Foams Algebraic Structures Derived from Essential …saito/talks/talk09Waseda.pdfAlgebraic...

AlgebraicStructures

Derived fromEssential

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Algebraic Structures Derived from EssentialSurfaces and Foams

Masahico Saito (U. of South Florida)Coauthor: J. Scott Carter

Waseda University, December 25, 2009

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

2 Essential surfaces

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Background

{ (1 + 1)-TQFTs } 1−1↔ { Commutative Frobenius algebras }A

Diagrammatics of Frobenius algebras, generating operators:

Transposition

Multiplication Comultiplication

Counit (Frobenius form)

Relations:

CommutativeUnit

Associativity Coassociativity Compatibility

Counit

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Problem

Problem: Characterize TQFTs of essential surface cobordismsand foams.

A A A A

We focus on thickened surfaces F × [0, 1] and sl(3)-foams, andinvestigate their algebraic structures.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Problem

Problem: Characterize TQFTs of essential surface cobordismsand foams.

A A A A

We focus on thickened surfaces F × [0, 1] and sl(3)-foams, andinvestigate their algebraic structures.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Motivations

A natural question as generalizations of (1 + 1)-TQFT.

Essential cobordisms are used in global Khovanovhomology by Asaeda–Przytycki-Sikora [APS] (for thickenedsurfaces F × [0, 1]), Turaev-Turner [TT] (unorientedTQFT) and Ishii-Tanaka [IT] (for virtual knots, cf.[Manturov]). They should have algebraic structures.

Foams appear in sl(3) Khovanov homology [Mackaay-Vaz].

By considering such algebraic structures of generalizedTQFTs, new generalizations of KhoHo may be found.

Possible refinements of Bar-Natan modules byAsaeda-Frohman, Kaiser.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Motivations

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Motivations

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Motivations

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Motivations

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Motivations

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Generating maps

Generating maps for some saddle points:

A E E A

These suggest module and comodule structures over Frobeniusalgebras.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Generating maps

Generating maps for some saddle points:

A E E A

These suggest module and comodule structures over Frobeniusalgebras.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Relations

Relations are checked case by case.For example, if a handle is present as depicted, we obtain arelation:

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Frobenius pairs

By checking all conditions in [APS],

Definition

We propose a commutative Frobenius pair (A,E ):

(i) A = (mA,∆A, ιA, εA) is a commutative Frobenius algebraover k .(ii) E is an A-bimodule and A-bicomodule, with the same rightand left actions and coactions.

(4)(1)

a ( b x )

( ab ) x

(2) (3)

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Frobenius pairs (cont.)

These should satisfy variety of conditions:

Roughly, the dotted line does not end at a trivalent vertex,while a solid line can. All relations that satisfy this conditionhold.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Mobius maps

Three maps

are called Mobius maps if they satisfy:

(1) (2) (3)

(5) (6) (7)

(8) (10)

corresponding to non-orientable cobordisms.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Mobius maps

For example, Case (4) of [APS] gives rise to one of theconditions:

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Examples

Example

For [APS]: A = Z[X ]/(X 2), E = 〈Y ,Z 〉Z withcorrespondence: 1↔ −, X ↔ +, Y ↔ −0, Z ↔ +0.Multiplications: XY = XZ = Y 2 = Z 2 = 0 and YZ = X .Comultiplications:

∆(1) = 1⊗ X + X ⊗ 1 + Y ⊗ Z + Z ⊗ Y ,

∆(X ) = X ⊗ X , ∆(Y ) = X ⊗ Y , ∆(Z ) = X ⊗ Z .

Mobius maps: ν(1) = Y + Z , ν(X ) = 0, ν(Y ) = ν(Z ) = X .

Example

For [TT]: A = E = Z2[X , λ±1]/(X 2 − hX ) with h = λ2. Allmutiplications and comultiplications are those of A. AllMobius maps are defined by multiplication by λ.

Note: (µ∆)(1) = h = λ2, .

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Examples

Example

For [APS]: A = Z[X ]/(X 2), E = 〈Y ,Z 〉Z withcorrespondence: 1↔ −, X ↔ +, Y ↔ −0, Z ↔ +0.Multiplications: XY = XZ = Y 2 = Z 2 = 0 and YZ = X .Comultiplications:

∆(1) = 1⊗ X + X ⊗ 1 + Y ⊗ Z + Z ⊗ Y ,

∆(X ) = X ⊗ X , ∆(Y ) = X ⊗ Y , ∆(Z ) = X ⊗ Z .

Mobius maps: ν(1) = Y + Z , ν(X ) = 0, ν(Y ) = ν(Z ) = X .

Example

For [TT]: A = E = Z2[X , λ±1]/(X 2 − hX ) with h = λ2. Allmutiplications and comultiplications are those of A. AllMobius maps are defined by multiplication by λ.

Note: (µ∆)(1) = h = λ2, .

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Virtual circles with poles

Smoothings for the Miyazawa polynomial

Smoothings of a virtual Hopf link

A circle with pairs of opposite poles are considered essential.The two smoothings correspond to a cobordism betweenessential and inessential circles.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Smoothings for the Miyazawa polynomial

Smoothings of a virtual Hopf link

A circle with pairs of opposite poles are considered essential.The two smoothings correspond to a cobordism betweenessential and inessential circles.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

(1) (3)(2)

Relations among poles in the Miyazawa polynomial

(3)(1) (2)

Hemmed cobordisms formed by poles

Cobordisms of virtual circles with poles have a structure ofcommutative Frobenius pairs with Mobius maps.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Constructions

A generalization of [APS]:

Theorem

Let A = Z[X , h, t]/(X 2 − hX − t) and E = 〈Y ,Z 〉. If (A,E ) isa commutative Frobenius pair with Mobius maps such thatµE

E ,E = ∆E ,EE = 0, then A must be A = Z[X , a]/(X − a)2. For

this A, operations are defined by

XY = aY , XZ = aZ , Y 2 = cYY (X − a),

YZ = cYZ (X − a), Z 2 = cZZ (X − a),

∆A,EE (Y ) = (X − a)⊗ Y , ∆A,E

E (Z ) = (X − a)⊗ Z ,

∆E ,EA (1) = dYY Y ⊗ Y + dYZ Y ⊗ Z + dYZ Z ⊗ Y

+dZZ Z ⊗ Z , ∆E ,EA (X ) = a ∆E ,E

A (1),

for some constants that satisfy certain conditions.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Constructions

[APS ] New

A Z[X ]/(X 2) Z[X , a]/(X − a)2

XY 0 aYXZ 0 aZY 2 0 cYY (X − a)Z 2 0 cZZ (X − a)YZ 0 cYZ (X − a)

∆A,EE (Y ) X ⊗ Y (X − a)⊗ Y

∆A,EE (Z ) X ⊗ Z (X − a)⊗ Z

∆E ,EA (1) Y ⊗ Z + Z ⊗ Y dYY Y ⊗ Y + dYZ Y ⊗ Z

+dYZ Z ⊗ Y + dZZ Z ⊗ Z

∆E ,EA (X ) 0 a ∆E ,E

A (1)νEE 0 0νAE (Y ) X eY (X − a)νAE (Z ) X eZ (X − a)νEA (1) Y + Z fY Y + fZ Z

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Constructions

A generalization of [TT]:

Theorem

Let A be a commutative Frobenius algebra with handle elementφ, such that there exists an element ξ ∈ A with ξ2 = φ. Thenthere exists a commutative Frobenius pair with Mobius maps.

Corollary

Let A = E = k[X ]/(X 2 − hX − t), where k = Z[a±1, b±1], andh = −2b−1(a− b−1), t = −b−2(a2 + h). Then (A,E ) givesrise to a commutative Frobenius pair with Mobius maps.

Proof. Let ξ = a + bX , and one computes thatξ2 = φ = 2X − h.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Constructions

A generalization of [TT]:

Theorem

Let A be a commutative Frobenius algebra with handle elementφ, such that there exists an element ξ ∈ A with ξ2 = φ. Thenthere exists a commutative Frobenius pair with Mobius maps.

Corollary

Let A = E = k[X ]/(X 2 − hX − t), where k = Z[a±1, b±1], andh = −2b−1(a− b−1), t = −b−2(a2 + h). Then (A,E ) givesrise to a commutative Frobenius pair with Mobius maps.

Proof. Let ξ = a + bX , and one computes thatξ2 = φ = 2X − h.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Constructions

A generalization of [IT]:

Theorem

Let A be a commutative Frobenius algebra with handle elementφ such that its handle element φ ∈ A is invertible, then thereexists a commutative Frobenius pair (A,E ) with Mobius maps.

Idea of Proof: Take E = A⊗ A and define maps as follows:

2ν0φ

e0e1 e2

φφ φ

There is a solution for constants: e0 = −1, e1 = e2 = −2,ν0 = 1, ν1 = −1 and ν2 = 0.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example

In [Mackaay-Vaz], the sl(3)-invariant was described by foams,with Frobenius algebra structure defined byA = Z[a, b, c][X ]/(X 3 − aX 2 − bX − c)with the multiplication and the unit are defined by those forpolynomials,the Frobenius form (counit) ε is defined byε(1) = ε(X ) = 0, ε(X 2) = −1,the comultiplication is computed as

∆(1) = −(1⊗ X 2 + X ⊗ X + X 2 ⊗ 1)

+a(1⊗ X + X ⊗ 1) + b(1⊗ 1),

∆(X ) = −(X ⊗ X 2 + X 2 ⊗ X ) + a(X ⊗ X )− c(1⊗ 1),

∆(X 2) = −(X 2 ⊗ X 2)− b(X ⊗ X )− c(1⊗ X + X ⊗ 1).

What does the branch line represent?

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example

∆(1) = −(1⊗ X 2 + X ⊗ X + X 2 ⊗ 1)

+a(1⊗ X + X ⊗ 1) + b(1⊗ 1),

∆(X ) = −(X ⊗ X 2 + X 2 ⊗ X ) + a(X ⊗ X )− c(1⊗ 1),

∆(X 2) = −(X 2 ⊗ X 2)− b(X ⊗ X )− c(1⊗ X + X ⊗ 1).

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example

∆(1) = −(1⊗ X 2 + X ⊗ X + X 2 ⊗ 1)

+a(1⊗ X + X ⊗ 1) + b(1⊗ 1),

∆(X ) = −(X ⊗ X 2 + X 2 ⊗ X ) + a(X ⊗ X )− c(1⊗ 1),

∆(X 2) = −(X 2 ⊗ X 2)− b(X ⊗ X )− c(1⊗ X + X ⊗ 1).

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example

∆(1) = −(1⊗ X 2 + X ⊗ X + X 2 ⊗ 1)

+a(1⊗ X + X ⊗ 1) + b(1⊗ 1),

∆(X ) = −(X ⊗ X 2 + X 2 ⊗ X ) + a(X ⊗ X )− c(1⊗ 1),

∆(X 2) = −(X 2 ⊗ X 2)− b(X ⊗ X )− c(1⊗ X + X ⊗ 1).

AlgebraicStructures

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MasahicoSaito

Motivations

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Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example

∆(1) = −(1⊗ X 2 + X ⊗ X + X 2 ⊗ 1)

+a(1⊗ X + X ⊗ 1) + b(1⊗ 1),

∆(X ) = −(X ⊗ X 2 + X 2 ⊗ X ) + a(X ⊗ X )− c(1⊗ 1),

∆(X 2) = −(X 2 ⊗ X 2)− b(X ⊗ X )− c(1⊗ X + X ⊗ 1).

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

Following our approach, evaluate the branch line operation byskein relation. For A = Z[X ]/(X N), for example, it looks like:

XiX N 1 iN

[X j ,X k ] =N−1∑i=0

θ(X i ,X j ,X k)X N−1−i

Theorem

The branch curve operation [ , ] is skew-symmetric andsatisfies the Jacobi identity:

[U, [V ,W ]] + [V , [W ,U]] + [W , [U,V ]].

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

Following our approach, evaluate the branch line operation byskein relation. For A = Z[X ]/(X N), for example, it looks like:

XiX N 1 iN

[X j ,X k ] =N−1∑i=0

θ(X i ,X j ,X k)X N−1−i

Theorem

The branch curve operation [ , ] is skew-symmetric andsatisfies the Jacobi identity:

[U, [V ,W ]] + [V , [W ,U]] + [W , [U,V ]].

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

Using the Frobenius form, define ∆ : A→ A⊗ A by∆(V ) =

∑[V , 1(1)]⊗ 1(2), where ∆(u) =

∑u(1) ⊗ u(2).

Similarly, for ∆, denote ∆(u) =∑

u((1)) ⊗ u((2)).

AlgebraicStructures

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Motivations

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Frobeniuspairs

Constructions

Lie algebras

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Skein modules

Example (cont.)

Theorem

For A = Z[a, b, c]/(X 3− aX 2−bX − c) with θ values as above,the map ∆ : A→ A⊗ A satisfies the following identities:

(m⊗ id)(id⊗∆) = ∆(1)(εµ)− τ,( (m⊗ id)(id⊗∆) )2 = id + ∆(1)(εµ),

m∆ = 2 id.

Same as sl(3) relations!

AlgebraicStructures

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MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

Theorem

For A = Z[a, b, c]/(X 3− aX 2−bX − c) with θ values as above,the map ∆ : A→ A⊗ A satisfies the following identities:

(m⊗ id)(id⊗∆) = ∆(1)(εµ)− τ,( (m⊗ id)(id⊗∆) )2 = id + ∆(1)(εµ),

m∆ = 2 id.

Same as sl(3) relations!

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

Relations to foam skeins:

Lie alg.Rep. of q-gp→ quantum knot inv.

Categorify by foams→ KhoHoTQFT and foams→ Lie alg.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Example (cont.)

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Lie algebras

Are there other TQFTs and theta foam values that produce Liealgebras?

Theorem

For any positive odd integer N > 1, there exists a TQFT thatinduces a Lie algebra structure along branch circles.

Sketch proof. Let A = R[X ]/(X N) for an odd integer N > 1.For N > 3, define

θ(X a,X b,X c) =

1 if a = 0, b + c = N, 1 < b < c ,−1 if a = 0, b + c = N, 1 < c < b,0 otherwise

and the same values for cyclic permutations.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Lie algebras

Theorem

θ(X a,X b,X c) =

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Lie algebras

Theorem

θ(X a,X b,X c) =

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Bialgebras

Let A = R[G ] be the group ringwith a commutative unital ring R.∆(x) = x ⊗ x : bialgebra comultiplication∆(x) =

∑x=yz y ⊗ z : Frobenius algebra comultiplication

In this case, foams admit both structures(saddle ↔ Frobenius, branch lines ↔ bialgebra)only when every element of G is of order 2.

Compatibility ∆(xy) = ∆(x)∆(y)

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Bialgebras

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Bialgebras

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Outline

1 Motivations

3 Frobenius pairs

4 Constructions

5 Foams

6 Lie algebras

7 Bialgebras

8 Skein modules

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Definition

M : a compact 3-manifoldFF(M) : free module on foams in MFS(M) : submodule generated by skein relations in thefigure, where a dot represents x ∈ A = R[x ]/(x2 − 1)F(M) = FF(M)/FS(M) : the surface skein module with A

(Saddle)

!( (1) )

(Closed foams)

An analogue of Bar-Natan skein module by Asaeda-Frohman,Kaiser.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Definition

M : a compact 3-manifoldFF(M) : free module on foams in MFS(M) : submodule generated by skein relations in thefigure, where a dot represents x ∈ A = R[x ]/(x2 − 1)F(M) = FF(M)/FS(M) : the surface skein module with A

(Saddle)

!( (1) )

(Closed foams)

An analogue of Bar-Natan skein module by Asaeda-Frohman,Kaiser.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Conclusion

Conclusion:

For essential cobordisms:

We proposed an algebraic structure, commutative Frobeniuspairs, that describes essential cobordisms in thickened surfacesand cobordisms formed by virtual circles with poles.

Constructions and new examples are provided, that may beuseful in further generalizing Khovanov homology for thickenedsurfaces and virtual knots.

For foams:

We studied Lie algebra and bialgebra structures along branchcircles.

Skein modules and more general foams need more study.

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Conclusion

Conclusion:

For foams:

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Conclusion

Conclusion:

For foams:

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Conclusion

Conclusion:

For foams:

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Conclusion

Conclusion:

For foams:

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Conclusion

Conclusion:

For foams:

AlgebraicStructures

Surfaces andFoams

MasahicoSaito

Motivations

Essentialsurfaces

Frobeniuspairs

Constructions

Lie algebras

Bialgebras

Skein modules

Conclusion

Conclusion:

For foams:

Foams Algebraic Structures Derived from Essential …saito/talks/talk09Waseda.pdfAlgebraic...

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