Tensor Products of A -algebras with Homotopy Inner Products (Joint - - PowerPoint PPT Presentation

Tensor Products of A∞-algebras with Homotopy Inner Products

(Joint work with Thomas Tradler, CUNY)

Ron Umble Millersville University

Lehigh Geometry and Topology Conference

May 25, 2013

Tensor Products of A-infinity Algebras

Let K = Kn denote Stasheff’s associahedra

Tensor Products of A-infinity Algebras

Let K = Kn denote Stasheff’s associahedra Let R be a commutative ring with unity

Tensor Products of A-infinity Algebras

Let K = Kn denote Stasheff’s associahedra Let R be a commutative ring with unity Let (A, {µn}) and (B, {νn}) be A∞-algebras over R

Tensor Products of A-infinity Algebras

Let K = Kn denote Stasheff’s associahedra Let R be a commutative ring with unity Let (A, {µn}) and (B, {νn}) be A∞-algebras over R A diagonal on cellular chains

∆K : C∗K → C∗K ⊗ C∗K was constructed by Saneblidze-U and Markl-Snider

Tensor Products of A-infinity Algebras

Let K = Kn denote Stasheff’s associahedra Let R be a commutative ring with unity Let (A, {µn}) and (B, {νn}) be A∞-algebras over R A diagonal on cellular chains

∆K : C∗K → C∗K ⊗ C∗K was constructed by Saneblidze-U and Markl-Snider

∆K induces an A∞-algebra structure {ϕn} on A ⊗ B

Tensor Products of A-infinity Algebras

Define ϕ1 = µ1 ⊗ 1 + 1 ⊗ ν1

Tensor Products of A-infinity Algebras

Define ϕ1 = µ1 ⊗ 1 + 1 ⊗ ν1 Given operadic representations of A∞-structures

• ζn : C∗K → Hom
• A⊗n, A
• n≥2
• ξn : C∗K → Hom
• B⊗n, B
• n≥2

Tensor Products of A-infinity Algebras

Define ϕ1 = µ1 ⊗ 1 + 1 ⊗ ν1 Given operadic representations of A∞-structures

• ζn : C∗K → Hom
• A⊗n, A
• n≥2
• ξn : C∗K → Hom
• B⊗n, B
• n≥2

Define a representation θn by the composition

C∗K

θn

−−→ Hom

• (A ⊗ B)⊗n , A ⊗ B
• ∆K ↓

↑≈ C∗K ⊗ C∗K

ζn⊗ξn

− → Hom (A⊗n, A) ⊗ Hom (B⊗n, B)

Tensor Products of A-infinity Algebras

Then θn sends the top-dimensional cell en−2 ⊂ Kn to

ϕn = (ζn ⊗ ξn) ∆K

• en−2

σn where σn : A⊗n ⊗ B⊗n → (A ⊗ B)⊗n is canonical permutation

Tensor Products of A-infinity Algebras

Then θn sends the top-dimensional cell en−2 ⊂ Kn to

ϕn = (ζn ⊗ ξn) ∆K

• en−2

σn where σn : A⊗n ⊗ B⊗n → (A ⊗ B)⊗n is canonical permutation

And lower-dimensional faces to ◦i-compositions

Tensor Products of A-infinity Algebras

Then θn sends the top-dimensional cell en−2 ⊂ Kn to

ϕn = (ζn ⊗ ξn) ∆K

• en−2

σn where σn : A⊗n ⊗ B⊗n → (A ⊗ B)⊗n is canonical permutation

And lower-dimensional faces to ◦i-compositions ϕ2 = (µ2 ⊗ ν2) σ2

Tensor Products of A-infinity Algebras

Then θn sends the top-dimensional cell en−2 ⊂ Kn to

ϕn = (ζn ⊗ ξn) ∆K

• en−2

σn where σn : A⊗n ⊗ B⊗n → (A ⊗ B)⊗n is canonical permutation

And lower-dimensional faces to ◦i-compositions ϕ2 = (µ2 ⊗ ν2) σ2 ϕ3 = µ2 (µ2 ⊗ 1) ⊗ ν3 + µ3 ⊗ ν2 (1 ⊗ ν2)

Tensor Products of A-infinity Algebras

Then θn sends the top-dimensional cell en−2 ⊂ Kn to

ϕn = (ζn ⊗ ξn) ∆K

• en−2

σn where σn : A⊗n ⊗ B⊗n → (A ⊗ B)⊗n is canonical permutation

And lower-dimensional faces to ◦i-compositions ϕ2 = (µ2 ⊗ ν2) σ2 ϕ3 = µ2 (µ2 ⊗ 1) ⊗ ν3 + µ3 ⊗ ν2 (1 ⊗ ν2) And so on...

Cyclic A-infinity Algebras

An A∞-algebra (A, {µn}) is cyclic if

A is equipped with a cyclically invariant inner product µn (a1, . . . , an) , an+1 = µn (a2, . . . , an+1) , a1

Cyclic A-infinity Algebras

An A∞-algebra (A, {µn}) is cyclic if

A is equipped with a cyclically invariant inner product µn (a1, . . . , an) , an+1 = µn (a2, . . . , an+1) , a1

What is the structure of A ⊗ B when A and B are cyclic?

Cyclic A-infinity Algebras

An A∞-algebra (A, {µn}) is cyclic if

A is equipped with a cyclically invariant inner product µn (a1, . . . , an) , an+1 = µn (a2, . . . , an+1) , a1

What is the structure of A ⊗ B when A and B are cyclic? Inner products −, −A and −, −B induce an inner product

a|b, c|dA⊗B = a, cA b, dB

Tensor Product of Cyclic A-infinity Algebras

The differential ϕ1 is cyclically invariant (ignoring signs):

ϕ1 (a|b) , c|d = µ1 (a) |b + a|ν1 (b) , c|d

Tensor Product of Cyclic A-infinity Algebras

The differential ϕ1 is cyclically invariant (ignoring signs):

ϕ1 (a|b) , c|d = µ1 (a) |b + a|ν1 (b) , c|d = µ1 (a) , cA b, dB + a, cA ν1 (b) , dB

Tensor Product of Cyclic A-infinity Algebras

The differential ϕ1 is cyclically invariant (ignoring signs):

ϕ1 (a|b) , c|d = µ1 (a) |b + a|ν1 (b) , c|d = µ1 (a) , cA b, dB + a, cA ν1 (b) , dB = µ1 (c) , aA d, bB + c, aA ν1 (d) , bB

Tensor Product of Cyclic A-infinity Algebras

The differential ϕ1 is cyclically invariant (ignoring signs):

ϕ1 (a|b) , c|d = µ1 (a) |b + a|ν1 (b) , c|d = µ1 (a) , cA b, dB + a, cA ν1 (b) , dB = µ1 (c) , aA d, bB + c, aA ν1 (d) , bB = µ1 (c) |d + c|ν1 (d) , a|b

Tensor Product of Cyclic A-infinity Algebras

The differential ϕ1 is cyclically invariant (ignoring signs):

ϕ1 (a|b) , c|d = µ1 (a) |b + a|ν1 (b) , c|d = µ1 (a) , cA b, dB + a, cA ν1 (b) , dB = µ1 (c) , aA d, bB + c, aA ν1 (d) , bB = µ1 (c) |d + c|ν1 (d) , a|b = ϕ1 (c|d) , a|b

Tensor Product of Cyclic A-infinity Algebras

The product ϕ2 is cyclically invariant:

ϕ2 (a|b, c|d) , e|f = µ2 (a, c) |ν2 (b, d) , e|f

Tensor Product of Cyclic A-infinity Algebras

The product ϕ2 is cyclically invariant:

ϕ2 (a|b, c|d) , e|f = µ2 (a, c) |ν2 (b, d) , e|f = µ2 (a, c) , eA ν2 (b, d) , f B

Tensor Product of Cyclic A-infinity Algebras

The product ϕ2 is cyclically invariant:

ϕ2 (a|b, c|d) , e|f = µ2 (a, c) |ν2 (b, d) , e|f = µ2 (a, c) , eA ν2 (b, d) , f B = µ2 (c, e) , aA ν2 (d, f ) , bB

Tensor Product of Cyclic A-infinity Algebras

The product ϕ2 is cyclically invariant:

ϕ2 (a|b, c|d) , e|f = µ2 (a, c) |ν2 (b, d) , e|f = µ2 (a, c) , eA ν2 (b, d) , f B = µ2 (c, e) , aA ν2 (d, f ) , bB = µ2 (c, e) |ν2 (d, f ) , a|b

Tensor Product of Cyclic A-infinity Algebras

The product ϕ2 is cyclically invariant:

ϕ2 (a|b, c|d) , e|f = µ2 (a, c) |ν2 (b, d) , e|f = µ2 (a, c) , eA ν2 (b, d) , f B = µ2 (c, e) , aA ν2 (d, f ) , bB = µ2 (c, e) |ν2 (d, f ) , a|b = ϕ2 (c|d, e|f ) , a|b

Tensor Product of Cyclic A-infinity Algebras

But ϕ3 is not cyclically invariant because...

Tensor Product of Cyclic A-infinity Algebras

But ϕ3 is not cyclically invariant because... ϕ3 (a|b, c|d, e|f ) , g|h = ϕ3 (c|d, e|f , g|h) , a|b implies

(1) µ2 (µ2 (a, c) , e) , gA = µ2 (µ2 (c, e) , g) , aA (2) ν2 (b, ν2 (d, f )) , hB = ν2 (d, ν2 (f , h)) , bB

Tensor Product of Cyclic A-infinity Algebras

But ϕ3 is not cyclically invariant because... ϕ3 (a|b, c|d, e|f ) , g|h = ϕ3 (c|d, e|f , g|h) , a|b implies

(1) µ2 (µ2 (a, c) , e) , gA = µ2 (µ2 (c, e) , g) , aA (2) ν2 (b, ν2 (d, f )) , hB = ν2 (d, ν2 (f , h)) , bB

Which only hold up to homotopy

Tensor Product of Cyclic A-infinity Algebras

Cyclicity and homotopy associativity give chain homotopies

Tensor Product of Cyclic A-infinity Algebras

Cyclicity and homotopy associativity give chain homotopies For relation (1):

µ2 (µ2 (a, c) , e) , g = [µ1, µ3] (a, c, e) ± µ2 (a, µ2 (c, e)) , g = [µ1, µ3] (a, c, e) , g ± µ2 (µ2 (c, e) , g) , a

Tensor Product of Cyclic A-infinity Algebras

Cyclicity and homotopy associativity give chain homotopies For relation (1):

µ2 (µ2 (a, c) , e) , g = [µ1, µ3] (a, c, e) ± µ2 (a, µ2 (c, e)) , g = [µ1, µ3] (a, c, e) , g ± µ2 (µ2 (c, e) , g) , a

Another application of cyclicity gives the chain homotopy

(µ3, − ◦ d) (a, c, e, g) = µ2 (µ2 (a, c) , e) , g ± µ2 (µ2 (c, e) , g) , a where d is the linear extension of µ1

Tensor Product of Cyclic A-infinity Algebras

Chain homotopies (1) and (2) induce a chain homotopy

̺2,0 : (A ⊗ B)⊗4 → R such that

• ̺2,0 ◦ d

(a|b, c|d, e|f , g|h) = ϕ3 (a|b, c|d, e|f ) , g|h − ϕ3 (c|d, e|f , g|h) , a|b

Tensor Product of Cyclic A-infinity Algebras

Chain homotopies (1) and (2) induce a chain homotopy

̺2,0 : (A ⊗ B)⊗4 → R such that

• ̺2,0 ◦ d

(a|b, c|d, e|f , g|h) = ϕ3 (a|b, c|d, e|f ) , g|h − ϕ3 (c|d, e|f , g|h) , a|b

̺2,0 extends to an infinite family of higher homotopies

• ̺k,l

Tensor Product of Cyclic A-infinity Algebras

Chain homotopies (1) and (2) induce a chain homotopy

̺2,0 : (A ⊗ B)⊗4 → R such that

• ̺2,0 ◦ d

(a|b, c|d, e|f , g|h) = ϕ3 (a|b, c|d, e|f ) , g|h − ϕ3 (c|d, e|f , g|h) , a|b

̺2,0 extends to an infinite family of higher homotopies

• ̺k,l
• Conclusion: The tensor product of cyclic A∞-algebras is

cyclic up to homotopy, and in fact...

Tensor Product of Cyclic A-infinity Algebras

Chain homotopies (1) and (2) induce a chain homotopy

̺2,0 : (A ⊗ B)⊗4 → R such that

• ̺2,0 ◦ d

(a|b, c|d, e|f , g|h) = ϕ3 (a|b, c|d, e|f ) , g|h − ϕ3 (c|d, e|f , g|h) , a|b

̺2,0 extends to an infinite family of higher homotopies

• ̺k,l
• Conclusion: The tensor product of cyclic A∞-algebras is

cyclic up to homotopy, and in fact...

∃ additional bimodule structure s.t.

• A ⊗ B, {ϕn} ,
• ̺k,l
• is an A∞-algebra with homotopy inner products (HIPs)

A-infinity Algebras with HIPs

Goal: Define tensor product of general A∞-algebras with HIPs

A-infinity Algebras with HIPs

Goal: Define tensor product of general A∞-algebras with HIPs An A∞-algebra with homotopy inner products consists of

A-infinity Algebras with HIPs

Goal: Define tensor product of general A∞-algebras with HIPs An A∞-algebra with homotopy inner products consists of

• 1. an A∞-algebra (A, {µn})

A-infinity Algebras with HIPs

Goal: Define tensor product of general A∞-algebras with HIPs An A∞-algebra with homotopy inner products consists of

• 1. an A∞-algebra (A, {µn})
• 2. a compatible family of higher inner products
• ̺j,k : A ⊗ A⊗j ⊗ A ⊗ A⊗k → R

A-infinity Algebras with HIPs

Goal: Define tensor product of general A∞-algebras with HIPs An A∞-algebra with homotopy inner products consists of

• 1. an A∞-algebra (A, {µn})
• 2. a compatible family of higher inner products
• ̺j,k : A ⊗ A⊗j ⊗ A ⊗ A⊗k → R
• 3. a compatible family of module maps
• λj,k : A⊗j ⊗ A ⊗ A⊗k → A

A-infinity Algebras with HIPs

Goal: Define tensor product of general A∞-algebras with HIPs An A∞-algebra with homotopy inner products consists of

• 1. an A∞-algebra (A, {µn})
• 2. a compatible family of higher inner products
• ̺j,k : A ⊗ A⊗j ⊗ A ⊗ A⊗k → R
• 3. a compatible family of module maps
• λj,k : A⊗j ⊗ A ⊗ A⊗k → A
• Structure relations encoded by a 3-colored operad C∗A

A-infinity Algebras with HIPs

Goal: Define tensor product of general A∞-algebras with HIPs An A∞-algebra with homotopy inner products consists of

• 1. an A∞-algebra (A, {µn})
• 2. a compatible family of higher inner products
• ̺j,k : A ⊗ A⊗j ⊗ A ⊗ A⊗k → R
• 3. a compatible family of module maps
• λj,k : A⊗j ⊗ A ⊗ A⊗k → A
• Structure relations encoded by a 3-colored operad C∗A

Identified with cellular chains of contractible pairahedra

The 3-colored operad CA

C∗A is generated by three types of planar diagrams Colors of leaves and root: Empty, thin, thick

• 1. Planar trees: Control A∞-algebra structure
• Thin leaves and root

The 3-colored operad CA

• 2. Module trees: Control homotopy bimodule structure
• Thick vertical root and leaf
• j thin leaves in left half-plane
• k thin leaves in right half-plane

The 3-colored operad CA

• 3. Inner product diagrams: Control HIP structure
• Empty root and two thick horizontal leaves
• j thin leaves in upper half-plane
• k thin leaves in lower half-plane

Operadic Structure of CA

Compose planar trees in the usual way

Operadic Structure of CA

Compose planar trees in the usual way Compose a module diagram M with a planar tree T by

attaching the root of T to a thin leaf of M

Operadic Structure of CA

Compose planar trees in the usual way Compose a module diagram M with a planar tree T by

attaching the root of T to a thin leaf of M

Compose module trees by

attaching thick root of 2nd to thick leaf of 1st

Operadic Structure of CA

Compose planar trees in the usual way Compose a module diagram M with a planar tree T by

attaching the root of T to a thin leaf of M

Compose module trees by

attaching thick root of 2nd to thick leaf of 1st

Compose an IP diagram I with a module tree M by

attaching thick root of M to a thick leaf of I

Operadic Structure of CA

Compose planar trees in the usual way Compose a module diagram M with a planar tree T by

attaching the root of T to a thin leaf of M

Compose module trees by

attaching thick root of 2nd to thick leaf of 1st

Compose an IP diagram I with a module tree M by

attaching thick root of M to a thick leaf of I

Two inner product diagrams cannot be composed

DG Module Structure of CA

Let D be a diagram — a generator of C∗A

DG Module Structure of CA

Let D be a diagram — a generator of C∗A L (D) = {Leaves of D}

DG Module Structure of CA

Let D be a diagram — a generator of C∗A L (D) = {Leaves of D} E (D) = {(Internal) edges of D}

DG Module Structure of CA

Let D be a diagram — a generator of C∗A L (D) = {Leaves of D} E (D) = {(Internal) edges of D} Degree:

|D| := #L (D) − #E (D) − 2

DG Module Structure of CA

Let D be a diagram — a generator of C∗A L (D) = {Leaves of D} E (D) = {(Internal) edges of D} Degree:

|D| := #L (D) − #E (D) − 2

Boundary: ∂C (D) :=

∑

D /e=D

D, where e is an edge of D

DG Module Structure of CA

Let D be a diagram — a generator of C∗A L (D) = {Leaves of D} E (D) = {(Internal) edges of D} Degree:

|D| := #L (D) − #E (D) − 2

Boundary: ∂C (D) :=

∑

D /e=D

D, where e is an edge of D

∂C (D) is the sum of all diagrams obtained from D by

inserting a single edge

Coloring in CA

0 = empty; 1 = thin; 2 = thick

Coloring in CA

0 = empty; 1 = thin; 2 = thick The coloring of a diagram D with n leaves is a pair

x × y = (x1, . . . , xn) × y ∈ Zn+1

3

• xi is the color of leaf i
• y is the color of the root

Coloring in CA

0 = empty; 1 = thin; 2 = thick The coloring of a diagram D with n leaves is a pair

x × y = (x1, . . . , xn) × y ∈ Zn+1

3

• xi is the color of leaf i
• y is the color of the root

C∗Ax y is generated by diagrams of coloring x × y

Coloring in CA

0 = empty; 1 = thin; 2 = thick The coloring of a diagram D with n leaves is a pair

x × y = (x1, . . . , xn) × y ∈ Zn+1

3

• xi is the color of leaf i
• y is the color of the root

C∗Ax y is generated by diagrams of coloring x × y Example: C∗A11···1 1

is generated by planar trees

Example

C∗A2112 generated by IP diagrams ↔ faces of pairahedron I2,0:

Cubical Subdivision of CA

Following the W -construction of Boardman and Vogt,

there is a cubical subdivision Q∗A of C∗A s.t.

Cubical Subdivision of CA

Following the W -construction of Boardman and Vogt,

there is a cubical subdivision Q∗A of C∗A s.t.

Q∗A is a 3-colored operad

Cubical Subdivision of CA

Following the W -construction of Boardman and Vogt,

there is a cubical subdivision Q∗A of C∗A s.t.

Q∗A is a 3-colored operad Q∗A is generated by all metric diagrams (D, g), where

• D is a generator of C∗A
• g : E (D) → {m, n} labels the (internal) edges of D

either “m” (metric) or “n” (non-metric)

Example

Cubical subdivision of I2,0 (only metric labels are displayed):

The 3-Colored Operad QA

When composing diagrams:

Label the new edge “n”

The 3-Colored Operad QA

When composing diagrams:

Label the new edge “n”

Degree:

|(D, g)| := # metric edges

The 3-Colored Operad QA

When composing diagrams:

Label the new edge “n”

Degree:

|(D, g)| := # metric edges

Boundary:

∂Q(D) := ∑

metric edges e

D/e + De where De is obtained from D by relabeling e non-metric

The 3-Colored Operad QA

When composing diagrams:

Label the new edge “n”

Degree:

|(D, g)| := # metric edges

Boundary:

∂Q(D) := ∑

metric edges e

D/e + De where De is obtained from D by relabeling e non-metric

Example - Boundary of a Metric Square

The Homotopy Equivalence q : CA –> QA

Let m denote the constant map m (e) = m

The Homotopy Equivalence q : CA –> QA

Let m denote the constant map m (e) = m C0A is generated by binary diagrams (#L = #E + 2)

The Homotopy Equivalence q : CA –> QA

Let m denote the constant map m (e) = m C0A is generated by binary diagrams (#L = #E + 2)

• Definition. On a corolla c ∈ C∗Ax

y define

q(c) =

∑

B∈C0Ax

y

(B, m)

The Homotopy Equivalence q : CA –> QA

Let m denote the constant map m (e) = m C0A is generated by binary diagrams (#L = #E + 2)

• Definition. On a corolla c ∈ C∗Ax

y define

q(c) =

∑

B∈C0Ax

y

(B, m)

A general diagram is a ◦i-composition of corollas

The Homotopy Equivalence q : CA –> QA

Let m denote the constant map m (e) = m C0A is generated by binary diagrams (#L = #E + 2)

• Definition. On a corolla c ∈ C∗Ax

y define

q(c) =

∑

B∈C0Ax

y

(B, m)

A general diagram is a ◦i-composition of corollas Extend q to ◦i-compositions multiplicatively:

q

• c ◦i c = q (c) ◦i q
• c

The Poset of Binary Diagrams in CA

Extend Tamari ordering on binary trees to binary diagrams

The Poset of Binary Diagrams

B denotes the poset of all binary diagrams

The Poset of Binary Diagrams

B denotes the poset of all binary diagrams BD ⊂ B is the vertex poset of a diagram D

The Poset of Binary Diagrams

B denotes the poset of all binary diagrams BD ⊂ B is the vertex poset of a diagram D BD has minimal element Dmin and maximal element Dmax

The Poset of Binary Diagrams

B denotes the poset of all binary diagrams BD ⊂ B is the vertex poset of a diagram D BD has minimal element Dmin and maximal element Dmax Example: BI2,0

The 2-Sided Homotopy Inverse p : QA –> CA

• Definition. On a fully metric (D, m) ∈ QkAx

y define

p(D, m) =

∑

S∈Ck Ax

y

Smax≤Dmin

S

The 2-Sided Homotopy Inverse p : QA –> CA

• Definition. On a fully metric (D, m) ∈ QkAx

y define

p(D, m) =

∑

S∈Ck Ax

y

Smax≤Dmin

S

Proposition

The 2-Sided Homotopy Inverse p : QA –> CA

• Definition. On a fully metric (D, m) ∈ QkAx

y define

p(D, m) =

∑

S∈Ck Ax

y

Smax≤Dmin

S

Proposition

• 1. On a corolla c ∈ Q∗A

p (c) = cmin ∈ Bc

The 2-Sided Homotopy Inverse p : QA –> CA

• Definition. On a fully metric (D, m) ∈ QkAx

y define

p(D, m) =

∑

S∈Ck Ax

y

Smax≤Dmin

S

Proposition

• 1. On a corolla c ∈ Q∗A

p (c) = cmin ∈ Bc

• 2. On a fully metric binary diagram (B, m)

p (B, m) =

• c,

if B = cmax for some corolla c 0,

• therwise

The 2-Sided Homotopy Inverse p : QA –> CA

A metric diagram is a ◦i-composition of fully metric diagrams

The 2-Sided Homotopy Inverse p : QA –> CA

A metric diagram is a ◦i-composition of fully metric diagrams Extend p to ◦i-compositions multiplicatively

p (D, m) ◦i

• D, m

= p (D, m) ◦i p

• D, m

The 2-Sided Homotopy Inverse p : QA –> CA

A metric diagram is a ◦i-composition of fully metric diagrams Extend p to ◦i-compositions multiplicatively

p (D, m) ◦i

• D, m

= p (D, m) ◦i p

• D, m
• Theorem

The 2-Sided Homotopy Inverse p : QA –> CA

A metric diagram is a ◦i-composition of fully metric diagrams Extend p to ◦i-compositions multiplicatively

p (D, m) ◦i

• D, m

= p (D, m) ◦i p

• D, m
• Theorem
• 1. p and q are chain maps

The 2-Sided Homotopy Inverse p : QA –> CA

A metric diagram is a ◦i-composition of fully metric diagrams Extend p to ◦i-compositions multiplicatively

p (D, m) ◦i

• D, m

= p (D, m) ◦i p

• D, m
• Theorem
• 1. p and q are chain maps
• 2. pq = Id and qp Id

The Diagonal on QA

Given (D, g) ∈ Q∗A, let X ⊆ {metric edges of D}

The Diagonal on QA

Given (D, g) ∈ Q∗A, let X ⊆ {metric edges of D} Let X = {metric edges of D} − X

The Diagonal on QA

Given (D, g) ∈ Q∗A, let X ⊆ {metric edges of D} Let X = {metric edges of D} − X Obtain D/X from D by contracting the edges of X

The Diagonal on QA

Given (D, g) ∈ Q∗A, let X ⊆ {metric edges of D} Let X = {metric edges of D} − X Obtain D/X from D by contracting the edges of X Obtain DX from D by relabeling the edges in X non-metric

The Diagonal on QA

Given (D, g) ∈ Q∗A, let X ⊆ {metric edges of D} Let X = {metric edges of D} − X Obtain D/X from D by contracting the edges of X Obtain DX from D by relabeling the edges in X non-metric Serre’s diagonal on I n induces a coassociative diagonal

∆Q : Q∗A → Q∗A ⊗ Q∗A

The Diagonal on QA

Given (D, g) ∈ Q∗A, let X ⊆ {metric edges of D} Let X = {metric edges of D} − X Obtain D/X from D by contracting the edges of X Obtain DX from D by relabeling the edges in X non-metric Serre’s diagonal on I n induces a coassociative diagonal

∆Q : Q∗A → Q∗A ⊗ Q∗A

Given by

∆Q(D) = ∑

X

D/X ⊗ DX

The Induced Diagonal on CA

∆Q induces a non-coassociative diagonal on C∗A

∆C : C∗A

q

− → Q∗A

∆Q

− → Q∗A ⊗ Q∗A

p⊗p

− → C∗A ⊗ C∗A

The Induced Diagonal on CA

∆Q induces a non-coassociative diagonal on C∗A

∆C : C∗A

q

− → Q∗A

∆Q

− → Q∗A ⊗ Q∗A

p⊗p

− → C∗A ⊗ C∗A

On a corolla c ∈ CkAx y :

∆C (c) =

∑

S⊗T ∈Ci Ax

y ⊗Ck−i Ax y

Smax≤Tmin

S ⊗ T

Examples

Tensor Products of A-infinity Algebras with HIPs

Given representations of A∞-algebras with HIPs

• φn : C∗A → Hom
• A⊗n, A
• n≥2
• ψn : C∗A → Hom
• B⊗n, B
• n≥2

Tensor Products of A-infinity Algebras with HIPs

Given representations of A∞-algebras with HIPs

• φn : C∗A → Hom
• A⊗n, A
• n≥2
• ψn : C∗A → Hom
• B⊗n, B
• n≥2

Define the representation εn to be the composition

C∗A

εn

−−→ Hom

• (A ⊗ B)⊗n , A ⊗ B
• ∆C ↓

↑≈ C∗A ⊗ C∗A

φn⊗ψn

− → Hom (A⊗n, A) ⊗ Hom (B⊗n, B)

Tensor Products of A-infinity Algebras with HIPs

Given representations of A∞-algebras with HIPs

• φn : C∗A → Hom
• A⊗n, A
• n≥2
• ψn : C∗A → Hom
• B⊗n, B
• n≥2

Define the representation εn to be the composition

C∗A

εn

−−→ Hom

• (A ⊗ B)⊗n , A ⊗ B
• ∆C ↓

↑≈ C∗A ⊗ C∗A

φn⊗ψn

− → Hom (A⊗n, A) ⊗ Hom (B⊗n, B)

(A ⊗ B, ϕ1, ε (C∗A)) is an A∞-algebra with HIPs

Tensor Products of A-infinity Algebras with HIPs

Given representations of A∞-algebras with HIPs

• φn : C∗A → Hom
• A⊗n, A
• n≥2
• ψn : C∗A → Hom
• B⊗n, B
• n≥2

Define the representation εn to be the composition

C∗A

εn

−−→ Hom

• (A ⊗ B)⊗n , A ⊗ B
• ∆C ↓

↑≈ C∗A ⊗ C∗A

φn⊗ψn

− → Hom (A⊗n, A) ⊗ Hom (B⊗n, B)

(A ⊗ B, ϕ1, ε (C∗A)) is an A∞-algebra with HIPs Paper to appear in TAMS

Thank you!