Braided Frobenius Algebras from certain Hopf Algebras
BBRAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS
MASAHICO SAITO AND EMANUELE ZAPPALA
Abstract.
A braided Frobenius algebra is a Frobenius algebra with braiding that commutes withthe operations, that are related to diagrams of compact surfaces with boundary expressed as ribbongraphs. A heap is a ternary operation exemplified by a group with the operation ( x, y, z ) (cid:55)→ xy − z ,that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of theheap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopfalgebras that admit integrals and cointegrals. For these Hopf algebras we show that the heapoperation induces a braiding, by means of a Yang-Baxter operator on the tensor product, whichsatisfies the required compatibility conditions. Diagrammatic methods are employed for provingcommutativity between the braiding and Frobenius operations. Contents
1. Introduction 12. Preliminary 32.1. Heaps 32.2. Hopf algebras 32.3. Frobenius algebras 42.4. The Yang-Baxter operator 53. Ternary self-distributive operations in coalgebras and braidings 54. Braidings and pairings in quantum heaps 75. Construction of braided Frobenius algebras 116. Twists in braided Frobenius algebras 13References 181.
Introduction
Frobenius algebras have been studied in recent decades in relation to 2-dimensional topologicalquantum field theories (TQFTs) [15], and to Khovanov homology [14] in knot theory, that is acategorification of the Jones polynomial [11]. Braid groups have been extensively used in relationto generalizations of the Jones polynomial, and braided monoidal categories have been developed tofurther extend knot invariants to ribbon graphs [20], that consist of disk vertices and ribbon edges.Spatial graphs with a move that corresponds to handle slides have been studied for handlebody-links[9]. Corresponding algebraic structures that have multiplication and braiding at the same time,with compatibility conditions, have also been studied [2, 17]. Compact surfaces with boundary canbe represented by ribbon graphs, and their moves [18] and their invariants [10] have been studied.For algebraic objects having both Frobenius and braiding structures, Frobenius objects in braidedmonoidal categories was proposed in [3], and relations to a certain tangle category were discussed. a r X i v : . [ m a t h . G T ] F e b MASAHICO SAITO AND EMANUELE ZAPPALA
Figure 1.
Axioms of a braided Frobenius algebraMotivated from these developments, in this paper, we present a construction of braided Frobeniusalgebras from certain Hopf algebras. A braided Frobenius algebra is a Frobenius object in thebraided strict monoidal category of modules over unital rings (Definition 5.1). Specifically, a braidedFrobenius algebra is a Frobenius algebra X = ( V, µ, η, ∆ , (cid:15) ) (multiplication, unit, comltiplication,counit) over a unital ring k , which commute with the braiding, as explicitly formulated below.This commutation is represented by diagrams depicted in Figure 1, where the multiplication andbraiding are represented by trivalent vertices and crossings, respectively, and these are part ofmoves for spatial graph diagrams. −1 u −1 x v y u −1 v z u −1 v −1 u −1 x y −1 z vuzy y u −1 z u −1 u −1 xvx y −1 z vvv ( ) x y −1 z v ( )( ) x vuzyx xx u Figure 2.
Heap operation and braid relationThe idea of the construction is based on heaps . A heap is an abstraction of a group endowed withthe ternary operation a × b × c (cid:55)→ T ( a, b, c ) = ab − c . It is computed that this operation on a groupsatisfies the ternary self-distributive law (TSD) T (( x, y, z ) , u, v ) = T ( T ( x, u, v ) , T ( y, u, v ) T ( z, u, v ))for all x, y, z, u, v . Binary self-distributive operations have been studied in relation to the Yang-Baxter operators through tensor categories (e.g., [1]). In [6] a diagrammatic interpretation of TSDwas given in terms of framed links, providing set-theoretic Yang-Baxter operators. The assignmentof heap elements on arcs and the heap operations to crossings are depicted in Figure 2, together withthe TSD property corresponding to a braid relation (the type III Reidemeister move in knot theory).In [5], the constructions of TSD operations from heaps were generalized to monoidal categories.Those in the category of finite dimensional Hopf algebras over a field are called quantum heaps . Weuse quantum heaps X to construct a Frobenius algebra structure on V = X ⊗ X that commute withbraiding induced from the TSD operations. A key method of proofs is extensive use of diagrams.The paper is organized as follows. In Section 2 we review basic definitions and facts regardingheap structures, Hopf algebras, Frobenius algebras and Yang-Baxter operators. In Section 3 wedeal with ternary self-distributive (TSD) structures in coalgebras, and construct a Yang-Baxteroperator associated to a TSD structure arising from quantum heaps in Hopf algebras. In Section 4(co)pairings are constructed that commute with braidings. These (co)pairing are used for (co)units RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 3 for Frobenius structures. In Section 5 we introduce the notion of braided Frobenius algebra andshow that there is a class of these structures arising from quantum heaps where a Frobenius algebrais defined via Hopf algebra (co)integrals. Section 6 discusses relations to compact surfeces withboundary embedded in 3-space, and issues of twists in braided Frobenius algebras.2.
Preliminary
In this section we review materials used in this paper.2.1.
Heaps.
We recall the definition and basic properties of heaps. Given a set X with a ternaryoperation [ − ], the set of equalities[[ x , x , x ] , x , x ] = [ x , [ x , x , x ] , x ] = [ x , x , [ x , x , x ]]is called para-associativity. The equations [ x, x, y ] = y and [ x, y, y ] = x are called the degeneracyconditions. A heap is a non-empty set with a ternary operation satisfying the para-associativityand the degeneracy conditions [5]. A typical example of a heap is a group G where the ternaryoperation is given by [ x, y, z ] = xy − z , which we call a group heap .Let X be a set with a ternary operation ( x, y, z ) (cid:55)→ T ( x, y, z ). The condition T (( x, y, z ) , u, v ) = T ( T ( x, u, v ) , T ( y, u, v ) T ( z, u, v )) for all x, y, z, u, v ∈ X , is called ternary self-distributivity , TSD forshort. It is known and easily checked that the heap operation ( x, y, z ) (cid:55)→ [ x, y, z ] = T ( x, y, z ) isternary self-distributive. We focus on the TSD property of heaps.2.2. Hopf algebras. A Hopf algebra ( X, µ, η, ∆ , (cid:15), S ) (a module over a unital ring k , multiplication,unit, comultiplication, counit, antipode, respectively), is defined as follows. First, a bialgebra X has a multiplication µ : X ⊗ X −→ X with unit η and a comultiplication ∆ : X −→ X ⊗ X withcounit (cid:15) such that the compatibility condition ∆ ◦ µ = ( µ ⊗ µ ) τ ◦ (∆ ⊗ ∆) holds. Then a Hopfalgebra is a bialgebra endowed with a map S : X −→ X , called antipode , satisfying the equations µ ◦ ( ⊗ S ) ◦ ∆ = η ◦ (cid:15) = µ ◦ ( S ⊗ ) ◦ ∆, called the antipode condition .The diagrammatic representation of the algebraic operations appearing in a Hopf algebra is givenin Figure 3. Diagrams are read from top to bottom. For example, the top two arcs of the trivalentvertex for µ (the leftmost diagram) represent X ⊗ X , the vertex represents µ , and the bottomarc represents X . In Figure 4 some of the defining axioms of a Hopf algebra are translated intodiagrammatic equalities. Specifically, diagrams represent (A) associativity of µ , (B) unit condition,(C), compatibility between µ and ∆, (D) the antipode condition. The coassociativity and counitconditions are represented by diagrams that are vertical mirrors of (A) and (B), respectively. ∆ηµ S ε Figure 3.
Operations of Hopf algebrasAny Hopf algebra satisfies the equality
Sµτ = µ ( S ⊗ S ), where τ denotes the transposition τ ( x ⊗ y ) = y ⊗ x for simple tensors. This equality is depicted in Figure 5. A Hopf algebra iscalled involutory if S = , the identity. It is known, [13] Theorem III.3.4, that if a Hopf algebrais commutative or cocommutative it follows that it is also involutory. In what follows, we will not MASAHICO SAITO AND EMANUELE ZAPPALA (D) (A) (C)(B)
Figure 4.
Axioms of Hopf algebrasmention that our Hopf algebras are involutory when they are (co)commutative, and freely applythe fact that S = without further mention.For the comultiplication, we use Sweedler’s notation ∆( x ) = x (1) ⊗ x (2) supressing the summation.Further, we use (∆ ⊗ )∆( x ) = ( x (11) ⊗ x (12) ) ⊗ x (2) and ( ⊗ ∆)∆( x ) = x (1) ⊗ ( x (21) ⊗ x (22) ), bothof which are also written as x (1) ⊗ x (2) ⊗ x (3) from the coassociativity. Figure 5.
Twisting µ with antipodesA left integral of X is an element λ ∈ X such that xλ = (cid:15) ( x ) λ for all x ∈ X . A right integral, a(two-sided) integral, cointegrals are defined similarly. Diagrams for integral conditions are depictedin Figure 6. The diagram (A) represents an integral, (B) represents the defining equation of a leftintegral, and similar for cointegrals in (C) and (D). The existence of integrals is a fundamental toolto endow a Hopf algebra with a Frobenius structure (defined below). It is known that the set ofintegrals of a free finite dimensional Hopf algebra over a PID admits a one dimensional space ofintegrals, see [16]. More generally, a finitely generated projective Hopf algebra over a ring admitsa left integral space of rank one [19]. Observe that when a Hopf algebra is (co)commutative, itfollows that a left (co)integral is also a right (co)integral. (A) (C) (D)(B) Figure 6.
Left (co)integral of Hopf algebras2.3.
Frobenius algebras.
We use the following definition: A
Frobenius algebra ( V, µ, η, ∆ , (cid:15) ) isan associative and coassociative coalgebra over a unital ring k with multiplication µ and co-multiplication ∆, respectively, with unit η : k → V and counit (cid:15) : V → k with the sameconditions as Hopf algebras, such that µ and ∆ satisfy the Frobenius compatibility condition:( µ ⊗ )( ⊗ ∆) = ∆ µ = ( ⊗ µ )(∆ ⊗ ). Thid condition is depicted in Figure 7. RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 5
Figure 7.
Frobenius compatibility condition2.4.
The Yang-Baxter operator.
Let X be a module over a ring and let R : X ⊗ X −→ X ⊗ X bean operator (i.e. a linear map). The Yang-Baxter equation, YBE for short, for R is the functionalequation ( R ⊗ ) ◦ ( ⊗ R ) ◦ ( R ⊗ ) = ( ⊗ R ) ◦ ( R ⊗ ) ◦ ( ⊗ R )where LHS and RHS are both endomorpshism of X ⊗ X ⊗ X . The YBE is well known to berepresented by the type III Reidemeister move in knot theory, and has been widely studied inlow-dimensional topology because it produces invariants of knots. If the operator R satisfies theYBE, then it is said to be a pre Yang-Baxter operator . If, in addition, R is invertible then we saythat R is a Yang-Baxter operator , YB operator for short.3.
Ternary self-distributive operations in coalgebras and braidings
In this section we provide a method of producing braidings from ternary self-distributive (TSD)operations.
Definition 3.1. [6] A morphism T : V ⊗ → V for a coalgebra V over a unital ring k is called ternary self-distributive (TSD for short) if it satisfies, when expressed in simple tensors, T ( T ( x ⊗ y ⊗ z ) ⊗ u ⊗ v )= T ( T ( x ⊗ u (11) ⊗ v (11) ) ⊗ T ( x ⊗ u (12) ⊗ v (12) ) ⊗ T ( x ⊗ u (2) ⊗ v (2) )) . Lemma 3.2. [6]
Let ( X, µ, ∆ , ι, (cid:15), S ) be an involutory Hopf algebra. Let T ( x ⊗ y ⊗ z ) = xS ( y ) z = µ ( µ ( x ⊗ S ( y )) ⊗ z ) expressed in simple tensors, where the concatenation denotes the multiplication.Then T is TSD. This construction is represented by the diagrams in Figure 8. µ( µ( ) ) yx z yx zT(x y z ) S( )yx z
Figure 8.
Quantum heap operation as a TSD
Definition 3.3.
A TSD morphism T : V ⊗ → V for a module V over a unital ring k is called invertible if it satisfies T ( T ( x ⊗ y (2) ⊗ z (2) ) ⊗ z (1) ⊗ y (1) ) = (cid:15) ( y ) (cid:15) ( z ) · x, for all x, y, z ∈ V . MASAHICO SAITO AND EMANUELE ZAPPALA
Lemma 3.4.
Let ( X, µ, ∆ , ι, (cid:15), S ) be an involutory Hopf algebra, and let T ( x ⊗ y ⊗ z ) = xS ( y ) z beas defined in Lemma 3.2. If ( X, ∆) is cocommutative, then T is invertible.Proof. One computes T ( T ( x ⊗ y (2) ⊗ z (2) ) ⊗ z (1) ⊗ y (1) )= xS ( y (2) ) z (2) S ( z (1) ) y (1) = xS ( y (2) ) S ( z (1) ) z (2) y (1) = (cid:15) ( z ) · xS ( y (2) ) y (1) = (cid:15) ( y ) (cid:15) ( z ) · x as desired. (cid:3) xzz x T(x )xyz (1) y (1) y (2) z (2) T(x ) y z z (1) y (1) z (2) y (2) y zx y Figure 9.
Hopf algebra maps corresponding to crossings
Lemma 3.5.
Let ( X, ∆) be a cocommutative coalgebra over a unital ring k . Let T : X ⊗ → X bean invertible TSD coalgebra morphism. Then the map β : X ⊗ → X ⊗ defined for simple tensorsby β ( x ⊗ y ⊗ z ) = y (1) ⊗ z (1) ⊗ T ( x ⊗ y (2) ⊗ z (2) ) is invertible with the inverse β − ( y ⊗ z ⊗ x ) = T ( x ⊗ z (2) ⊗ y (2) ) ⊗ y (1) ⊗ z (1) , so that β β − = and β − β = .Proof. The proof is an application of the invertibility condition of T . On simple tensors we have β − β ( x ⊗ y ⊗ z ) = β − ( y (1) ⊗ z (1) ⊗ T ( x ⊗ y (2) ⊗ z (2) ))= T ( T ( x ⊗ y (2) ⊗ z (2) ) ⊗ z (12) ⊗ y (12) ) ⊗ y (11) ⊗ z (11) = T ( T ( x ⊗ y (22) ⊗ z (22) ) ⊗ z (21) ⊗ y (21) ) ⊗ y (1) ⊗ z (1) = (cid:15) ( y (2) ) (cid:15) ( z (2) ) x ⊗ y (1) ⊗ z (1) = x ⊗ y ⊗ z, which shows that β − β = . Similar considerations imply that β β − = as well. (cid:3) Diagrammatic representations of morphisms β and β − in Lemma 3.5 are depicted in the leftand right of Figure 9, respectively. The first equality β − β = in the lemma is represented byFigure 10. Figure 10.
The type II Reidemeister move with a single under-arc and double over-arcs.
RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 7
Lemma 3.6.
Let ( X, ∆) be a cocommutative coalgebra over a unital ring k . Let T : X ⊗ → X be an invertible TSD coalgebra morphism. Let V = X ⊗ X be endowed with the tensor coalgebrastructure induced by ( X, ∆) . Then the map β : V ⊗ → V ⊗ defined for simple tensors by β (( x ⊗ x (cid:48) ) ⊗ ( y ⊗ z )) := ( y (1) ⊗ z (1) ) ⊗ T ( x ⊗ y (21) ⊗ z (21) ) ⊗ T ( x (cid:48) ⊗ y (22) ⊗ z (22) ) satisfies the YBE. Furthermore, there is an inverse β − (( y ⊗ z ) ⊗ ( x ⊗ x (cid:48) )) = ( T ( x ⊗ z (21) ⊗ y (21) ) ⊗ T ( x (cid:48) ⊗ z (22) ⊗ y (22) )) ⊗ y (1) ⊗ z (1) . Proof.
We show that the YBE holds on simple tensors. Let x, y, z, w, u, v ∈ X , then the LHS ofthe YBE is computed as( β ⊗ )( ⊗ β )( β ⊗ )( x ⊗ y ⊗ z ⊗ w ⊗ u ⊗ v )= u (1) ⊗ v (1) ⊗ T ( z (1) ⊗ u (2) ⊗ v (2) ) ⊗ T ( w (1) ⊗ u (3) ⊗ v (3) ) ⊗ T ( T ( x ⊗ z (2) ⊗ w (2) ) ⊗ u (4) ⊗ v (4) ) ⊗ T ( T ( y ⊗ z (3) ⊗ w (3) ) ⊗ u (5) ⊗ v (5) ) . The RHS computed on x ⊗ y ⊗ z ⊗ w ⊗ u ⊗ v gives( ⊗ β ) ◦ ( β ⊗ ) ◦ ( ⊗ β )( x ⊗ y ⊗ z ⊗ w ⊗ u ⊗ v )= u (1) ⊗ v (1) ⊗ T ( z (1) ⊗ u (4) ⊗ v (4) ) ⊗ T ( w (1) ⊗ u (7) ⊗ v (7) ) ⊗ T ( T ( x ⊗ u (2) ⊗ v (2) ) ⊗ T ( z (2) ⊗ u (5) ⊗ v (5) ) ⊗ T ( w (2) ⊗ u (8) ⊗ v (8) )) ⊗ T ( T ( y ⊗ u (3) ⊗ v (3) ) ⊗ T ( z (3) ⊗ u (6) ⊗ v (6) ) ⊗ T ( w (3) ⊗ u (9) ⊗ v (9) )) . Rearranging terms by means of the cocommutativity of ∆, using the fact that T is a coalgebramorphism and applying the TSD property of T , we see that the two terms coincide, showing that β satisfies the YBE.To show that β is invertible observe that, since ∆ is cocommutative, one has β = ( β ⊗ ) ◦ ( ⊗ β ).Similar considerations allow us to write β − as composition of terms where β − appears. Aniteration of Lemma 3.5 then shows that β − is the inverse of β . (cid:3) Figure 9 shows the diagrammatic interpretation of the braiding and its inverse in Lemma 3.6on a single edge of a ribbon. The full braiding, as well as its inverse, is obtained by repeating theprocedure on both edges that delimit a ribbon.
Lemma 3.7.
Let ( X, µ, η, ∆ , (cid:15), S ) be an involutory Hopf algebra. Then the map β : X ⊗ → X ⊗ defined on simple tensors as x ⊗ y ⊗ z ⊗ w (cid:55)→ z (1) ⊗ w (1) ⊗ xS ( z (2) ) w (2) ⊗ yS ( z (3) ) w (3) is a Yang-Baxter operator.Proof. The statement follows directly by applying Lemma 3.6 to the quantum heap constructionof Lemma 3.2. The invertibility follows from Lemma 3.4. (cid:3) Braidings and pairings in quantum heaps
In this section we introduce pairings and copairings that commute with braiding constructed inthe preceding section. We construct such (co)pairing using integrals of Hopf algebras.
MASAHICO SAITO AND EMANUELE ZAPPALA
Figure 11.
The switchback property
Definition 4.1.
A pairing ∪ : V ⊗ V → k and a copairing ∩ : k → V ⊗ V in a module V over aunital ring k are said to have (or satisfy) the switchback property if they satisfy the equalities( ∪ ⊗ )( ⊗ ∩ ) = = ( ⊗ ∪ )( ∩ ⊗ ) . The conditions imply that ∪ is non-singular. :=:= Figure 12.
Defining cup and cap by left (co)integrals
Definition 4.2.
Let (
X, µ, η, ∆ , (cid:15), S ) be a finitely generated projective Hopf algebra over a (unital)ring k . Then X has an integral and a cointegral [19]. Let us indicate them by λ and γ , respectively.We define a cup on X by ∪ := λµ ( ⊗ S ) and ∩ := ∆ γ , as depicted in Figure 12.For a Hopf algebra ( X, µ, η, ∆ , (cid:15), S ), the following module P ( H ∗ ) was considered in [19]. Let χ : X ∗ → X ∗ ⊗ X be a right X -comodule structure on X ∗ defined by the left H ∗ -module structure.Then P ( H ∗ ) was defined by P ( H ∗ ) = { x ∗ ∈ X ∗ | χ ( x ∗ ) = x ∗ ⊗ } . Lemma 4.3.
Let ( X, µ, η, ∆ , (cid:15), S ) be a finitely generated projective Hopf algebra over a ring k , suchthat P ( X ∗ ) ∼ = k , and ∪ , ∩ be as in Definition 4.2. Then ∪ and ∩ satisfy the switchback property.Proof. In [19], it is proved, under the assumptions, that there exists an integral λ and cointegral γ such that ∪ (cid:48) = λµ and ∩ (cid:48) = ( S ⊗ )∆ γ satisfy the switchback property. It then follows that so do ∪ and ∩ in Definition 4.2 as well. (cid:3) Since we use this lemma extensively from here forward, we will assume that every Hopf algebrasatisfies the assumption of this lemma. As pointed out in [19], the condition that P ( H ∗ ) ∼ = k isautomatically satisfied when pic( k ) = 0. This is the case for instance when k is a PID or a localring. In particular, one obtains the result of Larson and Sweedler in [16], where the ground ring istaken to be a PID. yvux yvux= Figure 13.
The passcup property
RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 9
Definition 4.4.
Let V be a coalgebra over a unital ring k , with ternary morphism (of coalgebras) T : V ⊗ → V . A pairing ∪ : V ⊗ V → k is said to have (or satisfy) the passcup property with respectto T if it satisfies( ⊗ ⊗ ∪ )( u (1) ⊗ v (1) ⊗ T ( x ⊗ u (2) ⊗ v (2) ) ⊗ y ) = ( ∪ ⊗ ⊗ )( x ⊗ T ( y ⊗ v (2) ⊗ u (2) ) ⊗ u (1) ⊗ v (1) )for all x, y, u, v ∈ V .The passcup property is depicted in Figure 13. Lemma 4.5.
Let ( X, µ, η, ∆ , (cid:15), S ) be a cocommutative Hopf algebra. Then the pairing ∪ defined inDefinition 4.2 satisfies the passcup property with respect to the TSD defined in Lemma 3.2.Proof. In order to prove the passcup property, we proceed as in Figure 14. The first equalitycorresponds to rewriting one negative crossing using the definition of inverse of quantum heapoperation T , the first arrow utilizes naturality of the switching map X ⊗ X → X ⊗ X , the secondarrow corresponds to the compatibility relation between the antipode S and the comultiplication∆ of X , the third arrow is given by redrawing the diagram using naturality of switching map, thefourth arrow corresponds to the fact that λ is both a right and left integral. Involutority is used atStep (3). This completes the proof of the passcup property. (cid:3) (4)(1) (2) (3) Figure 14.
Proof of the passcup property
Lemma 4.6.
Let ( X, µ, η, ∆ , (cid:15), S ) be a commutative and cocommutative Hopf algebra and set V = X ⊗ X . Then the cup and cap defined in Definition 12 commute with the braiding defined inLemma 3.7. Specifically, it holds that ( ⊗ ∪ ) β = ∪ ⊗ and ( ∪ ⊗ ) β = ⊗ ∪ as morphisms V ⊗ V → V , ( ⊗ ∩ ) β = ∩ ⊗ and β ( ∩ ⊗ ) = ⊗ ∩ as morphisms V → V ⊗ V .Proof. Diagrammatic sketch proofs are found in Figures 15 through 18. In Figure 15, ( ⊗ ∪ ) β = ∪ ⊗ is proved by Lemmas 4.5 and 3.5 successively. Other equalities are proved as depicted, usingHopf algebra axioms and the definition of integrals. In Figure 16, other than axioms, commutativityis used in the 4th equality, and cocommutativity is used in the 5th equality. While the diagrams inFigures 16 and 17 treat the single-stranded case, an iteration of the diagrammatic proof implies thecase with two edges sliding. Cocommutativity is used in the 3rd and 5th equalities in Figure 17, and3rd equality in Figure 18. Observe that the equalities Sη = η and (cid:15)S = (cid:15) , which are consequencesof S being an anti-homomorphism, have been used. (cid:3) Figure 15.
Proof of ( ⊗ ∪ ) β = ∪ ⊗ Figure 16.
Proof of ( ∪ ⊗ ) β = ⊗ ∪ Figure 17.
Proof of ( ⊗ ∩ ) β = ∩ ⊗ Figure 18.
Proof of β ( ∩ ⊗ ) = ⊗ ∩ RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 11 Construction of braided Frobenius algebras
In [3], a braided Frobenius object is defined to be a Frobenius object in a braided monoidalcategory. A monoidal category has (tensor) products among objects, with some other data andconditions, such as unitors and associators, corresponding to units and associativity of algebras.A monoidal category is strict if its associators and left/right unitors are identity natural transfor-mations. A braided monoidal category has a braiding between (tensor) products of two objects,that are functorial. Hence it is natural to define a braided Frobenius algebra to be a Frobenuisobject in the braided strict monoidal category of finitely generated modules over a unital ring. Thisdefinition is equivalent to having a braiding β that commutes with all defining data of a Frobeniusalgebra, that corresponds to functoriality of braiding. Definition 5.1. (cf. [3]) A braided Frobenius algebra is a Frobenuis object in the braided strictmonoidal category of finitely generated projective modules over a unital ring. Specifically, a braidedFrobenius algebra is a Frobenius algebra X = ( V, µ, η, ∆ , (cid:15) ) (multiplication, unit, comltiplication,counit) over unital ring k , which commute with the braiding, as follows:( µ ⊗ )( ⊗ β )( β ⊗ ) = β ⊗ ( ⊗ µ ) , ( ⊗ µ )( β ⊗ )( ⊗ β ) = β ⊗ ( µ ⊗ ) , (∆ ⊗ ) β = ( β ⊗ )( β ⊗ )( ⊗ ∆) , ( ⊗ ∆) β = ( β ⊗ )( β ⊗ )(∆ ⊗ ) , ( ⊗ η ) β = η ⊗ , β ( η ⊗ ) = ⊗ η, ( ⊗ (cid:15) ) β = (cid:15) ⊗ , ( (cid:15) ⊗ ) β = ⊗ (cid:15). Figure 19.
Commutation of (co)unit and braidingThe commuting conditions for a braided Frobenius algebra for multiplication are depicted inFigure 1. Those for comultiplication are represented by the upside down diagrams. The commutingconditions for the (co)unit are depicted in Figure 19.We now proceed to construct a family of braided Frobenius algebras from a class of Hopf algebras.We mention that the monoid structure in the next theorem also appears in Sections 4 and 5 of [8],under the name of pair of pants monoid , for dagger pivotal categories. In our construction, the factthat Frobenius monoids (e.g. algebras) are self-dual allows us to discard the duality in X ∗ ⊗ X . Theorem 5.2.
Let ( X, µ, η, ∆ , (cid:15), S ) be a commutative and cocommutative Hopf algebra. Then V = X ⊗ X has a braided Frobenius algebra structure.Proof. Since X is an involutory Hopf algebra, applying Lemma 3.7 it follows that X ⊗ X hasa braiding β that is induced by the quantum heap structure of X . We define a product µ ⊗ : X ⊗ ⊗ X ⊗ → X ⊗ by means of ∪ as µ ⊗ := ⊗ ∪ ⊗ . The coproduct ∆ ⊗ : X ⊗ → X ⊗ ⊗ X ⊗ is obtained from ∩ by the definition∆ ⊗ := ⊗ ∩ ⊗ . Product is associative since µ ⊗ ◦ ( µ ⊗ ⊗ ⊗ ) = ( ⊗ ∪ ⊗ ) ◦ ( ⊗ ∪ ⊗ ⊗ ⊗ )= ⊗ ∪ ⊗ ∪ ⊗ = ( ⊗ ∪ ⊗ ) ◦ ( ⊗ ⊗ ⊗ ∪ ⊗ )= µ ⊗ ◦ ( ⊗ ⊗ µ ⊗ ) . Similarly, we see that ∆ ⊗ is coassociative. The fact that µ ⊗ and ∆ ⊗ satisfy the Frobenius lawsis seen directly, as we have that ∆ ⊗ ◦ µ ⊗ , ( ⊗ ⊗ µ ⊗ ) ◦ (∆ ⊗ ⊗ ⊗ ) and ( µ ⊗ ⊗ ⊗ ) ◦ ( ⊗ ∆ ⊗ )evaluated on simple tensors x ⊗ y ⊗ z ⊗ w all equal ∪ ( y ⊗ z ) · x ⊗ ∩ (1) ⊗ w, where we have used · to separate an element of the ground ring k from elements of X ⊗ to avoidconfusion. = Figure 20.
The unit axiomThe unit η ⊗ : k → X ⊗ ⊗ X ⊗ is defined by ∩ . The unit condition follows from the switchbackcondition, as depicted in Figure 20. The counit (cid:15) ⊗ is defined by (cid:15) ⊗ = ∪ and the counit conditionfollows similarly. Hence X ⊗ X is endowed with a Frobenius structure and a braiding induced fromthe quantum heap operation. Figure 21.
Braided Frobenius conditions for a doubled Hopf algebraTo complete the proof we need to show that braiding and Frobenius morphisms commute in thesense of Definition 5.1. The commutations between (co)units and braiding follow from Lemma 4.6(see Figures 15 through 18). For doubled strands, the commutations between multiplication andbraiding are depicted in Figure 21. These follow from commutations between counits and braiding.The commutations between comultiplication and braiding are represented by the upside down (thevertical mirror) figures of Figure 21, and follow from commutations between units and braiding. (cid:3)
Example 5.3.
Let X = k [ G ] be a group ring of a group heap G with the TSD operation definedby linearlization of the group heap operation T ( x ⊗ y ⊗ z ) := xy − z for x, y, z ∈ G . Endow X with the Hopf algebra structure, where µ is defined by the linearlized group multiplication, group∆( x ) = x ⊗ x for x ∈ G , unit defined by η (1) = e ∈ G (the identity element), and counit definedby (cid:15) ( x ) = 1 for x ∈ G . The integral is defined by (cid:80) x ∈ G x and cointegral by e (cid:55)→ e (cid:54) = g (cid:55)→ T , and for group elements RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 13 β (( x ⊗ y ) ⊗ ( u ⊗ v )) = ( u ⊗ v ) ⊗ ( xu − v ⊗ yu − v ). Thus the braiding is the linearlization of groupheap braiding as depicted in Figure 2. If the group G is abelian, X satisfies the assumption ofTheorem 5.2. Moreover, so does the dual Hopf algebra k [ G ] ∗ . Example 5.4.
Let k be a PID or a local ring of characteristic p . Then the truncated polynomialalgebra H = k [ X ] / ( X p k ) is a finitely genereated free (hence projective) Hopf algebra for any k ≥ H satisfies P ( X ∗ ) ∼ = k since k is either a PID or a local ring. We cantherefore apply Lemma 4.3 and Theorem 5.2, since H is commutative and cocommutative. Explic-itly, the algebra structure of H is determined by multiplication of polynomials, the comultiplicationis obtained extending ∆( X ) = 1 ⊗ X + X ⊗ H is truncated at a power of the characteristic of the ground ring), the counit isdefined by (cid:15) (1) = 1, (cid:15) ( X ) = 0 and the antipode is given by S ( X ) = − X . This construction can begeneralized to truncated polynomial algebras with more than one indeterminate.We note that considering local rings gives a wider class of objects with respect to that of PID’sin [16]. For instance, the ring Z p [ Y , Y ] / ( Y , Y ) is a local ring that is not a PID to which theprevious construction can be applied.6. Twists in braided Frobenius algebras
In this section we introduce twists in braided Frobenius algebras, and discuss relations to tortilecategory structure and surfaces with boundary embedded in 3-space.
Definition 6.1.
Let ( V, ∆ , (cid:15) ) be a finite dimensional coalgebra over a field k with a TSD operation T : V ⊗ → V (Definition 3.1). Then the operation θ : V ⊗ V → defined by θ ( x ⊗ y ) = T ( x (1) ⊗ x (2) ⊗ y (2) ) ⊗ T ( y (1) ⊗ x (3) ⊗ y (3) )is called a twist by T . Remark 6.2.
The twisting introduced in Definition 6.1 is motivated from a “quantum” version ofthe core quandle [7] operation ( x, y ) (cid:55)→ yx − y defined on groups. In fact we have x ⊗ y (cid:55)→ x (1) S ( x (2) ) y (2) ⊗ y (1) S ( x (3) ) y (3) = (cid:15) ( x (1) ) y (2) ⊗ y (1) S ( x (2) ) y (3) = y (2) ⊗ y (1) S ( x ) y (3) , where the second term in the tensor product can be identified with the core quandle operationbetween y (1) and x . Figure 22.
Twisting a ribbonThe operation θ is written by maps as follows. Fix a basis { e i : i = 1 , . . . , n } for V , and definethe pairing ∨ : V ⊗ V ∗ → k for the dual space V ∗ by ∨ ( x i ⊗ x ∗ j ) = δ i,j with the Kronecker’s delta,and copairing ∧ : k → V ⊗ V ∗ by ∧ (1) = (cid:80) ni =1 x i ⊗ x ∗ i . Then θ is written as θ = ( ⊗ ⊗ ∨ )( ⊗ ⊗ ∨ ⊗ )( β ⊗ ⊗ )( ⊗ ⊗ ∧ ⊗ )( ⊗ ⊗ ∧ )with the braiding β induced from T (Lemma 3.6). Diagrammatically, θ is represented by Figure 22,and corresponds to a full twist as in the right of the figure. In the figure, the maxima and minimacorresponds to ∧ and ∨ , respectively, and indicated by such notations, to distinguish them from ∩ and ∪ . Proposition 6.3.
Let ( V, ∆ , (cid:15) ) be a cocommutative coalgebra over a unital ring k with a TSDoperation T : V ⊗ → V (Definition 3.1). Let θ be the twist in Definition 6.1. Then θ commuteswith the braiding β induced from T . Specifically, we have β ( θ ⊗ ) = ( ⊗ θ ) β and β ( ⊗ θ ) = ( θ ⊗ ) β .Proof. On simple tensors we have β ( θ ⊗ )( x ⊗ y ⊗ z ⊗ w ) = β ( T ( x (1) ⊗ x (2) ⊗ y (2) ) ⊗ T ( y (1) ⊗ x (3) ⊗ y (3) ) ⊗ z ⊗ w )= z (1) ⊗ w (1) ⊗ T ( T ( x (1) ⊗ x (2) ⊗ y (2) ) ⊗ z (2) ⊗ w (2) ) ⊗ T ( T ( y (1) ⊗ x (3) ⊗ y (3) ) ⊗ z (3) ⊗ w (3) ) , and also( ⊗ θ ) β ( x ⊗ y ⊗ z ⊗ w ) = ( ⊗ θ )( z (1) ⊗ w (1) ⊗ T ( x ⊗ z (2) ⊗ w (2) ) ⊗ T ( y ⊗ z (3) ⊗ w (3) ))= z (1) ⊗ w (1) ⊗ T ( T ( x (1) ⊗ z (21) ⊗ w (21) ) ⊗ T ( x (2) ⊗ z (22) ⊗ w (22) ) ⊗ T ( y (2) ⊗ z (32) ⊗ w (32) )) ⊗ T ( T ( y (1) ⊗ z (31) ⊗ w (31) ) ⊗ T ( x (3) ⊗ z (23) ⊗ w (23) ) ⊗ T ( y (3) ⊗ z (33) ⊗ w (33) )) , where the fact that, by definition, T is a coalgebra morphism has been applied. Applying cocom-mutativity of ∆ we can rearrange the z and w terms in such a way that β ( θ ⊗ )( x ⊗ y ⊗ z ⊗ w )and ( ⊗ θ ) β ( x ⊗ y ⊗ z ⊗ w ) differ by an application of the TSD condition of T utilized twice. Thisshows the equality β ( θ ⊗ ) = ( ⊗ θ ) β .Let us now consider the equation β ( ⊗ θ ) = ( θ ⊗ ) β . For the LHS we have β ( ⊗ θ )( x ⊗ y ⊗ z ⊗ w ) = T ( z (11) ⊗ z (21) ⊗ w (21) ) ⊗ T ( w (11) ⊗ z (31) ⊗ w (31) ) ⊗ T ( x ⊗ T ( z (12) ⊗ z (22) ⊗ w (22) ) ⊗ T ( w (12) ⊗ z (32) ⊗ w (32) ) ⊗ T ( y ⊗ T ( z (13) ⊗ z (23) ⊗ w (23) ) ⊗ T ( w (13) ⊗ z (33) ⊗ w (33) ) , while for the RHS we have( θ ⊗ ) β ( x ⊗ y ⊗ z ⊗ w ) = T ( z (11) ⊗ z (12) ⊗ w (12) ) ⊗ T ( w (11) ⊗ z (13) ⊗ w (13) ) ⊗ T ( x ⊗ z (2) ⊗ w (2) ) ⊗ T ( y ⊗ z (3) ⊗ w (3) ) . To complete the proof we see that it is enough to show the equality T ( x ⊗ T ( z (1) ⊗ z (2) ⊗ w (2) ) ⊗ T ( w (1) ⊗ z (3) ⊗ w (3) )) = T ( x ⊗ z ⊗ w ) . (1)We have T ( x ⊗ T ( z (1) ⊗ z (2) ⊗ w (2) ) ⊗ T ( w (1) ⊗ z (3) ⊗ w (3) ))= (cid:15) ( z (2) ) (cid:15) ( w (2) ) · T ( x ⊗ T ( z (1) ⊗ z (3) ⊗ w (3) ) ⊗ T ( w (1) ⊗ z (4) ⊗ w (4) ))= T ( T ( T ( x ⊗ w (3) ⊗ z (3) ) ⊗ z (2) ⊗ w (2) ) ⊗ T ( z (1) ⊗ z (4) ⊗ w (4) ) ⊗ T ( w (1) ⊗ z (5) ⊗ w (5) )) , RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 15 where the first equality uses the definition of counit (cid:15) , and the second equality makes use of theinvertibility condition of T . Let us now apply the TSD property of T to the terms T ( x ⊗ w (2) ⊗ z (2) ), z (1) , w (1) , z (3) and w (3) , where we set T ( x ⊗ w (3) ⊗ z (3) ) = q for convenience. We get T ( T ( q ⊗ z (2) ⊗ w (2) ) ⊗ T ( z (1) ⊗ z (4) ⊗ w (4) ) ⊗ T ( w (1) ⊗ z (5) ⊗ w (5) ))= T ( T ( q ⊗ z (2) ⊗ w (2) ) ⊗ z (1) ⊗ w (1) ) = (cid:15) ( z (1) ) (cid:15) ( w (1) ) · T ( x ⊗ z (2) ⊗ w (2) ) = T ( x ⊗ z ⊗ w ) , where invertibility of T , as well as cocommutativity of ∆, has been used in the second equality.This shows that Equation (1) holds. (cid:3) Figure 23.
Twisting a doubled ribbon
Remark 6.4.
Here we discuss relations to the tortile category . A braided monoidal category iscalled tortile [12] (or ribbon [8]) if there is a morphism θ X called a twist for every object X suchthat θ X,Y = β Y,X β X,Y ( θ X ⊗ θ Y ) for all objects X, Y , where β denotes the braiding.Let ( V, ∆ , (cid:15) ) be a finite dimensional coalgebra over a field k with a TSD operation T : V ⊗ → V (Definition 3.1). Then Proposition 6.3 implies that the subcategory generated by V in the categoryof braided monoidal category of finite dimensional coalgebras with TSDs forms a tortile category.The twist θ ⊗ k on V ⊗ k is defined by parallel loops, that are defined by taking k -fold parallel ribbons.The case k = 2 is depicted in Figure 23 left. The equality θ V,V = β V,V β V,V ( θ V ⊗ θ V ) is indicatedin the figure. The fact that full twist of parallel strings form a tortile category is pointed out in[12]. In [12] the twists are defined by parallel loops, that topologically correspond to full twists ofparallel strings, using dual spaces. Thus the construction of this twists are obtained by applyingthe twists in [12] to braiding defined by TSD operations on coalgebras. (A) (C)(B) Figure 24.
Commutation between a twist and multiplication
Proposition 6.5.
Let X be as in Theorem 5.2, and let V = X ⊗ X denote the associated braidedFrobenius structure on the doubled vector space. Let θ be the twist in Definition 6.1. Then thetwist θ commutes with the multiplication and comultiplication. This means, with notations as inRemark 6.4, that θ V µ ⊗ = µ ⊗ θ V,V and ∆ ⊗ θ V = θ V,V ∆ ⊗ hold. The commutation between the twist and multiplication is depicted in the left equality ( A ) = ( B )of Figure 24. The right equality ( B ) = ( C ) is a consequence of Figure 23. We note that theresulting equality ( A ) = ( C ) corresponds diagrammatically to twisting the trivalent vertex by onefull twist. Proof.
We verify equality θ V µ ⊗ = µ ⊗ θ V,V on simple tensors x ⊗ y ⊗ z ⊗ w . For the LHS we have θ V µ ⊗ ( x ⊗ y ⊗ z ⊗ w ) = γ ( yS ( z )) · w (2) ⊗ w (1) S ( x ) w (3) . The RHS is given as µ ⊗ θ V,V ( x ⊗ y ⊗ z ⊗ w ) = γ ( y (1) S ( x (3) ) y (3) S ( z (3) ) w (3) S ( w (4) ) z (4) S ( y (4) ) x (4) S ( z (1) )) · x (1) S ( x (2) ) y (2) S ( z (2) ) w (2) ⊗ w (1) S ( x (5) ) y (5) S ( z (5) ) w (5) = γ ( y (1) S ( z (1) )) · y (2) S ( z (2) ) w (2) ⊗ w (1) S ( x ) y (3) S ( z (3) ) w (3) = γ ( yS ( z )) · w (2) ⊗ w (1) S ( x ) w (3) , where the first equality is obtained by unraveling the definitions, the second equality is a multi-ple application of the counit axiom, and the third equality follows by applying the definition ofcointegral γ twice. Equality ∆ ⊗ θ V = θ V,V ∆ ⊗ is proven on simple tensors in a similar fashion. (cid:3) Figure 25.
Twisting a ribbon by a loop
Remark 6.6.
For the braided Frobenius algebra V constructed in Theorem 5.2, a twist Θ can bedefined using ∩ and ∪ instead of ∧ and ∨ as depicted in Figure 25 left. Specifically,Θ = ( ⊗ ⊗ ∪ )( ⊗ ⊗ ∪ ⊗ )( β ⊗ ⊗ )( ⊗ ⊗ ∩ ⊗ )( ⊗ ⊗ ∩ )with the braiding β induced from T (Lemma 3.6). Since all maps that appear in this formulacommute with the braiding β from earlier lemmas, Θ commute with β . By the same argument asRemark 6.4, we obtain a tortile category from V . Similarly, Θ commutes with µ and ∆. A sketchproof of the commutation between µ and Θ is depicted in Figure 26.We close the paper with remarks on invariants of embedded surfaces with boundary. It is ofinterest to find invariants of compact orientable surfaces with boundary represented by ribbon graphdiagrams, as considered in [18], in a way analogous to quantum invariants using braided Frobeniusalgebras. In this approach, a height function is fixed on the plane, and building blocks of diagramsconsist of cups and caps in addition to crossings and trivalent vertices. Although a complete set ofmoves for ribbon graph diagrams for certain embedded surfaces was given in [18], height functionswere not considered. It is desirable to have a list of additional moves. For example, the passcup RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 17
Figure 26.
Sketch picture proof of commutationmove and passcap move (the upside down of passcup) are such moves, and they are satisfied bybraided Frobenius algebras constructed in this paper. Another move depicted in Figure 27 isalso satisfied from Frobenius algebra axioms. Although most moves in [18] for orientable surfaces(without half twists), with appropriate choices of height functions, are satisfied by our resultingbraided Frobenius algebras, it is not clear at this time whether the equation corresponding to themove depicted in Figure 28 is satisfied, in general, under our construction. However, it may besatisfied by some specific examples, and may provide invariants for such surfaces.
Figure 27.
Conversion of µ to ∆ through ∩ Figure 28.
Canceling a pair of loopsFor non-orientable surfaces, ribbon graph diagrams [18] contain half-twists, and there is a moveof twisting a vertex as indicated in Figure 29, that involve half twists of ribbons merging at avertex. From the topological correspondence of the twist θ to a full twists as in Figure 25, such ahypothetical half twist, which we denote by √ θ , would be required to satisfy √ θ ◦ √ θ = θ (thusthe notation). We have not found such a morphism in braided Frobenius algebras constructed inTheorem 5.2, and raise a question: For the twists ( θ and Θ) defined in this section for the braidedFrobenius algebras constructed in Theorem 5.2, are there half twists √ θ and √ Θ ? We point outa curious fact that the composition of a half-twist of a vertex in Figure 29 twice is a full twist of avertex represented by Figure 24, which is satisfied by the braided Frobenius algebras constructedin this paper.
Figure 29.
Twisting a vertex of a ribbon
Acknowledgements.
We are thankful to J. Scott Carter and Atsushi Ishii for valuable infor-mation. MS was supported in part by NSF DMS-1800443. EZ was supported by the EstonianResearch Council via the Mobilitas Pluss scheme, grant MOBJD679.
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RAIDED FROBENIUS ALGEBRAS FROM CERTAIN HOPF ALGEBRAS 19
Department of Mathematics, University of South Florida, Tampa, FL 33620, U.S.A.
Email address : [email protected] Institute of Mathematics and Statistics, University of Tartu, Narva mnt 18, 51009 Tartu, Estonia
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