Higher Loday Algebras

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L

Norbert Poncin

University of Luxembourg

Geometry of Manifolds and Mathematical Physics Krakow 2011

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Higher (Lie and) Loday (Leibniz) algebras:

1 Coalgebraic approach

2 Category theoretical standpoint

3 Operadic view

4 Supergeometric framework

L

L

C

L

:

DGLA

(L,d)

→p

←i (V,d^′) pi=idV,ip∼^h idL

(L,d) :DGLA⇒(V,d^′) :L∞algebra THEOREM

(L,d)and(V,d^′)homotopy equivalent ds. AnL∞structure on L induces anL∞structure on V .

⇝Importance ofL∞in BRST, closed string theory

L

: DQ

P

(V,[−,−]) :GLA,𝜋 ∈V¹,[𝜋, 𝜋] =0 𝔤:=𝜈V[[𝜈]],(𝔤,[−,−], ∂_𝜋): DGLA

𝜋_𝜈 =𝜋+𝜎, s.th. 𝜎∈MC(𝔤) ={s∈𝔤¹:∂_𝜋s+¹₂[s,s] =0}

Def(𝔤) =MC(𝔤)/G(𝔤) Def:L_∞→Set THEOREM

If F :L→L^′ is a Qiso, thenDef(F) :Def(L)→Def(L^′)is bijective.

L= (T_poly(M) = Γ(∧TM),[−,−]^SN,0): DGLA L^′= (D_poly(M)⊂ ℒ_∙(C^∞(M)),[−,−]^G, ∂𝜇): DGLA

L

𝜋p: (sV)^∨p→sV, 𝜋:S(sV)→sV

L∞(V)

∥ +

{(𝜋₁, 𝜋₂, 𝜋₃, . . .), multilin., AS,∣𝜋_p∣=2−p}

CoDiff¹(S(sV))

∥ +

CoDer¹(S(sV))

≃

≃

1)𝜋₁²=0 2)𝜋₁der. of𝜋₂ 3)𝜋₂Jacobi id. modulo homotopy Example: (𝜋₁, 𝜋₂,0, . . .)DGLA

seq. conditions

Q²=0= [Q,Q]

≃

L

𝜋p: (sV)^∨p→sV, 𝜋:S(sV)→sV

L∞(V)

∥ +

{(𝜋₁, 𝜋₂, 𝜋₃, . . .), multilin., AS,∣𝜋_p∣=2−p}

CoDiff¹(S(sV))

∥ +

CoDer¹(S(sV))

≃

≃

1)𝜋₁²=0 2)𝜋₁der. of𝜋₂ 3)𝜋₂Jacobi id. modulo homotopy Example: (𝜋₁, 𝜋₂,0, . . .)DGLA

seq. conditions

Q²=0= [Q,Q]

≃

L

𝜋p: (sV)^∨p→sV, 𝜋:S(sV)→sV

L∞(V)

∥ +

{(𝜋₁, 𝜋₂, 𝜋₃, . . .), multilin., AS,∣𝜋_p∣=2−p}

CoDiff¹(S(sV))

∥ +

CoDer¹(S(sV))

≃

≃

1)𝜋₁²=0 2)𝜋₁der. of𝜋₂ 3)𝜋₂Jacobi id. modulo homotopy Example: (𝜋₁, 𝜋₂,0, . . .)DGLA

seq. conditions ≃ Q²=0= [Q,Q]

Lod

Lod∞(V)

∥ +

@

@

{(𝜋₁, 𝜋₂, 𝜋₃, . . .), multilin., AS,∣𝜋_p∣=2−p}

CoDiff¹(T(sV))

∥ +

CoDer¹(T(sV))

≃?

≃?

1)𝜋²₁=0 2)𝜋₁der. of𝜋₂ 3)𝜋₂Jacobi id. modulo homotopy Example:(𝜋₁, 𝜋₂,0, . . .)DGLodA

? seq. conditions ≃^? Q²=0= [Q,Q]

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THEOREM

Δ(v₁⊗. . .⊗v_p−1⊗v_p) =

∑p−2 i=1

∑

𝜎∈sh(i,p−1−i)±v_𝜎(1)⊗. . .⊗v_𝜎(i)⊗

v_𝜎(i+1)⊗. . .⊗ v_𝜎(p−1)⊗vp

(T(V),Δ):twistedcoassociative coalgebra, i.e.

(id⊗

Δ)Δ = (Δ⊗

id)Δ +(T⊗

id)(Δ⊗ id)Δ

Lod

THEOREM

C_n=

∑

i+j=n+1

∑

k≥j

∑

𝜎∈sh(k−j,j−1)

𝜒(𝜎)(−1)(k+i−j)(j−1)(−1)^{j(v𝜎(1)+...+v𝜎(k−j))} 𝜋_i(v_𝜎(1), ...,v_𝜎(k−j), 𝜋_j(v_{𝜎(k+1−j)}, ...,v_𝜎(k−1),v_k),v_k+1, ...,v_n) =0,

𝜒(𝜎) :Koszul sign, n≥1.

Lod

Lod∞(V)

∥ +

{seq. 𝜋= (𝜋₁, 𝜋₂, . . .)multilin.,∣𝜋_p∣=2−p}

[𝜋, 𝜋]^Stem=0?

Cochain space={seq. multilin.}

[−,−]^Stem?

⇝Lod∞cohomology ?

Lod

Lod∞(V)

∥ +

{seq.𝜋 = (𝜋₁, 𝜋₂, . . .)multilin.,∣𝜋_p∣=2−p}

CoDiff¹(T(sV))

∥ +

CoDer¹(T(sV))

≃

≃

[𝜋, 𝜋]^Stem=0 ≃ Q²=0= [Q,Q]

Cochain space={seq. multilin.} ≃ CoDer(T(sV))

≃ [−,−]

[−,−]^Stem

⇝Lod∞cohomology

S

THEOREM

[𝜋, 𝜌]^Stem=∑

q≥1

∑

s+t=q+1

(−1)^{1+(s−1)⟨e1,𝜌⟩}[𝜋_s, 𝜌_t]^Stem,

[A,B]^Stem=j_AB−(−1)⟨(A,a),(B,b)⟩j_BA,

j_BA= (−1)^⟨A,B⟩∑

I∪J∪K=N^(a+b+1) I,J<K,∣J∣=b

(−1)^{⟨B,VI⟩+b∣I∣}(−1)^(I;J)𝜀V(I;J)

A(V_I ⊗B(V_J⊗v_k1)⊗V_K∖k1)

D

THEOREM

[𝜋,−]^Stemencodes

graded Loday (DT), graded Lie (LMS), graded Poisson, graded Jacobi (GM, up to isom) cohomologies

Loday infinity, Lie infinity (FP),2n-ary graded Loday and Lie(MV, VV) cohomologies

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C

-

Sets⇝categories Maps⇝functors

Equations⇝natural isomorphisms + coherence laws

1 vs≃linear set⇝2vs≃linear category

2 LA≃vs + bilinear map + AS, Jacobi

⇝

L2A≃2vsL+ bilinear functor[−,−]+ trilinear natural isomorphismJ : [−,[−,−]]+↻⇒ 0 + coherence laws Semistrict L2A [Baez, Crans, ’04], weak L2A [Roytenberg,

’07]

2

2 category (rough description):

objectsA,1-morphismsf :A→B,2-morphisms 𝛼:f ⇒f^′ (homological algebra, string theory)

vertical composition:f,f^′,f^′′∈Hom(A,B),𝛼:f ⇒f^′, 𝛽 :f^′ ⇒f^′′,

𝛼∙𝛽 :f ⇒f^′′

functorial horizontal composition:

∘:Hom(A,B)×Hom(B,C)→Hom(A,C)is a bifunctor, i.e.

is s.th. if𝛼:f ⇒f^′,𝛽 :g⇒g^′, then 𝛼∘𝛽 :f ∘g ⇒f^′∘g^′

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∙ _CC∙ _CC∙

∙ ^//_EE

∙

~~

∙ ^{// //} _CC∙

2

Lie2Alg

Objects: L2A(L,[−,−],J),(L^′,[−,−]^′,J^′), . . . 1-morphisms:

lin funF :L→L^′

bilin natural isoF₂: [−,−]^′∘(F ⊗F)⇒F ∘[−,−]

respect of Jacobiators 2-morphisms:

lin natural transfo𝜃:F :L→L^′⇒G:L→L^′ respect ofF₂andG₂

2vs:s,t :L₁→L₀ + 1,∘; 2term:𝜋₁:V₁→V₀; 𝜋₁=t∣_kers Lie2Alg

≃2TermL_∞

2

Lie2Alg

Objects: L2A(L,[−,−],J),(L^′,[−,−]^′,J^′), . . . 1-morphisms:

lin funF :L→L^′

bilin natural isoF₂: [−,−]^′∘(F ⊗F)⇒F ∘[−,−]

respect of Jacobiators 2-morphisms:

lin natural transfo𝜃:F :L→L^′⇒G:L→L^′ respect ofF₂andG₂

2vs:s,t :L₁→L₀ + 1,∘; 2term:𝜋₁:V₁→V₀; 𝜋₁=t∣_kers Lie2Alg≃2TermL_∞

2

Lod2Alg

THEOREM

Lod2Alg≃2TermLod_∞ Suppression of AS = destruction of simplifications Conceptual approach to and explicit formulae for 1- and 2-morphisms

C

.

[KHUDAVERDYAN, MANDAL, P, ’10]

Conjecture 1:LnA≃nTermL∞

L2A ∼ catL fun[−,−] transJ coh

L3A :∼ 2-catL 2-fun[−,−] 2-transJ 2-modI ?coh?

↕ ↕ (1) ↕ (2) ↕ (3) ↕ (4) ↕ (5)

3TermL∞ ∼ V_i, 𝜋1,C₁ 𝜋2,C₂ 𝜋3,C₃ 𝜋4,C₄ C₅ Answer 1a: (5) involves more than 100 terms

Answer 1b: (2) is “[−,−]respects∘ ↔C₂” and holds true only under two (acceptable)conditions

C

.

Fact:Vectn-Catis a symmetric monoidal category L×L^′ → L^′′

⊠ ↓ ↗ L⊠L^′

!⊠:L×L^′ →L⊠L^′ is not ann-functor ! (★)

Conjecture 2: Use

𝒩 :VectCat→s(Vect),N :s(Vect)→C⁺(Vect), EZ:N(S)⊗N(T)↔N(S⊗T) :AW

Answer 2:Obstruction = defect (★)

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D

5 definitions:

classical, partial compositions, type of algebras, functorial, combinatorial

algebra→matrices, operad→algebra operad:

vsP(n),n∈ℕ^∗, of abstractn-ary operations -T_n

S_n-module structure onP(n) linear composition maps

𝛾_i1...ik :P(k)⊗P(i₁)⊗. . .⊗P(i_k)→P(i₁+. . .+i_k), which are associative, respect theS-action (and possibly the unit)

D

5 definitions:

classical, partial compositions, type of algebras, functorial, combinatorial

algebra→matrices, operad→algebra operad:

vsP(n),n∈ℕ^∗, of abstractn-ary operations -T_n S_n-module structure onP(n)

linear composition maps

𝛾_i1...ik :P(k)⊗P(i₁)⊗. . .⊗P(i_k)→P(i₁+. . .+i_k), which are associative, respect theS-action (and possibly the unit)

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Example:

AA

P(1) =𝕂T₁

P(2) =𝕂T₂(12) +𝕂T₂(21)(encoding of symmetries) P(3) =𝕂T₂(T₂,T₁) +𝕂T₂(T₁,T₂)(encoding of defining relations)

P(n) =𝕂[S_n] General case:

type of algebras

generating operations with symmetries⇝S_n-moduleM(n) defining relations⇝ideal (R)

operadP=ℱ(M)/(R)

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A: D(GA)A, C: D(GA)C, suspensions + reductions understood Homalg(T(C),A)≃Hom_𝕂(C,A)≃Homcoalg(C,T^c(A)) ΩC: DAcobar complex, BA: DC bar complex

d₂(a₁⊗. . .⊗an) =∑

±a₁⊗. . .⊗𝜇(a_i,a_i+1)⊗. . .⊗an

HomDA(ΩC,A)≃TW(C,A)≃HomDC(C,BA) Qiso(ΩC,A)≃Kos(C,A)≃Qiso(C,BA)

d𝛼 :C⊗A^Δ⊗id→ C⊗C⊗A^{id⊗𝛼⊗id}→ C⊗A⊗A^id→^⊗𝜇C⊗A

ΩBA→^∼ A: bar-cobar resolution of A

K

C(V,R) =𝕂⊕V ⊕⊕

k≥2

∩

i+j+2=k

V^⊗i⊗R⊗V^⊗j ⊂T^c(V) Koszul dual coalgebra ofA(V,R):

A¡ =C(sV,s²R)⊃𝕂⊕sV TW(A¡,A)∋𝛼:A¡ ↠sV →V ↣A

If𝛼∈Kos(A¡,A), then

ΩA¡ →^∼ A

Extension of this minimal model ofKoszul algebras[Priddy, ’70]

toKoszul operads[Ginzburg, Kapranov, ’94]

I

End(V): endomorphism operad on a vsV

P→End(V): operadic morphism,typeP algebra structure onV

IfP is aKoszul operad,

P_∞:=

ΩP¡ →^∼ P

↘ ↓

End(V)

FromΩℒie¡andΩℒod¡we recoverL∞andLod∞structures onV

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End(V): endomorphism operad on a vsV

P→End(V): operadic morphism,typeP algebra structure onV

IfP is aKoszul operad,

P_∞:=ΩP¡ →^∼ P

↘ ↓

End(V)

FromΩℒie¡andΩℒod¡we recoverL∞andLod∞structures onV

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L

Lod

Leibniz (Loday) algebras

★Origin: periodicity phenomena inK-theory

★Examples:

1 Courant-Dorfman bracket, Courant algebroid bracket

2 Loday bracket associated with a Nambu-Poisson structure

3 Derived brackets Loday infinity algebras

★Origin: universal GLA and cohomologies

★Examples:

1 Fulton-MacPherson type compactification [Kontsevich, ’03]

[Merkulov, ’09]

2 Formal deformation of a DGLodA and derived brackets [Uchino, ’09]

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K

Ginzburg-Kapranov, ’94:

P∞-algebra onV ⇌Q∈Der1(ℱ_P^gr_!(sV^∗)),Q²=0 Example:

L∞-algebra⇌homological vf on apointed formal smfd Geometric ‘example’:

L∞-algebroid⇌homological vf on asplitℕ-mfd Particular case:

LAD⇌homological vf on asplit smfd

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Find a concept ofLoday algebroidthat is close to the notion of Lie algebroid

contains Courant algebroids as special case reduces to a Loday algebra over a point

includes a differentiability condition on both arguments, and interpret it ashomological vf on a supercommutative mfd

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:

‘Definition’: ALoday algebroidis a Loday bracket[−,−]on sections of a vbE together with aleft and right anchor J. Grabowski, G. Marmo:

Ifrk(E) =1,[−,−]is AS and 1st order Ifrk(E)>1,[−,−]is ‘locally’ aLAD bracket

‘No’ new examples⇝modify ‘definition’

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:

𝜌: Γ(E)→DerC^∞(M),C^∞(M)-linear

∂_x(fY) =∂_xf Y +f ∂_xY,∂_x²(fY) =∂²_xf Y +2∂_xf∂_xY +f ∂_x²Y [X,fY] =f[X,Y] +𝜌(X)f Y

[Xⁱe_i,fY^je_j] =

Xⁱa^k_ijfY^j e_k+Xⁱ𝜌^a_i∂af Y^je_j+Xⁱ𝜌^a_i f ∂aY^j e_j−Yⁱ𝜌^a_i∂aX^j e_j 𝜌(X)(df⊗Y) =Xⁱ𝜌^ak_ij ∂af Y^je_k

𝜌: Γ(E)^C

∞(M)−lin

−→ Γ(TM)⊗_C∞(M)End_C^∞_(M)Γ(E) 𝜌:E →TM⊗EndE

Cohomology theory⇝representation,traditional left anchor

D

DEFINITION

ALoday algebroid(LodAD) is a Loday bracket on sections of a vb E→M together with two bundle maps𝜌:E→TMand

𝛼:E→TM⊗EndE such that

[X,fY] =f[X,Y] +𝜌(X)f Y and

[fX,Y] =f[X,Y]−𝜌(Y)f X+𝛼(Y)(df ⊗X).

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EXAMPLES

Loday algebra

(twisted) Courant-Dorfman (TM⊕T^∗M)

Grassmann-Dorfman (TM⊕ ∧T^∗M orE⊕ ∧E^∗)

Leibniz algebroid associated to a Nambu-Poisson structure Courant algebroid

...

E

L

(I)

(E,[−,−], 𝜌)⇌Q∈Der1(Γ(∧E^∗),∧),Q²=0 (E,[−,−], 𝜌)→LAD cohomology operator∂𝜌

∂𝜌is the C-E operator restricted to

∧_C∞(M)(Γ(E),C^∞(M)) = Γ(∧E^∗)

∂_𝜌is the Loday operator restricted to Lin_C^∞_(M)

D

(Γ(E),C^∞(M)) = Γ(⊗E^∗)

=:D(E)

(D⋔Δ)(X₁, . . . ,X_p+q)

= ∑

𝜎∈sh(p,q)

sign(𝜎)D(X_𝜎1, . . . ,X_𝜎p) Δ(X_𝜎p+1, . . . ,X_𝜎p+q)

E

L

(I)

(E,[−,−], 𝜌)⇌Q∈Der1(Γ(∧E^∗),∧),Q²=0 (E,[−,−], 𝜌)→LAD cohomology operator∂𝜌

∂𝜌is the C-E operator restricted to

∧_C∞(M)(Γ(E),C^∞(M)) = Γ(∧E^∗)

∂_𝜌is the Loday operator restricted to

Lin_C^∞_(M)

D(Γ(E),C^∞(M))

= Γ(⊗E^∗)

=:D(E)

(D⋔Δ)(X₁, . . . ,X_p+q)

= ∑

𝜎∈sh(p,q)

sign(𝜎)D(X_𝜎1, . . . ,X_𝜎p) Δ(X_𝜎p+1, . . . ,X_𝜎p+q)

E

L

(I)

(E,[−,−], 𝜌)⇌Q∈Der1(Γ(∧E^∗),∧),Q²=0 (E,[−,−], 𝜌)→LAD cohomology operator∂𝜌

∂𝜌is the C-E operator restricted to

∧_C∞(M)(Γ(E),C^∞(M)) = Γ(∧E^∗)

∂_𝜌is the Loday operator restricted to

Lin_C^∞_(M)

D(Γ(E),C^∞(M))

= Γ(⊗E^∗)

=:D(E)

(D⋔Δ)(X₁, . . . ,X_p+q)

= ∑

𝜎∈sh(p,q)

sign(𝜎)D(X_𝜎1, . . . ,X_𝜎p) Δ(X_𝜎p+1, . . . ,X_𝜎p+q)

E

L

(II)

(E,[−,−], 𝜌, 𝛼)⇌Q∈Der1(D(E),⋔),Q²=0

𝒟^k(E)⊂D^k(E):k-DO of deg 0 in last section and tot degk −1 𝒟^k(E)⋔𝒟^ℓ(E)⊂ 𝒟^k+ℓ(E): reduced shuffle algebra

∂_𝜌𝒟^k(E)⊂ 𝒟^k+1(E): LodAD subcomplex

E

L

(III)

(E,[−,−], 𝜌, 𝛼)⇌Q∈Der1(𝒟(E),⋔),Q²=0

𝜌(X)f :=⟨Qf,X⟩

⟨ℓ,[X,Y]⟩=⟨X,Q⟨ℓ,Y⟩⟩ − ⟨Y,Q⟨ℓ,X⟩⟩ −(Qℓ)(X,Y)

Q(𝒟²(E))↮Q(𝒟⁰(E)⊕ 𝒟¹(E))

𝒟er₁(𝒟(E),⋔)⊂Der1(𝒟(E),⋔): deg 1 der + appropriate rel (E,[−,−], 𝜌, 𝛼)⇌Q∈ 𝒟er₁(𝒟(E),⋔),Q²=0

(E,[−,−], 𝜌, 𝛼)⇌[Q]∈ 𝒟er₁(𝒟(E),⋔),Q²=0

L

AD

THEOREM

There is a 1-to-1 correspondence between LodAD structures on a vb E andequivalence classes of homological vfs

Q∈ 𝒟er₁(𝒟(E),⋔),Q²=0 of the supercommutative mfd(𝒟(E),⋔).

Remarks: Homological vfQ⇝ Cartan calculusfor(D(E),⋔)

LodAD bracket isderived bracketgiven by gLa (Der(D(E),⋔),[−,−]_c)and its interior der[Q,−]_c

R

1 M. Ammar, P,Coalgebraic Approach to the Loday Infinity Category, Stem Differential for 2n-ary Graded and Homotopy Algebras, Ann. Inst. Fourier, 60(1), 2010, pp 321-353

2 D. Khudaverdyan, A. Mandal, P,Equivalence of the 2-categories of categorified Loday algebras and 2-term Loday infinity algebras, draft

3 D. Khudaverdyan, A. Mandal, P,Higher categorified algebras versus bounded homotopy algebras, Theo. Appl.

Cat., Vol. 25, No. 10, 2011, pp 251-275

4 P. Hilger, A. Mandal, P,Lectures on Operad Theory, preprint

5 J. Grabowski, D. Khudaverdyan, P,Loday algebroids and their supergeometric interpretation, preprint arXiv:

1103.5852

Thank you!