Deformation theory of algebraic structures

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Deformation theory of algebraic structures

Second Congr`es Canada-France

June 2008

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Data:

An algebraic structure P A vector space A

Goal: Study the deformation theory of P -structures on A

Define a dg Lie algebra g(P , A) [Deligne]

Maurer-Cartan elements ←→ P-structures on A

H

^•

(g(P , A)) : obstruction space for the possible deformations

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Paradigm: Associative algebras

•LetAbe aK-module, considerEndA:={Hom(A^⊗n,A)}n≥0. Forf ∈Hom(A^⊗n,A),g∈Hom(A^⊗m,A)

f ?g:=

n

X

i=1

± g

???

i

f :::: ⁼

n

X

i=1

±f ◦ig.

Theorem (Gerstenhaber)

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

(EndV,[,])is a Lie algebra.

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•Associative algebra structure onA:

µ : A^⊗2→A , ??

??

?? ₋? ????? ^{= 0} ⁱⁿ ^Hom(A^⊗3^,^A)^⇐⇒ ^{µ ? µ}^{= 0} ^⇐⇒ ^{[µ, µ] = 0} In this case,

dµ(f) := [µ,f] satisfies dµ(f)²= 0.

Explicitly, forf ∈Hom(A^⊗n,A)

dµ(f) =

n

X

i=1

±

DDDD zzµzz

i

f :::: ^±

f CCCC wwww

µ

>>> ^±

f {{ {{ GGGG µ

:::: ∈Hom(A^⊗n+1,A) Hom( ,A)−→^d^µ Hom(A,A)−→^d^µ Hom(A^⊗2,A)−→ · · ·^d^µ

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•Deformation complex of the associative algebra structureµonA: C^•(Ass,A) := (EndA,dµ,[,]) dg Lie algebra (twisted byµ).

.Interpretation ofH⁰,H¹,H²in terms of deformations ofµ

→Deformation Quantization[Kontsevich]

•More operations onC^•(Ass,A) : . Cup product∪: associative operation . Braces operations

→Deligne Conjecture

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•(A,d) dg module =⇒EndA dg module

D(f) :=

n

X

i=1

d

i

f

:::: −(−1)^|f^| f

????

d

(EndA,D, ?) is a dg preLie algebra and (EndA,D,[,]) is a dg Lie algebra.

(A,d, µ) is a dg associative algebra⇐⇒

Dµ+µ ? µ= 0 ⇐⇒ Dµ+1

2[µ, µ] = 0 :Maurer-Cartan equation

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General solutions:

µ∈ {Hom(V^⊗n,V)}n≥2, µn:V^⊗n→V Dµ+µ ? µ= 0⇐⇒

n= 2 : d >>>

+ 8888 ^d

= EEEE yyyy d

: d is a derivation forµ2

n= 3 : ????₋? ????? ⁼^D(µ³^{) :} ^µ²is associative up to the homotopyµ3

n : X

i+j=n+1 i,j≥2

±?????_i?_j

₌_D(µ

n)

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Definition (Stasheff)

Maurer-Cartan elementµin (EndA,D, ?)⇐⇒

associative algebra up to homotopyorA∞-algebrastructure on (A,d, µ={µn}n).

Viewpoint : An associative algebra = very particularA∞-algebra.

Once again,dµ(f) :=D(f) + [µ,f] satisfies d_µ²= 0.

=⇒ (EndA,dµ,[,]) dg Lie algebra (twisted byµ)

→ deformation complex of A∞-algebra structures

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•Other kind of algebraic structures :

. Lie algebras, commutative algebras, Poisson algebras, Gerstenhaber algebras, PreLie algebras, Batalin-Vilkovisky algebras, ...

. Lie bialgebras, Frobenius bialgebras, associative bialgebras, ...

For any algebraic structureP:

dg moduleA

dg Lie algebra : g(P,A)

Maurer-Cartan elements= homotopyP −algebra structure onA

twisted dg Lie : deformation complex (cohomology)

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Operations • @ ~~@_• @ ~~@_•

~~ @@ @ ~~@_•

~~ @@

Composition •

•

@ ~~ @@ ~~@_• LL L • •

•rrr

@ ~~@_•

@@

@@ ^•

~~~~

• •

~~ @@

@ ~~ @@ ~~@_•

@@

@@ ^•

~~~~ ^•

• •

~~ @@ ^•

Monoidal category

(Vect,⊗) (S-Mod,◦) (S-biMod,c) (S-biMod,) Monoid Associative Algebra^R⊗R→R

P◦P→P Operad

PcP→P Properad

PP→P Prop

Modules Modules Algebras Bialgebras Bialgebras

Examples Associative, Lie,

Gerstenhaber algebras

Lie, associative bialgebras

Free monoid

Ladders

(Tensor module) Trees ^Connected_graphs Graphs

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Properadic homological algebra

C dg coproperadC → CcC ,P a dg properadPcP → P.

Theorem (Merkulov-V.)

. Hom(C,P) :={Hom(C(m,n),P(m,n)}m,nis a dg properad : convolution properad

. (Hom_S(C,P),D,[,])is a dg Lie algebra

Hom_S(C,P) is a “symmetric non-differentialB∞-algebra” :

→non-symmetric operations{x1, . . . ,xk}{y1, . . . ,yl}.

Solution of the Maurer-Cartan equation : twisting morphismTw(C,P).

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Theorem (Merkulov-V., V.)

Hom

dg properads

(Ω(C), P) ∼ = Tw(C, P ) ∼ = Hom

dg coproperads

(C, B(P))

Proof. Cobar construction of a coproperad : Ω(C) := (F (s

⁻¹

C), ¯ d ), where

F : free properad (graphs) s

⁻¹

: homological desuspension C ¯ : augmentation coideal

d : unique coderivation which extends

C

DDD zzz

DD

D ^∆

0

−→

C

_E_E yyyyy yyyyy

yy EE

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Theorem (V.)

Canonical bar-cobar resolution (counit of adjunction)

Ω(B (P )) − → P

^∼

Definition

Quadratic resolution : P

_∞

= Ω(C) − → P

^∼

Homotopy P -algebra : P

_∞

-algebra

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Deformation dg Lie algebra

P∞= Ω(C)MMMMMM^∼MM&& //^P

EndA

g(P,A) := (Hom_S(C,EndA),D,[,])

MC(g(P,A)) =Tw(C,EndA)←→homotopyP-algebra structures onA

Particular case : α:C0→EndA,α(Cn) = 0←→strictP-algebra structures onA Example : P=Ass,C=Ass^∗

∆⁰ O ??OO?OO

ooooo

= X

i+j=n+1 i,j≥2

±????_i??_j

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Problems:

Well defined ?

How to makeCorP∞explicit?

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Homotopy theory for properads

Problem : No model category structure onP-bialgebras over a properad (no coproduct)

Theorem (Merkulov-V.)

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

=⇒g(P,A)well defined in the homotopy category of dg Lie algebras

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Definition

Quasi-free properad (F(X ), d )

F(X ) free properad

d derivation ←→ d

_|X

: X → F(X )

d

_|X

:

^{@ ~~}^@

•

~~ @@

7→ X

@ ~~@

•

@@

•

~~~~

• •

~~ @@

: not only two vertices Theorem (Merkulov-V.)

Cofibrant properads = rectract of quasi-free properads

= ⇒ Quasi-free properads are cofibrant

Any properad admits a quasi-free resolution

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Explicit quasi-free resolution

•

Koszul duality theory

Associative algebras [Priddy], Operads [Ginzburg-Kapranov], Properads [V.]) WhenP=F(V)/(R) withR⊂ F(V)⁽²⁾: quadratic relations

C=P^!^∗=, withP^!:=F(V^∗)/(R^⊥) d ←→∆⁰C=γ_P^t¡

small criterion Examples

P=Ass,C=Ass^∗: Hochschild cohomology of associative algebras P=Lie,C=Com^∗: Chevalley-Eilenberg cohomology of Lie algebras P=Com,C=Lie^∗: Harrison cohomology of commutative algebras

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•

Linear Koszul duality theory

[Galvez-Tonks-V.]

WhenP=F(V)/(R) withR⊂ F(V)⁽²⁾⊕ F(V)⁽¹⁾ : quadratic and linear relations Example : Batalin-Vilkovisky algebras

BB B _•|||

∆

−

∆AAA

• − <<< ^∆

•}}}

| {z }

2

= BB B }}}

[,]

| {z }

1

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•

Homotopy Koszul

(F(C∞),d) quasi-free properad =⇒ C∞is ahomotopy coproperadC∞→ F(C∞) WhenP=F(V)/(R) withR⊂ F(V)^(≥2)

Example : P=BiAss

Theorem (Merkulov-V.)

g(BiAss,A)∼=Gerstenhaber-Shack bicomplex

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Deformation L

_∞

-algebra

P∞=F(CN∞N)NNNN^∼NN'' //^P

EndA

g(P,A) := (Hom_S(C∞,EndA),D,{ln}n) : L∞−algebra Well defined in the homotopy category ofL∞-algebras

Example : P=BiAss

Theorem (Merkulov-V.)

L∞-algebra structure on Gerstenhaber-Shack bicomplex

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Deformation theory of morphisms of properads

P∞=F(CL∞LL)LLL^∼LLL//&&^P

Q

g(P,Q) := (Hom_S(C∞,Q),D,{ln}n) : L∞−algebra Extension of Quillen-Illusie deformation theory of morphisms

Der(P∞,Q)∼=Hom_S(C∞,Q) ∼= HomP−bimodules( L^P∞

| {z }

Cotangent complex

,Q)

based on the module of K¨ahler differentials of the properadP.

Example : P=Ass,Q=Poisson: Invariant of knots [Vassiliev, Turchine]