Algebraic Structures Derived from Foams

Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.


Introduction
Frobenius algebras have been used extensively in the study of categorification of the Jones polynomial [10], via 2-dimensional Topological Quantum Field Theory (2D-TQFT, [11]). For categorifications of other knot invariants, 2-dimensional complexes called foams have been used instead [9,13]. Although 2D-TQFT has been characterized [11] in terms of commutative Frobenius algebras, foams have not been algebraically characterized in terms of TQFT. Relations to Lie algebras, for example, have been suggested [9,13] through their boundaries which are called webs and that are trivalent graphs. Foams have branch curves along which three sheets meet. Similar complexes appear as spines of 3-manifolds, and have been used for quantum invariants [4,5,7,14].
Herein we study the types of algebraic operations that appear along the branch curves of foams in relation to 2D-TQFT. Recall that a 2D-TQFT is a functor from the category of 2-dimensional cobordisms to a category of R-modules (for some suitable ring R) that assigns an R-module to each connected component (circle) on the boundary of a surface, and an R-module homomorphism to a surface. In the case of a foam, we examine the associated algebraic operations that might be associated to branching circles in relation to the Frobenius algebra structure that occurs on the unbranched surfaces. Specifically, we identify and study Lie algebra and bialgebra structures in relation to branch curves, and study their relations to the Frobenius algebra structure.
After reviewing necessary materials in Section 2, a Lie algebra structure for the branch curves is studied in Section 3, and comultiplications of bialgebras are examined in Section 4. The foam skein theory based on the bialgebra case is also defined in Section 4.

Preliminary
Algebraic structures we investigate include Frobenius algebras, Lie algebras and bialgebras. We restrict to the following situations.
A Lie algebra is a module A over a unital commutative ring R with a binary operation [ , ] : A Frobenius algebra is an algebra over R (that comes with associative linear multiplication µ : A ⊗ A → A and unit η : R → A) with a non-degenerate form ǫ : A → R that is associative (ǫ(x ⊗ yz) = ǫ(xy ⊗ z) for x, y, z ∈ A). There is an induced co-associative comultiplication ∆ : A → A ⊗ A. See [3] for diagrams for Frobenius algebras which we will use in this paper. A bialgebra is an algebra A over R with a comultiplication ∆ : A → A ⊗ A that is an algebra homomorphism (∆(xy) = ∆(x)∆(y)) and a counit ǫ : A → R such that (ǫ ⊗ id)∆ = id = (id ⊗ ǫ)∆. The following are typical examples.
Diagrammatically, this is represented by a "neck cutting" relation [2], which we call a ∆(1)-relation to distinguish the specific relation given in [2] for N = 2. See the right of Fig. 4 for a diagrammatic representation of the ∆(1)-relation in this case.
In general, the ∆(1)-relation is also described as follows (see [9,11]). For a commutative Frobenius algebra A over a unital ring R of finite rank and with a non-degenerate Frobenius form ǫ, there is a basis {x i } and a dual basis {y i }, i = 1, . . . , n, such that ǫ(x i y i ) = δ i,j , the Kronecker delta, and x = i y i ǫ(x i x). This situation is depicted in Fig. 1, where the identity map x → x in the LHS corresponds to the annular cobordism in the left of the figure, and the sum involving the Frobenius form ǫ is depicted in the right of the figure.  [13] as follows. The multiplication and the unit are defined by those for polynomials, the Frobenius form (counit) ǫ is defined by ǫ(1) = ǫ(X) = 0, ǫ(X 2 ) = −1. The comultiplication is accordingly computed as Figure 2: Operation on a branch circle We fix a 2D-TQFT such that a connected circle corresponds to A. For TQFTs we refer to [11]. We follow definitions of foams in [9,13], except that facets of foams are decorated by basis elements of A, in a general way as in [6]. A foam without boundary is called closed.
We briefly summarize their definitions. Foam A is the category of formal linear combination over R of cobordisms of compact 2-dimensional complexes in 3-space with the following data.  7) Values θ(α, β, γ) ∈ A of the theta foam, as depicted in Fig. 3 are specified.
In [9,13], it was shown that the values in A of closed foams are well-defined for values of the theta foams, as long as the cyclic symmetry condition θ(α, β, γ) = θ(β, γ, α) = θ(γ, α, β) is satisfied. β α γ For the cyclic order along the oriented branch circle as depicted, make a correspondence between the facet labeled 1, 2, 3, respectively, to the first, second, and the target factor of A ⊗ A → A. Thus the cobordism near a branch circle as depicted in the figure induces a linear map A ⊗ A → A under the chosen TQFT and the values of theta foams. Denote this map by m : A ⊗ A → A. The goal of this paper is to investigate this map.
In terms of maps among tensor products of As, we use planar graphs regularly used in knot theory, as well as Frobenius algebras as in [3]. In particular, the Frobenius form (the counit) is depicted by a maximum, unit by a minimum, (co)multiplications by trivalent vertices. In this convention, diagrams are read from bottom to top, corresponding to the domain and range of maps. The map m corresponding to theta foams has a specified cyclic order, as indicated on the right of Fig. 2. The map m is defined with this specific order, and the map with the opposite order, depicted by a diagram with the opposite arrow, represent the map m • τ , where τ :

Lie algebras
In this section we show that there are infinitely many TQFTs under which Lie algebra structures are induced from the branch circle operation. Since our goal is to exhibit a Lie bracket, in this section we use the notation [ , ] : as well as all cyclic permutations of such (a, b, c). Finally define θ(a, b, c) = 0 for all the other cases. For N = 3, replace the conditions 1 < b < c and 1 < c < b, respectively, by b < c and c < b. We show that these theta foam values induce Lie brackets as desired. The operation [X j , X k ] is evaluated, using the ∆(1)-relation, by This calculation is depicted in Fig. 4 hence it is sufficient to prove that the sum of the right-hand sides is zero. Case 1: j + k + ℓ > N + 1.
In this case, the second factors of the RHS are zero, so that all terms are zero. Case 2: j + k + ℓ ≤ N + 1 and k + ℓ > N .
This case implies that j = 0 and k + ℓ = N + 1. Since N + 1 is even, k and ℓ have the same parity. The first factor θ(N − (k + ℓ), k, ℓ) is 0 since N − (k + ℓ) = −1. (When the arguments of θ are out of range, then θ = 0. ) Suppose k = ℓ = (N + 1)/2. Then the second and the third terms are as desired. Hence assume k < ℓ without loss of generality. For the second and third terms, we have as desired. Case 3: j + k + ℓ ≤ N + 1 and k + ℓ ≤ N .
Since θ vanishes unless one of the entries is 0, the first factors of the RHS are zero if N > k + ℓ and 0 < j < k < ℓ. hence we may assume that k + ℓ = N or j = 0.
Finally, suppose that j = 0, so that k + ℓ = N . The RHS becomes: If 1 < j, then the first argument of all the second factors is negative, so the sum is 0. If j = 1, then each term has a factor that is 0.
Since the original motivation came from the foams in [9,13], we examine the Frobenius algebra in [13] closely. In this case, the multiplication that is induced by branch circles also satisfies the Jacobi identity.

Proposition 3.2 Let
with Frobenius structure defined as in Example 2.2 from [13]. The branch curve operation [ , ] is skew-symmetric and satisfies the Jacobi identity: for any U, V, W ∈ A.
Proof. This is confirmed by calculations. From the axioms of A and the theta foam values that are given in [13]: θ(1, X, X 2 ) = θ(X 2 , 1, X) = θ(X, X 2 , 1) = 1 = −θ(1, X 2 , X) = −θ(X, 1, X 2 ) = −θ(X 2 , X, 1) while θ = 0 for any other arguments, we compute using the ∆(1) relation for Example 2.2: Then one computes as desired. In general, we consider cyclic permutations of X j , X k , and X ℓ in the expression [X j , [X k , X ℓ ]]. Since the bracket is skew-symmetric, then we need only consider the cases in which j, k, and ℓ are distinct. The remaining case follows by skew-symmetry. We define the operation ∆ : A → A ⊗ A that is associated to the left of Fig. 5, a diagram that is up-side down of Fig. 2, in which one circle branches into two from bottom to top. A cyclic order is specified in the figure. If we specify the ordered tensor factors assigned to each sheet by A i , Figure 5: Upside-down operation i = 1, 2, 3, then the operation is defined as ∆ : A planar diagram representing this operation is depicted in the right of the figure. Imitating Sweedler notation ∆(u) = u (1) ⊗ u (2) for comultiplication, we denote ∆(u) = u ((1)) ⊗ u ( (2)) . The next lemma relates this operation to the unit map, and diagrammatic formulations are given in Fig. 6.
The map ∆ is computed as follows.  The following relations hold for maps in Frobenius algebras and maps associated to branch circles. Here we used the notation m instead of [ , ] to formulate in tensor products. The equalities are diagrammatically represented in Fig. 8.
Proof. The first and the third equalities are verified by calculations on basis elements. For all X i and X j , it is computed as [X i , X j ((1)) ] ⊗ X j ((2)) = X j ⊗ X i + ǫ(X i+j )∆(1). The second relation is diagrammatically computed as in Fig. 9. Note that the handle element ǫµ∆(1) is 3.
(εµ∆)(1) 2 Figure 9: Proof of the skein relation Remark 3.6 The skein relations stated in Proposition 3.5, as planar diagrams (instead of surface skein relation), coincide with those described in [9] as a description of Kuperberg's invariant [12], with the choice of q = 1.
Thus, the operation at branch curve of the foam used to categorify the quantum sl(3) invariant satisfies the skein relations at the classical limit of the invariant. Figure 10: A surface skein relation in [9,13] Remark 3.7 The second relation in Proposition 3.5 is related to the local surface skein relation in [9,13] as follows. Their local relation is depicted in Fig. 10. Notice the negative signs, as well as resemblance to our relation. After performing their relations locally, move the holes of each term along the S 1 factor to the other side. Then one obtains a tube connecting two sheets. Then perform the bamboo cutting relation, that is computed by applying ∆(1)-relation three times. In this case, one computes that it is the negative of the original bamboo segment. These negative signs cancel, and we obtain our equation. Thus, our relation follows from theirs, or algebraically as we have shown.
We also point out that the second and the third relation in Proposition 3.5 have interpretations in Foam A . One simply takes the product of these diagrams with S 1 to obtain foams, and the equalities hold in Foam A . The first equality, however, is not realized in Foam A , as the intersection of surfaces are not allowed in Foam A .

Bialgebras
In this section, we investigate functors whose image of branch curves induce bialgebra structure for group algebras. Let G be a group. Let A = R[G] be the group ring with a commutative unital ring R. It is well known that A has a commutative Hopf algebra structure defined as follows (see, for example, [11]). Define ∆ : A → A ⊗ A by linearly extending ∆(x) = x ⊗ x. (This is different from the comultiplication as a Frobenius algebra ∆(x) = x=yz y ⊗ z.) The unit map is defined as the same as the Frobenius unit map η(1) = 1 G , where 1 G is the identity element of G. (The counit map as a Frobenius algebra is defined by ǫ(1 G ) = 1 and ǫ(x) = 0 for x = 1 G .) The following shows that there is a strong requirement for group algebras to give bialgebra structures through branch curves. Proof. The ∆(1)-relation is written as ∆(1) = y∈G y ⊗ y −1 , and the reducing ∆ into the theta foam is depicted in Fig. 12. For ∆(x) = x ⊗ x to hold in the figure, we have y = z = x, and the value of the theta foam being θ(x, y −1 , z −1 ) = 1 for y = z = x and 0 otherwise.
For θ to satisfy the cyclic symmetry, this condition is satisfied if and only if x −1 = x (x having order 2) for any x ∈ G, and in this case, the theta foam values are determined by θ(x, x, x) = 1 for any x ∈ G and 0 otherwise.  The condition of a bialgebra that the comultiplication is an algebra homomorphism (also called a compatibility condition) ∆(ab) = ∆(a)∆(b) for a, b ∈ A, is represented by surfaces in Fig. 13.  , skein modules for 3-manifolds based on embedded surfaces modulo the surface skein relations described in [2] were defined and studied. Surface skein modules were generalized in [6] using general commutative Frobenius algebras. Such notions are directly generalized to foams, with various skein relations at hand. Skein modules for sl(3) foams are analogously defined using the local skein relations given in [9,13], for example.
Here we propose local skein relations based on the foams in Proposition 4.1 with the bialgebra on branch curves for the group ring Z[x]/(x 2 − 1). Considering that the move characteristic to bialgebras is the compatibility condition as depicted in Fig. 13, we take a local change that happens at the saddle point of this move, as depicted in the top of Fig. 14 (labeled as saddle) as a local surface skein relation. Other relations in Fig. 14 are those coming from Frobenius algebra structure and the theta foam values as before.
Thus the skein module F(M ) in this case can be defined to be the isotopy classes of foams in a given 3-manifold M modulo the local surface skein relations in Fig. 14. Although computations of this skein module in general is out of the scope of this paper, it seems interesting to look into relations between foams and the topology of 3-manifolds.