Deformation theory of morphisms by theoryman

VIEWS: 29 PAGES: 25

									Deformation theory of morphisms
              Bruno Vallette

      Trends in Noncommutative Geometry




                    0-0
1 Paradigm : Associative algebras
• Let V be a K-module, consider EndV := {Hom(V ⊗n , V )}n≥0 .
For f   ∈ Hom(V ⊗n , V ) and g ∈ Hom(V ⊗m , V ), binary product
                                   cc
                                      c 
                           n       XX g Ô                n
               f    g :=         ±
                                     XX i ÔÔ =                 ±f ◦i g.
                                         Ô
                           i=1             f             i=1




Degree convention :
|f | = n − 1, |g| =   m − 1, so |f         g| = |f | + |g|, that is |     | = 0.

Theorem (Gerstenhaber).

          (f   g)   h−f          (g   h) = (f       h)    g−f       (h    g)
                      Assoc (f, g, h)      = Assoc (f, h, g)
(EndV, ) is a preLie algebra.
=⇒ with [f, g] := f g − (−1)|f |.|g| g              f,
(EndV, [ ]) is a Lie algebra.

• Associative algebra structure on V :

                                      ⊗2
                                                c
                                                c
                           µ : V           →V ,
               c  c c
               c  cc c
                c  − c = 0
                 c                            in   Hom(V ⊗3 , V )
                       
                    ⇐⇒ µ          µ = 0 ⇐⇒ [µ, µ] = 0
In this case,
                      dµ (f ) := [µ, f ] verifies dµ (f )2 = 0.
Explicitly, for f   ∈ Hom(V ⊗n , V )

                      cc         aa      }       ee
                         c        a } }}          ee ÑÑÑ
                n     XX µ Ô         f b       X      f
                        XX i ÔÔ ±      bb ÔÔ ± XXX Ð
dµ (f ) =           ±       Ô           b Ô
                                            Ô      ÐÐ
                                                   Ð
             i=1             f                    µ                µ


∈ Hom(V ⊗n+1 , V )
                        dµ                   dµ                     dµ
      Hom(K, V ) −→ Hom(V, V ) −→ Hom(V ⊗2 , V ) −→ · · ·
                  −             −                 −
Hochschild cohomology of the associative algebra V “with coefficients into itself”
(C • (Ass, V ), dµ , [ , ]) dg Lie algebra (twisted by µ).
Deformation complex of the associative structure µ
(Interpretation of H0 , H1 , H2 in terms of formal deformations : see Konstevich)

Operations on C • (Ass, V ) :
          Cup product ∪ : associative operation
          Deligne Conjecture
• (V, d) dg module, EndV is a dg module

                                 WW d Õ                   aa
                           n       WW i ÕÕ                  a ÑÑÑ
                                       Õ                       f
             D(f ) :=               f        − (−1)|f |
                          i=1
                                                               d
(EndV, D, ) is a dg preLie algebra and (EndV, D, [ ]) is a dg Lie algebra.
(V, d, µ) is a dg associative algebra ⇐⇒


                                         1
  Dµ + µ      µ = 0 ⇐⇒ Dµ +                [µ, µ] = 0 : Maurer-Cartan equation
                                         2
General solutions :
µ ∈ {Hom(V ⊗n , V )}n≥1 ,              µn : V ⊗n → V    with   µ1 = d.
Dµ + µ µ = 0 ⇐⇒

                              d         WW   d  hh z
                                  b ÕÕ            hzz
            n=2           :        bÕ + WWÐ Ð =
                                   bÕ      Ð
                                                               d
                       c  c c
                       c  cc c
                 n=3 :  c  − c  = D(µ3 )
                         c    
µ2 is associative up to the homotopy µ3

                                       cc i 
                                         cc 
                n     :               ±  j  = D(µn )
                                           cc 
                              i+j=n+1        
                               i,j≥2


Definition (Stasheff).
A Maurer-Cartan element µ is an associative algebra up to homotopy or
A∞ -algebra structure on (V, d, µ = {µn }n ).

Viewpoint : An associative algebra = very particular A∞ -algebra.
Once again, dµ (f ) := D(f ) + [µ, f ] verifies d2 = 0. µ
(C • (Ass, V ), dµ , [ , ]) dg Lie algebra (twisted by µ)
defines the cohomology of an A∞ -algebra.


Same interpretation of all the H • in terms of deformations of µ.


• Homological perturbation lemma

                                                 i    /            s
                          (V, dV )         o              (W, dW )         h
                                                 p

         p ◦ i = IdV          and         i ◦ p − IdW = dW ◦ h + h ◦ dW
V is a deformation retract of W .

Theorem (Kadeishvili, Merkulov, Kontsevich-Soibelman, Markl).
If ν = {νn }n is an A∞ -algebra structure on W , than

                                          i    dd            i ii
                                                                       ii
                                                                               i
                                                                                       ~
                                                                                           i
                                                 dd      yy
                                                        yy
                                                                        i
                                                                                    ~~~
                                                     ν2                        ν3
                                                        pp
                                                         ph
                                                                        yy
                                                                       h
             µn =                                         p            yy
                      planar trees with                       ν2
                          n leaves


                                                              p
defines an A∞ -algebra on V such that i, p and h extends to morphisms and
homotopy in the category of A∞ -algebras.
• Other kind of algebraic structures :
        Lie, commutative, Poisson, Gerstenhaber, PreLie, BV algebras.

        Lie bialgebras, associative bialgebras.


For any type of (bi)algebras


                     dg module V



                                 
                     dg Lie algebra



                                 
 Maurer-Cartan elements   = (bi)algebra∞ structure on V   /   Homological perturbation lemma




                             
    twisted dg Lie : deformation complex (cohomology)
2 ......., ........., .....
                                                 d ~
                                                  d~
                                                   •                    d ~
                                                                         d~
 Operations                 •                                             •
                                               no symmetry

                                           d ~ d ~
                                            d~     d~            d ~ d ~
                                                                  d~     d~
                                            •vv • r•              •vv • r•
                            •                  v rr                  v rr
 Composition                                       •                      •
                            •
                                                 planar               non-planar

 Monoidal category    (Vect, ⊗)               (gVect, ◦)           (S-Mod, ◦)
 Monoid              A⊗A→A                   P ◦P →P              P ◦P →P
                                              Non-symmetric
 Modules                Modules                 algebras
                                                                      Algebras
                                                                   Lie, commutative,
 Examples                                 associative algebras   Gerstenhaber algebras
                          Ladders                Planar
 Free monoid          (Tensor module)            trees
                                                                        Trees

                           d ~
                            d~                 d ~
                                                d~
 Operations
                            ~•d
                           ~ d                  ~•d
                                               ~ d
                       d ~
                        d~                 d ~ d ~
                                            d~      d~
                        •d ~
                          d •               •d ~
                                              d • •
                           d
                           ~                   d
                                               ~
 Composition
                         ~ d•d
                          ~
                        • ~ d                ~ d•d •
                                              ~
                                            • ~ d
                           ~                   ~
 Monoidal category   (S-biMod, c )          (S-biMod, )
 Monoid              P cP→P                 P P→P
 Modules               (Bial)gebras          (Bial)gebras
                       Lie, associative
 Examples                 bialgebras
                         Connected
 Free monoid                graphs
                                                Graphs
2   Operads, properads, props
                                                  d ~
                                                   d~
                                                    •                     d ~
                                                                           d~
Operations                 •                                                •
                                                no symmetry

                                            d ~ d ~
                                             d~     d              d ~ d ~
                                                                    d~     d
                                             •vvv• rr•~             •vvv• rr•~
Composition
                           •                      r •                    r  •
                           •
                                                  planar                non-planar

Monoidal category    (Vect, ⊗)                 (gVect, ◦)             (S-Mod, ◦)
                      Associative              Non-symmetric
Monoid                 algebras                   operads
                                                                        Operads
                                               Non-symmetric
Modules               Modules                    algebras
                                                                        Algebras
                                                                     Lie, commutative,
Examples                                   associative algebras    Gerstenhaber algebras
                        Ladders                   Planar
Free monoid         (Tensor module)                trees
                                                                          Trees
                          d ~
                           d~                    d ~
                                                  d~
Operations
                           ~•d
                          ~ d                     ~•d
                                                 ~ d
                      d ~
                       d~                    d ~ d ~
                                              d~      d~
                       •d ~
                         d •                  •d ~
                                                d • •
                          d
                          ~                      d
                                                 ~
Composition
                        ~ d•d
                         ~
                       • ~ d                   ~ d•d •
                                                ~
                                              • ~ d
                          ~                      ~
Monoidal category   (S-biMod,         c)      (S-biMod,        )
Monoid                Properads                    Props
Modules              (Bial)gebras              (Bial)gebras
                      Lie, associative
Examples                 bialgebras
                        Connected
Free monoid                graphs
                                                  Graphs
3 Homological algebra for prop(erad)s
• Recall for associative (co)algebras [Cartan, Eilenberg, MacLane, Moore, ...].

Pair of adjoint functors :

bar construction    B : {dg algebras}         {dg coalgebras} : Ω cobar construction
               ¯
B(A) := (T c (sA), δ), where
      • T c : cofree connected coalgebra (tensor module)
      • s homological suspension
         ¯
      • A augmentation ideal
      • δ unique coderivation which extends the product of A
                                               −1
                          ¯
                    T c (sA)         ¯    s
                                          −→ ¯      ¯ sµ   ¯
                                   (sA)⊗2 − − s(A ⊗ A) −→ sA
                                                        −

Explicitly,

      δ(a1 ⊗ · · · an ) =          ±a1 ⊗ · · · ⊗ µ(ai , ai+1 ) ⊗ · · · ⊗ an .
                               i
                               
                               
                         an    
                               
                                                   an
                               
                               
                          .    
                          .    
                          .                         .
                                                    .
                                                    .
                               
                         ai    
                               
                    δ
                     
                                =
                                             ± µ(ai , ai+1 )
                        ai+1            i
                               
                               
                                                    .
                                                    .
                          .                         .
                          .    
                          .    
                               
                                                   a1
                               
                         a1    
                               


                                            `
Contracting internal edges : Graph homology a la Kontsevich


• For operads, pair of adjoint functors [Ginzburg-Kapranov, Getzler-Jones]
bar construction   B : {dg operads}           {dg cooperads} : Ω cobar construction
               ¯
B(P) := (F c (sP), δ), where
     • F c : cofree connected cooperad (trees)
     • s homological suspension
       ¯
     • P augmentation ideal
     • δ unique coderivation which extends the partial product of P
        (composition of two operations)
                                            −1
                     ¯
                     c
                 F (sP)             ¯    s
                                         −→ ¯      ¯ sγ
                                  (sP)⊗2 − − s(P ⊗ P) −→ sP
                                                       −  ¯
Explicitely,

                                                                      €€€
    ff                 q
  
     ff
                  ww
                    ww qqq ||
                               ||       
                                                                           €€€
                                                                              € ||||
              p2           p3                              uuu               p
                 hh
                   hh                                          uuu          qq 3
                         {{                                               qqq
                                       
δ
                       {{              =
                                        
                                                     ±
                        p1                                     γ(p1 ⊗ p2 )
                                       


                                            `
Contracting internal edges : Graph homology a la Kontsevich


• Where do these constructions come from conceptually ?

(C, ∆) coalgebra, (A, µ) algebra; f, g : C → A
                                                 g
                                            C        /A
                                                             µ
                 f       g := C     ∆
                                        /   ⊗            ⊗       /A
                                                 f
                                            C        /A
(Hom(C, A), ) associative convolution algebra.

Theorem (Merkulov-V.).
For C a dg coprop(erad) and P a dg prop(erad),
Hom(C, P) is a dg prop(erad) called the convolution prop(erad).

Corollary (Merkulov-V.).      (Hom(C, P), [ , ]) is a dg Lie algebra.
Tw(C, P) := set of Maurer-Cartan elements in (Hom(C, P), [ ,        ]) :
set of Twisting morphisms (cochains).
Tw(−, −) is a bifunctor, try to represent it.

Definition. Bar construction of a prop(erad) :                  ¯
                                                B(P) := (F c (sP), δ), where
      • F c : cofree connected coprop(erad) (graphs)
      • s homological suspension
        ¯
      • P augmentation ideal
      • δ unique coderivation which extends the partial product of P ,
          composition of two operations


                                      x
                                      x   xxx             p
                                 x
                                   p2
                                x p γ p      x         ppp
                        p xxxx
                        p xx   xx       →
                                        −    γ(p1 , p2 )
                                                       xxx
                          p1              pppp            x
                        x
                        x    p
                             p
          Remark : The number of internal edges is not relevant.

Explicitly,

                 p                                qq
                  p      xx                           qq               w
                                                        qq          www
          
          
                      p3
                         pp           
                                                                 ww
                          pp                               p3
          
                            p  x
                                x     
                                      
         d                  p2       =        ±
          
                        xxx p
                          xx    p     
                                      
          
                      xxx
                       xx
                                      
                                      
                                                       γ(p1 ⊗ p2 )
                                                                  iii
                   p1                                yyy           iii
                  x p
                  x     p                           yyy
Recover particular cases : Associative algebras, operads.


Cobar construction Ω(C) is dual.

Theorem (Merkulov, V.).

 Homdg prop(erad)s (Ω(C), P) ∼ Tw(C, P) ∼ Homdg coprop(erad)s (C, B(P))
                             =          =

Representation of Tw(−, −) and adjunction.

P ROOF.

   Homprop(erad)s (Ω(C), P)        =                             ¯
                                           Homprop(erad)s (F(s−1 C), P)
                                   ∼
                                   =                ¯
                                           HomS (C, P) ⊂ HomS (C, P)
                                               −1                 −1



                                                     S
   Homdg prop(erad)s (Ω(C), P)         =     MC(Hom      (C, P)) = Tw(C, P)



Theorem (V.). Canonical bar-cobar resolution
                                              ∼
                              Ω(B(P)) − P
                                      →

• Minimal models

Definition. Minimal model for P

                                              ∼
                                        →
                              (F(X), ∂) − P
where

          F(X) is a (quasi)-free properad
          ∂ is a derivation ←→ ∂|X : X → F (X)
                           d ~
                            d~
                      d ~
                       d~   •d ~
                              d •
                    : ~d →     d
                               ~
             ∂|X
                      ~
                        •d
                             ~ d•d :
                              ~
                            • ~ d
                                                   Vertex expansion

                               ~
                                                                     ∼
Definition. A quadratic model is a minimal model               →
                                                    (F(X), ∂) − P such that

           ∂|X : X → F (X)(2)              :     graphs with 2 vertices

                        d ~
                         d~
                 d ~
                  d~     •dd
                            dd
                 ~~•d →
                     d
                             ~•d
                            ~ d
The number of vertices is relevant, not the number of internal edges.


If P has a quadratic model, it is called a Koszul properad.
In this case, X = C is a coproperad and (F(X), ∂) =           Ω(C)
• Koszul duality theory
  (associative algebras [Priddy], operads [Ginzburg-Kapranov], properads [V.])
   provides a method to

        compute X    = P ¡ : Koszul dual
        make ∂ explicit
                                                                         ∼
        criterion to prove the quasi-isomorphism F (X)              →
                                                          = Ω(P ¡ ) − P .
        (acyclicity of a small chain complex : the Koszul complex)


=⇒ Graph homology [Kontsevich, Markl-Voronov]
4 Deformation complex of a morphism
   f
  →
P − Q morphism of prop(erad)s (Q is a representation of P ).

Example.
V dg module, EndV := {Hom(V ⊗n , V ⊗m )}n,m is a dg prop(erad)
(composition of multilinear functions)

Definition.
A structure of P -gebra on V is a morphism of prop(erad)s P      → EndV .

Recall   Quillen : “(commutative) algebraic geometry”
deformation complex of morphisms of commutative algebras (cotangent complex,
model category structure).


       comm. algebras     →     ass. algebras   → operads → prop(erad)s
                          →     Noncommutative geometry           [Nonlinear]

Theorem (Merkulov-V.).

       • The category of dg prop(erad)s is a cofibrantly generated model
          category structure

       • Quasi-free prop(erad)s are cofibrant

P ROOF.
                F : dg S-bimodules              dg prop(erad)s   : U
Definition (Deformation complex).

                   cofibrant resolution   : (R, ∂)
                                                    ∼
                                                            /P
                                                 ii
                                                   ii
                                                     ii f
                                                       i" 
                                                             Q

                         C • (P, Q) := (Der(R, Q), ∂ ∗ )
• Well defined :
Extension of Quillen theory of commutative rings to prop(erad)s
To                 O

                    
         I        /P          /Q

     DerI (O, Q)         ∼
                         =    Homdg prop(erad)s /P (O,         P      Q            )
                                                         Eilenberg-MacLane space

                         ∼
                         =    HomP−bimodules (P      O    ΩO/I       O    P , Q),
                                                    Cotangent complex

where ΩO/I is the module of Kahler differentials of the prop(erad) O .
                             ¨
                                           f
(Read the properties of the morphism P    →
                                          − Q on the cotangent complex)
Quillen adjunction =⇒ Total derived functor (non-additive)
                   =⇒ well defined in the homotopy categories
• Explicitly,
                                            ¯ →     ∼
When P is Koszul :      R = Ω(P ¡ ) = F(s−1 P ¡ ) − P quadratic model
In this case,
                       ¯                  ¯
C • (P, Q) = Der(F(s−1 P ¡ ), Q) = HomS (P ¡ , Q) ⊂ HomS (P ¡ , Q)
                                      •−1
Theorem (Merkulov-V.).
  HomS (P ¡ , Q) is a non-symmetric dg prop(erad)
  =⇒ it is a dg Lie algebra.
                                   f
Proposition (Merkulov-V.).       →
                               P − Q is a morphism of dg prop(erad)s iff
¯           f
f : P ¡ → P − Q is a Maurer-Cartan element in HomS (P ¡ , Q)
            →
In this case,

                                                     ¯
                 (C • (P, Q), d) ⊂ (HomS (P ¡ , Q), [f , −])
                                                twisted dg Lie algebra


• Examples
P = Ass, P ¡ = Ass∨ and C • (Ass, EndV ) = EndV :
     Hochschild cohomology of associative algebras
P = Ass, P ¡ = Ass∨ and C • (Ass, Poisson) :
     Invariant of knots [Vassiliev, Turchine]
P = Lie, P ¡ = Com∨ and C • (Lie, EndV ) :
     Chevalley-Eilenberg cohomology of Lie algebras
P = Com, P ¡ = Lie∨ and C • (Com, EndV ) :
     Harrison cohomology of commutative algebras
P = BiLie, P ¡ = F rob∨ and C • (BiLie, EndV ) :
                      3
     Ciccoli-Guerra cohomology of Lie bialgebras
P = BiAss, not Koszul and C • (BiAss, EndV ) :
      Gerstenhaber-Shack bicomplex ???
P   Koszul =⇒ P quadratic but BiAss not quadratic

                                            1      2
                             1   c 2
                                 c
                                 •        t t utt
                                         vt uttt t
                                         t • t u•t
                                          v u tt
                                 ÕWW −
                                 •
                                 Õ
                                           •    •
                             1       2      1      2

Interpretation of H0 , H1 , H2 in terms of formal deformations

Definition. A P∞ -gebra (or homotopy P -gebra structure) on V is a
Maurer-Cartan element in HomS (P ¡ , EndV )

• Examples
P = Ass, homotopy associative algebra [Stasheff]
P = Lie, homotopy Lie algebra [Stasheff, Hinich-Schetchman, Kontsevich]
P = Com, C∞ -algebra [Stasheff]
P = Gerstenhaber, G∞ -algebra [Getzler-Jones]
P = BiLie, homotopy Lie bialgebra [Gan]
P = BiAss ...

(HomS (P ¡ , EndV ), [f, −]) : cohomology of P∞ -algebras
Interpretation of H• in terms of deformations
                                                  f
                                             →
• Operations on the deformation complex of P − Q

When P is a Koszul operad,

        Hom(P ¡ , Q) ← Hom(P ¡ , P) ∼ P ! ⊗ P ← P ! ◦ P
                                    =
                                                 Manin complex    Manin products
                                                tangent complex

Example :   Ass ◦ Ass = Ass =⇒ cup product
Theorem (V.).   P finitely generated binary non-symmetric Koszul operad

                   Little disk operad   ←− • −→ C • (P, Q)

Generalized Deligne conjecture (P   = Ass, Q = EndV )

P ROOF.
          C • (P, Q) = Hom(P ¡ , Q) non-symmetric operad =⇒ braces
          operations

          Ass → P ! ◦ P =⇒ cup product ∪
          ∂ = [∪, −]
          (McClure-Smith)



Examples : 4 infinite families of operads (Koszul by poset method)
5 Beyond the Koszul case
                                    ∼
Minimal model for P : (F(X), ∂)     →
                                    − P
where ∂|X : X → F (X)

                         d ~
                          d~
                    d ~
                     d~   •d ~
                            d •
                  : ~d →     d
                             ~
           ∂|X
                    ~
                      •d
                           ~ d•d :
                            ~
                          • ~ d
                                             not necessarily quadratic

                             ~
∂ 2 = 0 =⇒ X is a homotopy coprop(erad).
(coassociative of the coprop(erad) holds ‘up to homotopy’)

Theorem (Merkulov-V.).
  For C a homotopy coprop(erad) and P a dg prop(erad),
  Hom(C, P) is a homotopy prop(erad)
  =⇒ (Hom(C, P), [ , ]) is a homotopy Lie algebra.

Proposition (Merkulov-V.).
  In this case, Hom(C, P) is a filtered homotopy Lie algebra
                                     1
  Maurer-Cartan elements :   n≥0 n! ln (f, . . . , f ) = 0
      f
     →
  P − Q is a morphism of dg prop(erad)s iff
  ¯           f
  f : C → P − Q is a Maurer-Cartan element in HomS (P ¡ , Q)
             →
  In this case,

                                                                  ¯
                   (C • (P, Q), d) ⊂ ( HomS (P ¡ , Q), lf )
                                         twisted homotopy Lie algebra
Theorem (Merkulov-V.). For every minimal model of BiAss, the deformation
complex

                          C • (BiAss, Q) ∼ Q
                                         =
and
      2` .
    uuu ` . . m−ssm
    1
               Ò1
       uuu ÒÒs
         `` ss
            `
            u
            s
            Ò
         ÒÒ uuu
       sss ``` uu
d          •        =

    sss Ò . .
       Ò.
    1   2         n 1
                   −      n

                                                                   1       2                −
                                                                                           m 1     m

               2` .
             uuu ` . . m−ssm                                    ÔXX Ô•XX
                                                                                   ...
                                                                                            ÔXX Ô•XX
                                                               Ô•X ÔÔ X                    Ô•X ÔÔ X
                  1
                 uuu ÒÒs
                  `` ss  Ò1                                    Ô                           Ô
n−2
                     ÙsS
                  ssÒuuuu
                      u
                      S
        i+1          •S
    (−1)                                             +

              sssÙÙ•X S ..u
                                                                       m                       m
i=0
             s.. Ù Ô X                                         r r ... u v
                                                                ru u                      iiU... × y
                                                                                           U
                i Ô X                                            •u                         ii × y
            1       Ô       n                                      quu vvv
                                                                    q
                                                                    •q
                                                                                             UUy
                                                                      v                     y
                                                                                              y
                                                                                             ××UU i
                                                                                             yiii
                                                                                              •

                                                                                          yy×...
                          i+1           i+2                           •
                                                                                           ×
                                                                       1                      −
                                                                                          23 n 1n
                          1           2               −
                                                     m 1       m


                       ÔXX ÔÔXX                       ÔXX ÔÔXX
                                              ...
                      Ô•X Ô•X
                      Ô                              Ô•X Ô•X
                                                     Ô
            n+1
+ (−1)
                                  m                      m
                        iiU... × y
                         U                      rr sss s•vs
                          ii × y
                           Uiy                    rr ... w v v
                                                       s
                           ××ii
                            y
                            U                       rw•w
                          yy UU i
                            •

                        yy×...
                                                     •
                         ×
                          12          n 1
                                       −                  n

                                                                           1                   iiU × y
                                                                                                U   −
                                                                                               23 m 1m
                                                                                                  ...
                                                                                                 ii × y
                                                                                                  U×y
                     XX i+2                                              wr
                                                                        w• r
                                                                       wssssrrr                    i
                                                                                                   U
                                                                                                  ×yii
                                                                                                 yy UU i
                                  i+1
                       X Ô m                                            •                          •
               1u i S •Ô
                 uuuSS Ô Ù s..s
                 ..                                                v•s ...
                                                                     v
                                                                    vs                         yy×...
                                                                                                ×
  n−2                uu`ss s
                        SsÙ
                         ÙÙ
                         uuu                                               n                       n
           i+1
                       s•`
                     ÒÒ u
                   sss. . `` uu
+     (−1)                                                +
                                                                    XX Ô XX Ô
                                                                     XÔ XÔ                     XX Ô XX Ô
                                                                                                XÔ XÔ
                ss ÒÒ
  i=0
                     .                                               •Ô •Ô                      •Ô •Ô
                      1       2               n 1
                                               −     n
                                                                                         ...
                                                                       1       2                 −
                                                                                                n 1    n

                          iiU...m−y
                          1U1
                                 ×
                                   1                      m
                            ii × y
                              UUy                        q
                                                         •q
                              ××ii
                              •y
                               i
                            yy UU i                    vv uq r
                                                     vvuuu u•r
                                                           •
                          yy×...
                           ×                        v ... u r
            m+1                    n                       n
+ (−1)
                        XX Ô XX Ô
                         XÔ XÔ                        XX Ô XX Ô
                                                       XÔ XÔ
                         •Ô •Ô                         •Ô •Ô
                                               ...
                              1           2               −
                                                         n 1    n
Corollary (Merkulov-V.).
  C • (BiAss, EndV ) = Gerstenhaber-Shack bicomplex
  There is a homotopy Lie agebra structure on the Gerstenhaber-Shack
  bicomplex which measures the deformation of structures of bialgebras
6 Homological perturbation lemma
Theorem (V.).
                                     i
                                         /         s
                      (V, dV )   o           (W, dW )   h
                                     p

If W is a homotopy P -gebra, than there is an induced homotopy P -gebra
structure on V .

• Examples
P   = Ass (Kontsevich-Soibelman)
P   = Lie (Costello, Goncharov, Mnev, ...)
P   = Com (Cheng-Getzler)
P   = Gerstenhaber ‘new’
P   = BiLie ‘new’
P   = BiAss TBC ...
References
 [1] B. Vallette, A Koszul duality for props, To appear in Trans. of Amer. Math.
     Soc..

 [2] S. Merkulov, B. Vallette, Deformation theory of representations of
     prop(erad)s, preprint.

 [3] B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne
     conjecture, to appear in The Journal fur die reine und angewandte
                                           ¨
     Mathematik.

 [4] B. Vallette, Homology of generalized partition posets, Journal of Pure and
     Applied Algebra, Volume 208, Issue 2, February 2007, 699-725.

 [5] B. Vallette, Free monoid in monoidal abelian categories, preprint.



Home Page :   http://math.unice.fr/∼brunov/

								
To top