐0 , ๐๐1 = โ
1
2[๐1, ๐1]
The second condition implies:
๐1๐01 = ๐1 โ ๐0.
๐0 ๐01 ๐1
๐๐01 = ๐01, ๐1 +
๐=0
โ๐ต๐๐!ad๐01
๐ ( ๐1 โ ๐0)
Bernoulli numbers are defined by theseries ๐ฅ
๐๐ฅ โ 1=
๐=0
โ๐ต๐๐!๐ฅ๐
๐๐ฅ โ 1
๐ฅ=
๐=0
โ1
๐ + 1 !๐ฅ๐Since
๐ต0 = 1, ๐ต1 = โ1
2, ๐ต2 =
1
6, ๐ต3 = 0, ๐ต4 = โ
1
30, ๐ต5 = 0, ๐ต6 =
1
42, โฏ
Sullivanโs question
Sullivanโs question
Case ๐ = 2
๐ ๐ โ1ฮ2 , ๐ = ๐ ๐0, ๐1, ๐2, ๐01, ๐02, ๐12, ๐012 , ๐
๐๐0 = ๐๐1 = ๐๐2 = 0
๐๐01 = ๐1 โ ๐0
๐๐02 = ๐2 โ ๐0
๐๐12 = ๐2 โ ๐1
๐๐012 = ๐12 โ ๐02 + ๐01
๐0
๐1
๐2
๐01
๐02
๐12๐012
MC
MC MC
LS
LS
LS
๐๐012 = ๐12 โ ๐02 + ๐01 +ฮจ
๐๐012 = ๐12 โ ๐01 โ โ๐02 โ [๐012, ๐0]
BCH
Sullivanโs question
The Baker-Campbell-Hausdorff formula
Let ๐ฅ, ๐ฆ be two non-commuting variables. ๐(๐ฅ, ๐ฆ)
The Baker-Campbell-Hausdorff formula ๐ฅ โ ๐ฆ is
the solution to ๐ง = log(exp ๐ฅ exp ๐ฆ )
Explicitely ๐ฅ โ ๐ฆ = ฯ๐=0โ ๐ง๐(๐ฅ, ๐ฆ)
๐ง๐ ๐ฅ, ๐ฆ =
๐=1
๐(โ1)๐โ1
๐
๐ฅ๐1๐ฆ๐1โฏ๐ฅ๐๐๐ฆ๐๐
๐1! ๐1!โฏ๐๐! ๐๐!
๐1 + ๐1 > 0; โฏ ; ๐๐+๐๐ > 0. ๐1 + ๐1 +โฏ+ ๐๐ + ๐๐ = ๐
Sullivanโs question
The Baker-Campbell-Hausdorff formula
The Baker-Campbell-Hausdorff formula satisfies the following properties:
โ is an associative product: ๐ฅ โ ๐ฆ โ ๐ง = ๐ฅ โ (๐ฆ โ ๐ง)
The inverse of ๐ฅ is โ๐ฅ, i.e. ๐ฅ โ โ๐ฅ = 0.
๐ฅ โ ๐ฆ can be written as a linear combination of
nested commutators
๐ฅ โ ๐ฆ = ๐ฅ + ๐ฆ +1
2๐ฅ, ๐ฆ +
1
12๐ฅ, ๐ฅ, ๐ฆ โ
1
12๐ฆ, ๐ฅ, ๐ฆ + โฏ
๐ฅ โ ๐ฆ โ ๐(๐ฅ, ๐ฆ) โ ๐(๐ฅ, ๐ฆ)
Theorem
๐ = { ๐๐ } ๐โฅ0 is a cosimplicial DGL
U. Buijs, Y. Fรฉlix, A. Murillo, D. Tanrรฉ. Israel J. of Math. 2019
Definition
< ๐ฟ >๐= ๐ป๐๐๐๐ท๐บ๐ฟ( ๐๐ , ๐ฟ)
cDGL โถ< โ >
Theorem
๐0 < ๐ฟ >โ MC(๐ฟ)/๐ข
< ๐ฟ > โ โ๐งโMC(๐ฟ)/๐ข < ๐ฟ๐ง >
๐๐ < ๐ฟ >โ ๐ป๐โ1(๐ฟ) ๐ > 1
๐1 < ๐ฟ >โ ๐ป0(๐ฟ)
For any cDGL ๐ฟ we have
1.
2.
๐๐ < ๐ฟ >โ ๐ป๐โ1(๐ฟ) ๐ > 1
๐1 < ๐ฟ >โ ๐ป0(๐ฟ)๐1 < ๐ฟ >ร ๐๐ < ๐ฟ > โถ ๐๐ < ๐ฟ >
๐
โ โ
๐ป0 ๐ฟ ร ๐ป๐โ1(๐ฟ) ๐ป๐โ1(๐ฟ)?
(๐ฅ , ๐ง ) ๐ad๐ฅ(๐ง)
Reduced
Complete
HomcDGL(๐โ, ๐ฟ)
Reduced
Nilpotent, finite-type
Nilpotent, finite-type
Complete๐โ๐ฐโ๐ขโ เดฅ๐(๐ฟ)
๐(๐ฟ) ๐
MC(๐ฟโจฮฉโ)Neisendorfer,
Pac. J. M., 1978.Getzlerโs
observation
Buijs, Fรฉlix, Murillo, Tanrรฉ, Preprint 2017
Work in progress