Hopf-Frobenius Algebras and a Simpler Drinfeld Double
BBob Coecke and Mathew Leifer (Eds.):Quantum Physics and Logic 2019 (QPL)EPTCS 318, 2020, pp. 150–180, doi:10.4204/EPTCS.318.10 c (cid:13)
J. Collins & R. Duncan
Hopf-Frobenius Algebras and a Simpler DrinfeldDouble
Joseph Collins Ross Duncan , [email protected] [email protected] Department of Computer and Information SciencesUniversity of Strathclyde26 Richmond Street, Glasgow, United Kingdom Cambridge Quantum Computing Ltd9a Bridge Street, Cambridge, United Kingdom
The zx -calculus and related theories are based on so-called interacting Frobenius algebras,where a pair of † -special commutative Frobenius algebras jointly form a pair of Hopf algebras.In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras , startingfrom a single Hopf algebra which is not necessarily commutative or cocommutative. Weprovide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobeniusalgebra, and show that every Hopf algebra in
FVect k is a Hopf-Frobenius algebra. Inaddition, we show that this construction is unique up to an invertible scalar. Due to this fact,Hopf-Frobenius algebras provide two canonical notions of duality, and give us a “dual” Hopfalgebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. Weuse this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double, but hasa much simpler presentation. In the monoidal categories approach to quantum theory [1, 13] Hopf algebras [32] have a centralrole in the formulation of complementary observables [12]. In this setting, a quantum observableis represented as special commutative † -Frobenius algebra; a pair of such observables are called strongly complementary if the algebra part of the first and the coalgebra part of the second jointlyform a Hopf algebra. In abstract form, this combination of structures has been studied under thename “interacting Frobenius algebras” [16] where it is shown that relatively weak commutationrules between the two Frobenius algebras produce the Hopf algebra structure. From a differentstarting point Bonchi et al [7] showed that a distributive law between two Hopf algebras yieldsa pair of Frobenius structures, an approach which has been generalised to provide a model ofPetri nets [6]. Given the similarity of the two structures it is appropriate to consider both asexemplars of a common family of Hopf-Frobenius algebras .In the above settings, the algebras considered are both commutative and cocommutative.However more general Hopf algebras, perhaps not even symmetric, are a ubiquitous structurein mathematical physics, finding applications in gauge theory [27], topological quantum fieldtheory [3] and topological quantum computing [8]. In this paper we take the first steps towardsgeneralising the concept of Hopf-Frobenius algebra to the non-commutative case, and openingthe door to applications of categorical quantum theory in other areas of physics.Loosely speaking, a Hopf-Frobenius algebra consists of two monoids and two comonoids suchthat one way of pairing a monoid with a comonoid gives two Frobenius algebras, and the otherpairing yields two Hopf algebras, with the additional condition that antipodes are constructed . Collins & R. Duncan
Frobenius AlgebraFrobenius AlgebraHopf Algebra Hopf Algebra= = Antipodes
Figure 1: The elements of a Hopf-Frobenius algebrafrom the Frobenius forms. This schema is illustrated in Figure 1. In Section 3 we give the precisedefinition of Hopf-Frobenius algebras and state the necessary and sufficient conditions to extenda Hopf algebra to a Hopf-Frobenius algebra in an arbitrary symmetric monoidal category. Itwas previously known that in
FVect k , the category of finite dimensional vector spaces, everyHopf algebra carries a Frobenius algebra on both its monoid [26] and its comonoid [14, 24]; infact every Hopf algebra in FVect k is Hopf-Frobenius. In Section 4 we briefly present someexamples which are not the usual abelian group algebras. In Section 5 we show the structure of aHopf-Frobenius algebra can be used to give a simpler version of the Drinfeld double construction. Acknowledgements
The authors wish to thank Dr Gabriella Böhm (Wigner Research Centrefor Physics) for her very kind email, and all of the help and input that she gave us. The authorsalso wish to thank the anonymous reviewer for their many useful and insightful comments. JosephCollins is supported by a Carnegie Trust PhD Scholarship.
We assume that the reader is familiar with strict symmetric monoidal categories and theirdiagrammatic notation; see Selinger [30] for a thorough treatment. We make the convention thatdiagrams are read from top to bottom. When we work with the dual of an object, we will opt toomit the object names from the wires except where doing so would create ambiguity. Instead, wewill assign an orientation to the wires: downwards for the original object, upwards for its dual.
Definition 2.1.
In a monoidal category C with objects A and B , B is left dual to A if thereexists morphisms d : I → A ⊗ B and e : B ⊗ A → I such that dA e A = A and dB e B = B In this circumstance A is right dual to B . Note that if C is symmetric then left duals and rightduals coincide.52 Hopf-Frobenius Algebras and a Simpler Drinfeld Double
The morphisms d and e are usually called the unit and counit; for reasons which will becomeobvious shortly we avoid that terminology and refer to them as the cap and the cup . Note that ifan object has a dual it is unique up to isomorphism (see Lemma C.1). Definition 2.2. A compact closed category [22] is a symmetric monoidal category where everyobject A has an assigned dual ( A ∗ , d A , e A ). In the graphical notation we depict the cup and capin the obvious way: d A := A A ∗ e A := AA ∗ Proposition 2.3 ([22]) . Let C be a compact closed category. By defining f ∗ : B ∗ → A ∗ as f ∗ := f the assignment of duals A A ∗ extends uniquely to a strong monoidal functor ( · ) ∗ : C op → C ,with natural isomorphisms ( A ⊗ B ) ∗ ∼ = B ∗ ⊗ A ∗ , A ∗∗ ∼ = A , and I ∗ ∼ = I and, further, d and e arenatural transformations. Remark 2.4.
Note that if A is its own dual, a further collection of coherence equations mustapply; see Selinger [29].The main foci of this work – Frobenius and Hopf algebras – combine the structure of amonoid and a comonoid on the same object. See Appendix C.2 for basic definitions. Definition 2.5. A Frobenius algebra in a symmetric monoidal category C consists of a monoidand a comonoid on the same object, obeying the Frobenius law, shown below on the left:= = =A Frobenius algebra is called special or separable when it obeys the equation above right, and quasi-special when it obeys the special equation up to an invertible scalar factor. A Frobeniusalgebra is commutative when its monoid is, and cocommutative when its comonoid is. Lemma 2.6.
Every Frobenius algebra induces a cup and a cap which make the object self-dual.Proof.
Given the Frobenius algebra ( , , , ) define the cup and cap as shown below. d := = e := =From here the snake equation follows directly.Definition 2.5, due to Carboni and Walters [9], has a pleasing symmetry between the monoidand comonoid parts. However, an older equivalent definition will be useful in later sections . See Fauser’s survey [17] for several equivalent definitions. . Collins & R. Duncan
Definition 2.7. A Frobenius algebra in a symmetric monoidal category C consists of a monoid( F, , ) and a
Frobenius form β : F ⊗ F → I , which admits an inverse, ¯ β : I → F ⊗ F , satisfying: β = β β ¯ β = = ¯ β β To see that Definition 2.5 implies this definition it suffices to take the cup and cap definedabove as β and ¯ β . For the converse, we dualise with β to get a comonoid. For a proof of howthis comonoid fulfills the Frobenius law, see Kock [23]Frobenius forms are far from unique: there is one for each invertible element of the monoid(see Appendix C.3).Special Frobenius algebras can be understood as arising from a distribution law of comonoidsover monoids [25]. In the other direction, distributing monoids over comonoids yields bialgebras. Note.
Unlike the preceding section, in our discussion of bialgebras and Hopf algebras, we willuse different colours for the monoid and comonoid parts of the structure.
Definition 2.8. A bialgebra in symmetric monoidal category C consists of a monoid and acomonoid on the same object, which jointly obey the copy , cocopy , bialgebra , and scalar lawsdepicted below.= = = =Note that the dashed box above represents an empty diagram. We may equivalently define abialgebra as a monoid and a comonoid such that the comonoid is a monoid homomorphism. A bialgebra morphism is an morphism of the object which is both a monoid homomorphism and acomonoid homomorphism. Remark 2.9.
Some works, notably on the zx -calculus [12, 2, 20] and related theories [16], thelast axiom is dropped and the other equations modified by a scalar factor, to give a scaledbialgebra . Here we use the standard definition: the Frobenius algebras we construct will not bespecial. Definition 2.10. A Hopf algebra consists of a bialgebra (
H, , , , ) and an endomorphism s : H → H called the antipode which satisfies the Hopf law : s := = =Where unambiguous, we abuse notation slightly and use H to refer the whole Hopf algebra.Following Street [31], we can define another Hopf algebra H op on the same object, having thesame unit and counit, but with the arguments of the multiplication and comultiplication swapped:
7→ 7→ Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Replacing only the comultiplication as above yields a bialgebra H σ which is not necessarily Hopf.We quote the following basic properties from Street [31]. Proposition 2.11.
For a Hopf algebra H :1. The antipode s is unique.2. s : H op → H is a bialgebra homomorphism, i.e. = = H σ is a Hopf algebra if and only if s is invertible, in which case the antipode of H σ is s − .4. If H is commutative or cocommutative then s ◦ s = id H . Definition 2.12.
Let (
H, , , , , ) be a Hopf algebra, and suppose that the object H has a left dual H ∗ . We define the dual Hopf algebra ( H ∗ , ∗ , ∗ , ∗ , ∗ , ∗ ) as : ∗ := ∗ := ∗ := ∗ := ∗ :=It’s straightforward to prove that H ∗ is indeed a Hopf algebra using the equations of Def 2.1In later sections it will be helpful to consider duals with respect to different cups and caps, inwhich case we will vary notation accordingly but the same construction is used in all cases. We now arrive at the main subject of this paper, Hopf-Frobenius algebras in an arbitrarysymmetric monoidal category C . These algebras generalise interacting Frobenius algebras [12, 16],and share the same gross structure. It will be helpful to introduce a weaker notion first. Definition 3.1. A pre-Hopf-Frobenius algebra or pre-HF algebra consists of an object H bearinga green monoid ( , ), a green comonoid ( , ), a red monoid ( , ), a red comonoid( , ) and an endomorphism such that ( , , , ) and ( , , , ) are Frobeniusalgebras, and ( , , , , ) is a Hopf algebra. Definition 3.2. A Hopf-Frobenius algebra , or
HF algebra , is a pre-Hopf-Frobenius algebra wheresatisfies the left equation below, = , = and with defined as in the right equation above, ( , , , , ) is a Hopf algebra.We refer to the four algebras that make up an HF algebra by the colour of their multiplication ,so that ( , , , , ) is the green Hopf algebra, ( , , , ) is the red
Frobenius algebra,etc. If you are reading this document in monochrome green will appear as light grey and red as dark grey. . Collins & R. Duncan C will denote asymmetric monoidal category, and H will denote a Hopf algebra ( H, , , , , ) in C .Omitted proofs are found in Appendix A.A key concept is that of an integral. Pareigis [28] proved that in FPMod R , the category offinitely generated projective modules over a commutative ring, a Hopf algebra has Frobeniusstructure when its space of integrals is isomorphic to the ring. More generally, Takeuchi [34]and Bespalov et al. [5] gave conditions for the space of integrals in certain braided monoidalcategories to be invertible. Definition 3.3. A left (co)integral on H is a copoint : H → I (resp. a point : I → H ),satisfying the equations: = = A right (co)integral is defined similarly. Definition 3.4. An integral Hopf algebra ( H, , ) is a Hopf algebra H equipped with a choiceof right integral , and left cointegral , such that ◦ = id I . Lemma 3.5.
Let ( H, , ) be an integral Hopf algebra. Then the following map is the inverse ofthe antipode. := -1 The statement of the above Lemma bares some similarities with Definition 2.1. In whatfollows, we will be generalising this definition to capture the situation that arises with integralHopf algebras.
Definition 3.6.
Let A and B be objects in a symmetric monoidal category C . A is a right halfdual of B if there exists morphisms : I → A ⊗ B and : B ⊗ A → I which satisfy the followingequation A A = A In this circumstance, B is a left half dual of A Half duals are a strict generalisation of duals in the sense of Definition 2.1. Unlike true duals,an object may have non-isomorphic half duals. For example, if B is left dual to A , with a section m : B , → C for some retraction m : C (cid:16) B , then C is a left half dual of A . Further, any integralHopf algebra ( H, , ) makes H left half dual to itself as follows. Definition 3.7.
Let (
H, , ) be an integral Hopf algebra, and define β := γ := β := γ := This is a generalisation of earlier work by Larson and Sweedler [26] showing that the space of integrals in
FVect k is always isomorphic to k . Hopf-Frobenius Algebras and a Simpler Drinfeld Double
With these definitions, γ and β make H half dual to itself, and γ and β make H half dual toitself but in a different way. We say that H is nondegenerate when γ and β are a cap and a cuprespectively, (c.f. Definition 2.2), making H fully dual to itself. Furthermore, in this situation, γ and β are also a cup and a cap, giving a different self-dual structure to H . Lemma 3.8.
Let ( H, , ) be an integral Hopf algebra. H is nondegenerate if and only if = Lemma 3.9.
Let ( H, , ) be an integral Hopf algebra. H is nondegenerate if and only if β is a Frobenius form for ( H, , ) , or equivalently, if and only if γ is a Frobenius form for ( H, , ) . Hence, if H is nondegenerate, then H admits a pre-HF algebra structure. Per Definition 2.7, is the counit of the green Frobenius algebra and the green comultiplicationis obtained by dualising with β . The red unit and multiplication are obtained in a similarmanner. Definition 3.10.
Let the object H have a right half dual H ∗ . The integral morphism I : H → H is defined as shown below. := I Note that this definition does not depend on the choice of half dual – see Lemma A.2If H is in FPMod R , then I may be seen as a map from H to the space of left cointegrals.In fact, it is the retraction of the natural injection from the space of left integrals into H . Assuch, it acts trivially on integrals, and for every element v ∈ H , I ( v ) is a left integral (which maybe 0). In Lemma 3.11 we show that this holds in the general case, where we have exchangedelements of a module for points p : I → H . Lemma 3.11.
Given a point p : I → H , and copoint q : H → I , the morphism I ◦ p is a leftcointegral, and q ◦ I is a right integral. In addition, p is a left cointegral if and only if I ◦ p = p ,and q is a right integral if and only if q ◦ I = q . Definition 3.12.
We say that a Hopf algebra satisfies the
Frobenius condition if there existsmaps and such that = and = . Collins & R. Duncan
Theorem 3.13. If H satisfies the Frobenius condition, then H admits a pre-HF algebra structurewith the Frobenius forms and their inverses as shown below. := := := := Further, ( H, , ) is an integral Hopf algebra.Proof. The Frobenius condition implies that
I ◦ = , and ◦ I = . Hence, by Lemma 3.11,is a left cointegral and is a right integral. Now, we only need to show that it is nondegenerate.Observe that == = == = = =(1.) (2.) (3.)(4.) (5.) where (1 . ) is due to the Frobenius condition, (2 . ) comes from associativity and (3 . ) comesfrom the fact that the antipode is a bialgebra homomorphism H op → H . The presence of halfduals gives us (4 . ), and (5 . ) is due to the Hopf law. We then get the following identity = This is the identity required to make (
H, , ) nondegenerate, and we have our result by Lemma3.9.58
Hopf-Frobenius Algebras and a Simpler Drinfeld Double
The explicit definitions of the green comonoid and red monoid structures are shown below. := :=:= :=
As the name suggests, H fulfilling the Frobenius condition is equivalent to H admitting aFrobenius algebra structure. To prove this, we must first prove the following intermediate lemma Lemma 3.14.
Let the object H have a right half dual H ∗ , where H is a Hopf algebra. H fulfillsthe Frobenius condition if and only if there is an equaliser of and if and only if there is a coequaliser of and Theorem 3.15.
Let H be a Hopf algebra. H satisfies the Frobenius condition if and only if H admits a Frobenius structure on its multiplication or its comultiplication. Hence, H fulfills theFrobenius condition if and only if it admits a pre-HF algebra structure.Proof. Clearly if H satisfies the Frobenius condition, then by Theorem 3.13 it admits a Frobeniusstructure. For the converse, suppose that H admits a Frobenius structure ( H, , , , ) onits multiplication. This provides a cup and a cap that makes H self dual. Set α := . Wewill show that α : I → H is a split equaliser of the diagram H H ⊗ H . Note that the lower morphism is simply . To show that α is a split equaliser, we mustfirst show that it is is a cone of the appropriate diagram. This follows from the properties of theFrobenius algebra = = = . Collins & R. Duncan α and . It is clear that is a retract of α , and is aretract of . The final condition for α to be a split equaliser is = Thus, α a split equaliser. By Lemma 3.14, this implies that H must satisfy the Frobeniuscondition. If H admits a Frobenius algebra on its comultiplication, then the same result holdsby duality. Hence, when H admits a pre-HF algebra structure, it fulfills the Frobenius condition,and by Theorem 3.13, we get our equivalence.It may be important to mention that if H admits a Frobenius structure, then this is notnecessarily the same structure as the one given by Theorem 3.13. In Proposition C.8, we showthat Frobenius structures are not unique. One may start with different Frobenius structures, andend up with the same pre-HF algebra. In the proof of Lemma 3.15, one constructs an integraland cointegral, and it is these that determine the appropriate Frobenius structure.Pareigis [28] showed that in FPMod R , a Hopf Algebra will admit a Frobenius algebra whenintegrals only differ by a scalar multiple. This is clear from Lemma 3.14. Under mild assumptions,this is equivalent to the Frobenius condition.In a monoidal category, an object A is said to have enough points if, for all morphisms f, g : A → B , we have ( ∀ x : I → A, f x = gx ) ⇒ f = g . Lemma 3.16.
Let ( H, , ) be an integral Hopf algebra and suppose that H has enough points.If every left cointegral (right integral) is a scalar multiple of (resp. ) then H fulfills theFrobenius condition Since
FPMod R (and FVect k ) are categories where every object has enough points, Lemma 3.16implies Pareigis’ condition is exactly the Frobenius condition.We may now consider the main theorem of the paper - when exactly does a Hopf algebraadmit a Hopf-Frobenius algebra? Theorem 3.17.
Let H be a Hopf algebra such that the object H has some weak right dual H ∗ . Then H admits a Hopf-Frobenius algebra structure if and only if H fulfills the Frobeniuscondition.Sketch of Proof. We explore this in full detail in the appendix. Here, we only outline a sketch ofthe proofIf H is a Hopf-Frobenius algebra, then it admits a Frobenius algebra, and therefore, byTheorem 3.15, it fulfills the Frobenius condition.Consider the converse. By Theorem 3.13, we know that if H fulfills the Frobenius condi-tion, then H admits a pre-HF algebra and ( H, , ) is an integral Hopf algebra. It followsfrom Lemma 3.5 that = , and we show in Lemma A.3 that this is true if and onlyif (
H, , ) is an integral Hopf algebra. Hence, H admits Hopf-Frobenius structure if and60 Hopf-Frobenius Algebras and a Simpler Drinfeld Double only if (
H, , , , , ) forms a Hopf algebra, where = . We begin by proving that(
H, , , , ) is a bialgebra, and then that = is the appropriate antipode to makethis bialgebra a Hopf algebra. H admits pre-HF algebra structure, so it has a structure that makes H self dual. Let ( · )be the duality defined by the green Frobenius algebra. The dual of a Hopf algebra is a Hopfalgebra, in the sense of Definition 2.12. Therefore, applying the dual to H will give us anotherHopf algebra. Lemma A.5 tells us that ! = ! =Set H := ( H, , , , , ) to be the Hopf algebra obtained when we apply ( · )to H . The above result tells us that ( H, , , , ) is equal to ( H ) σ when viewed as abialgebra. Therefore, by Proposition 2.11 we only need to show that has an inverse. But theduality operation, ( · ) , maps isomorphisms to isomorphisms, so since is invertible, := ( − )will be the antipode of ( H, , , , ). All that is left is to show that = , and this isaccomplished by simple calculation.Let us summarise the various equivalent conditions for a Hopf algebra to be Hopf-Frobenius.
Theorem 3.18.
Let H be a Hopf algebra. The following conditions are equivalent • H admits a Hopf-Frobenius algebra structure • H admits a pre-HF algebra structure • H fulfills the Frobenius condition • H admits a Frobenius algebra structure on the multiplication or the comultiplication • H admits an equaliser of and • H admits an integral algebra structure, ( H, , ) , and H is nondegenerate • H admits an integral algebra structure, ( H, , ) , and ◦ ◦ = 1 I We finish this section by asking how canonical this structure is. Frobenius structure in generalis non-canonical (cf. Proposition C.8). Despite this, we find that Hopf-Frobenius structure iscanonical, as follows.
Lemma 3.19.
Let H admit a Hopf-Frobenius algebra structure. Then this structure is uniqueup to invertible scalar. Combined with the results of Larson and Sweedler [26], Pareigis [28], and Lemma 3.16, Theo-rem 3.17 implies that any Hopf algebra in
FVect k is Hopf-Frobenius. This allows the directextension of [16] to non-abelian group algebras, but there are plenty of other examples. Webriefly mention some examples which are neither commutative nor cocommutative. . Collins & R. Duncan Example 4.1.
Let k be a field with a primitive n th root of unity z . The Taft Hopf algebras [33] are a family of Hopf algebras in
FVect k whose antipodes have order 2 n . Generically, thealgebra ( H, µ, , ∆ , (cid:15), s ) is generated by elements x and g , such that x n = 0, g n = 1, and gx = zxg .The coalgebra is defined ∆( x ) = 1 ⊗ x + x ⊗ g , and ∆( g ) = g ⊗ g , with (cid:15) ( x ) = 0 and (cid:15) ( g ) = 1. Theantipode is s ( x ) = − xg − , s ( g ) = g , and the rest of the structure follows from the Hopf algebraaxioms. We may see that H has the basis x α g β , where 0 ≤ α, β, ≤ n −
1, so this will imply that H is n dimensional. One can calculate that the left integral of H is n X i =1 z − i g i x n − and the right cointegral is the functional that takes x n − to 1 and every other basis element to 0.We explicitly construct the HF algebra of the Taft Hopf algebra when n = 2 in the appendix. Example 4.2.
Hopf algebras which arise as the quantum enveloping algebra of Lie algebras area type of quantum group. Since these are infinite dimensional, they cannot be Hopf-Frobeniusalgebras. However their finite dimensional quotients will be Hopf-Frobenius. See Kassel [21] foran example.Moving away from
FVect k , we consider Rel , the category of sets and relations.
Example 4.3.
Let G be an infinite group. Following Hasegawa [19] we can construct its groupalgebra in Rel . The integral is { ( ?, g ) | g ∈ G } and the cointegral is the singleton (1 , ? ). Theconstruction detailed in Theorem 3.13 recovers the expected multiplication and comultiplicationrelations: := a ( b, c ) such that a = bc := ( a, b ) ( a if a = b ∅ otherwiseWe look forward to discovering more exotic examples. Braided categories of modules over a Hopf algebra are widely used in physics, where they givesolutions to the Yang-Baxter equation and in low dimensional topology, where they are used tofind invariants. However the category of modules over a Hopf algebra is braided if and only ifthe Hopf algebra is quasi-triangular . The
Drinfeld double [15] is a construction that takes a Hopfalgebra H in FVect k , and produces a quasi-triangular Hopf algebra D ( H ) on the object H ⊗ H ∗ .In this section we use the self-duality of a Hopf-Frobenius algebra to construct the canonicalisomorphism H ∼ = H ∗ and thus define a simpler version of the Drinfeld double on H ⊗ H .We will assume that C is a compact closed category. We denote the green and red Hopfalgebras of H as H and H respectively. We use the generalisation of Drinfeld’s originalconstruction to symmetric monoidal categories, due to Chen [10]. Definition 5.1.
Let H be a HF algebra on C . By Proposition C.1, we may define an isomorphism: H → H ∗ , with inverse -1 : H ∗ → H as := , := -1 Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Lemma 5.2.
The morphism is a Hopf algebra homomorphism between H σ and H ∗ . Remark 5.3.
The morphism is the canonical isomorphism between the compact closedstructure and the red dual structure given to us by the Hopf-Frobenius structure, in the senseof Proposition C.1. By Lemma 3.19, since the Hopf-Frobenius structure is canonical, the redand green Frobenius structures are also canonical, and by extension, the red and green dualstructures on H are also canonical. Therefore, whenever H admits a Hopf-Frobenius structureon a compact closed category, we may construct up to a unique invertible scalar. Definition 5.4.
A Hopf algebra H is quasi-triangular if there exists a universal R -matrix R : I → H ⊗ H such that • R is invertible with respect to • = R R • = R R R = R RR , All cocommutative Hopf algebras are quasi-triangular, with ⊗ as the universal R -matrix.This definition is motivated by the following theorem [21]. Theorem 5.5.
The category of modules over a Hopf algebra is braided if and only if the Hopfalgebra is quasi-triangular.
Definition 5.6.
Let H be a Hopf algebra in C with an invertible antipode. The Drinfeld double of H , denoted D ( H ) = ( H ⊗ H ∗ , µ, , ∆ , (cid:15), s ), is a Hopf algebra defined in the following manner: ∆ := (cid:15) := 1 :=* * * s := µ := * ** * − ∗ )( * Theorem 5.7 (Drinfeld[15, 10]) . D ( H ) is quasi-triangular, with the universal R -matrix shownbelow. -1 Our goal is to use the Hopf-Frobenius structure to get a Hopf algebra that is isomorphic tothe Drinfeld double, but is easier to do diagrammatic reasoning with.We will now use the Hopf-Frobenius structure to derive a Hopf algebra isomorphic to theDrinfeld double. Consider the composite of the map 1 ⊗ with the multiplication of the Drinfelddouble: . Collins & R. Duncan Lemma 5.8. = * ** = -1-1-1-1 Definition 5.9.
Let H be a HF algebra. The red Drinfeld double , denoted D ( H ) = ( H ⊗ H, µ , , ∆ , (cid:15) , s ), is a Hopf algebra on the object H ⊗ H with structure maps ∆ := (cid:15) := 1 := s := µ := Corollary 5.10. D ( H ) is a quasi-triangular Hopf algebra isomorphic to the Drinfeld double,with universal R-matrix -1 We have generalised the notions of interacting Frobenius algebras [12, 16] and interacting Hopfalgebras [7] to the non-commutative case, and in the process shown that they are rather commonstructures. This work could be viewed as an extension of classical results showing that concreteHopf algebras over finite dimensional vector spaces are also Frobenius algebras [26]. Anotherperspective is that we make precise how much ambient symmetry is required to obtain a Hopf-Frobenius algebra. The original setting of interacting Frobenius algebras [12] was a † -compactcategory, which provides a lot of duality on top of the commutative algebras themselves. We showthat none of this structure is necessary: all that is required is one-sided half-dual for the carrierobject. The major question that remains is to pin down exactly when the Frobenius conditionholds; as Lemma 3.16 shows, this is tightly related to the existence of integrals. Compact closuredoes not suffice to guarantee this: in FPMod R there are Hopf algebras which are not Frobenius.While we have established that Hopf algebras are frequently Hopf-Frobenius, the resultingFrobenius algebras need not be well behaved (commutative, dagger, special) as in the originalquantum setting [11]. It remains to investigate what Frobenius structures arise from “interesting”Hopf algebras, and whether they have any application in the categorical quantum mechanicsprogramme, or conversely, how HF algebras may be applied in the study of quantum groups.64 Hopf-Frobenius Algebras and a Simpler Drinfeld Double
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A Proofs omitted from the main body of the paper
Lemma 3.5.
Let ( H, , ) be an integral Hopf algebra. Then the following map is the inverse ofthe antipode. := -1 Proof.
From the definition of − , we see that = = == -1 This implies that H σ is a Hopf algebra with − as the antipode, and it follows from Proposition2.11 that the antipode of H σ is the inverse of the antipode of H . However, for the sake of claritywe will replicate the proof. We show that − ◦ = 1 as follows == = -1 -1 = -1 It follows from this that = -1-1 = = -1 = By a similar argument, − ◦ = 1.68 Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Lemma A.1. = Proof.
Observe that = = == = = =(1.) (2.) where (1 . ) comes from the bialgebra rule and (2 . ) comes from the Hopf law. Lemma 3.8.
Let ( H, , ) be an integral Hopf algebra. γ is a right inverse for β if and only if = . Collins & R. Duncan Proof.
Suppose that H is nondegenerate. Then = = = Consider the converse. Then we may characterise the unit as follows: = = =
This implies that = = -1 This then allows us to show the following, where (1.) comes from Lemma A.1, and (2 . ) is due tothe definition of cointegrals. = = = == (1.) (2.) Hence, H is nondegenerate, and we have our result. Lemma 3.9.
Let ( H, , ) be an integral Hopf algebra. H is nondegenerate if and only if β is a Frobenius form for ( H, , ) , or equivalently, if and only if γ is a Frobenius form for ( H, , ) . Hence, if H is nondegenerate, then H admits a pre-HF algebra structure.Proof. If H is nondegenerate, then the conditions of Definition 2.7 are satisfied.Conversely, suppose that β is a Frobenius form; then there exists some ¯ β such that ¯ β = Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Appealing to Lemma 3.5 we have ¯ β =¯ β = hence, γ is the right inverse of β , and H is nondegenerate. The proof for γ is similar. Lemma A.2.
When H has two half dual structures, , and , , then the integral mor-phisms coincide.Proof. = = Lemma 3.11.
Given a point p : I → H , and copoint q : H → I , the morphism I ◦ p is a leftcointegral, and q ◦ I is a right integral. In addition, p is a left cointegral if and only if I ◦ p = p ,and q is a right integral if and only if q ◦ I = q .Proof. Out goal is to show that, for all points p , I = I p p If we are able to prove the following, then the result will follow. I = I As such, we may begin by composing I with . . Collins & R. Duncan = == == =(1.) (2.)(3.) (4.) (5.) Where (1 . ) comes from Lemma A.1, (2.) comes from the presence of half duals. The antipodeis a bialgebra homomorphism H op → H , by Proposition 2.11, which gives us (3.). Associativitygives us (4.), and (5.) is due to the presence of half duals. Hence, we have proved our result.Out result also tells us that if I ◦ p = p , then p is a left cointegral. For the converse, let bea left cointegral. We then get = = The proof for right integrals is similar.
Lemma 3.14.
Let the object H have a right half dual H ∗ , where H is a Hopf algebra. H fulfillsthe Frobenius condition if and only if there is an equaliser of and Hopf-Frobenius Algebras and a Simpler Drinfeld Double if and only if there is a coequaliser of and
Proof.
Suppose that H fulfills the Frobenius condition. Then ( H, , ) is an integral Hopf algebraby Theorem 3.13, so = We shall actually prove that is a split equaliser. To do so, we need to find a retract of and. The Frobenius condition tells us that is the retract of , and we can easily calculatethat the morphism is a retract of . To show that must be a split equaliser,all we need to show now is = but this follows immediately from the assumption that the Frobenius condition is satisfied. Thus,we have one direction. Showing that is a split coequaliser follows dually.For the other direction, note that by Lemma 3.11 we have I = I where I is the integral morphism. Thus, I is a cone of the appropriate diagram. We are assumingthat is an equaliser, so there is a unique morphism : H → I such that = Also, since is a cointegral, by Lemma 3.11 we get that =
I ◦ = ◦ ◦ . Since is anequaliser, ◦ = 1 I . Hence, the Frobenius condition is satisfied. It is clear that if we assumethat we have a coequaliser, , this will also imply the Frobenius condition by duality. . Collins & R. Duncan Lemma 3.16.
Let ( H, , ) be an integral Hopf algebra and suppose that H has enough points.If every left cointegral (right integral) is a scalar multiple of (resp. ) then H fulfills theFrobenius conditionProof. If we can show that a = a then, since H has enough points, the result will follow. By Lemma 3.11, for all points a , I ◦ a isa cointegral. Then, by hypothesis there exists a scalar k : I → I such that I ◦ a = ⊗ k a = k Hence, if we can show that ◦ a = k , we will have our result. Observe that, since is an integral, ◦ I = , so we get the following. a = ka = = k and the result follows. Theorem 3.17.
Let H be a Hopf algebra such that the object H has some weak right dual H ∗ . Then H admits a Hopf-Frobenius algebra structure if and only if H fulfills the Frobeniuscondition.Proof. If H is a Hopf-Frobenius algebra, then it admits a Frobenius algebra, and therefore, byTheorem 3.15, it fulfills the Frobenius condition.Consider the converse. In what follows, we will prove the Theorem by first proving someintermediary lemmas. If H fulfills the Frobenius condition, then this is equivalent, by Theorem3.13, to H admitting a pre-HF structure such that ( H, , ) is an integral Hopf algebra. Webegin by proving that = .
Lemma A.3.
Let H admit a pre-HF algebra structure; ( H, , ) is an integral Hopf algebra ifand only if = . Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Proof.
The implication in one direction follows from Lemma 3.5. Suppose the converse. Notethat = is equivalent to − = . We use the fact that the antipode is a bialgebrahomomorphism to get the following. == It follows from this that that we may express similarly. == This allows us to show that is a left cointegral. = = =
The proof that is a right integral is similar. We only need to show that ◦ = 1 I , butthis follows from above. = = = It follows immediately from this Lemma that when H fulfills the Frobenius condition, theantipode is the canonical isomorphism that maps from one dual structure to the other, in thesense of Proposition C.1. We record this fact as a Corollary. Corollary A.4.
Let H admit a pre-HF algebra structure, such that ( H, , ) is an integralHopf algebra. Then = -1 = To prove the Theorem, we must show that (
H, , , , , ) forms a Hopf algebra, where= . We will accomplish this by showing first that (
H, , , , ) forms a bialgebra,and then that is the appropriate antipode. Recall that the dual of a Hopf algebra is a Hopfalgebra, in the sense of Definition 2.12. By using the dual structure of the green Frobeniusalgebra, we get the following:
Lemma A.5.
Let H admit a pre-HF algebra structure such that ( H, , ) is an integral Hopfalgebra, and let ( · ) be the duality defined by the green Frobenius algebra (cf. Lemma 2.6). Then: ! = (cid:16) (cid:17) = ! = (cid:16) (cid:17) = . Collins & R. Duncan Proof.
The first two statements are clear from the definition of the green dual. For the thirdstatement, we see that = = == ( ) (1.) -1 -1 where (1.) comes from Corollary A.4. The final statement follows from above, as ( ) will bethe unit of ( ) . Units of monoids are unique, so ( ) = .We now have that H := ( H, , , , , ) is a Hopf algebra. By the above Lemma,(
H, , , , ) is simply ( H ) σ when viewed as a bialgebra. Hence, by Proposition 2.11, toshow that this is a Hopf algebra, we only need to show that is invertible, and that it is equalto . But ( · ) preserves inverses, so we know that = ( ) − . All that remains is showingthat has the appropriate form, and this is done by straightforward calculation. = = Hence, = ( ) − = . Therefore, if H fulfills the Frobenius condition, the H admits aHopf-Frobenius algebra structure. Corollary A.6.
Let H admit a pre-HF algebra structure, such that ( H, , ) is an integralHopf algebra. Then = -1 = Lemma 3.19.
Let H admit a Hopf-Frobenius algebra structure. Then this structure is uniqueup to invertible scalar.Proof. Suppose that (
H, , , , , ) admits two Hopf-Frobenius structures, (
H, , , , , )and (
H, , , , , ). Recall that we refer to (
H, , , , , ) as the green Hopf algebra.The Hopf-Frobenius structures share a green Hopf algebra, so the respective units and counits ofthese Hopf algebras must be left cointegrals and right integrals of the green Hopf algebra. Itfollows from Lemma 3.14 that there exists unique scalars, k, l : I → I , such that = ⊗ k and= ⊗ l . Since these are both Hopf algebras, we get the following = = = kl kl Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Thus, k and l are mutually inverse. We now only need to show that the other structure mapsare scalar multiples of each other. We see that = follows from the above proof, as = = k = = k -1 Finally, this will imply that the multiplication and comultiplication maps only differ by aninvertible scalar. Note that as = , their inverses will also coincide. Recall how isconstructed, and observe that = = -1 = -1 =* k -1 k -1 where (*) comes from Corollary A.6. The same is true for and . Hence, if H admits twoHopf-Frobenius algebra structures, then they will only differ by an invertible scalar factor. Lemma 5.2.
The morphism is a Hopf algebra homomorphism between H σ and H ∗ .Proof. We will only show that is a homomorphism for ∗ , the rest of the structure mapswill have similar proofs. We first note that, by Corollary A.4 = = = -1 -1 Hence, we see that * = = = = . Collins & R. Duncan
Lemma 5.8. = * ** = -1-1-1-1 Proof.
This is clear from the definition of , Lemma 5.2 and Corollary A.4. We explicitly spellout the first statement here. = * ** * = = The proof of the second statement follows immediately from the definition of -1 . B Taft Hopf algebra for n = 2 Here we shall state the Hopf-Frobenius algebra for the 4 dimensional Taft Hopf algebra explicitly.It is generated by g and x , and has the structure1 x g gx x g gx ⊗ x x − gx x ⊗ x + x ⊗ g x x gxg g gx x g g ⊗ g g g ggx gx − x gx g ⊗ gx + gx ⊗ gx gx − x FVect k is a compact closed category, so the integral projection is the map xgxg I x − gx Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Hence, the element x − gx is a left cointegral, and the right integral is the delta function for x , δ x . Hence, by Theorem 3.13, these shall be our unit and counit respectively. It is now possible toconstruct the resulting Hopf-Frobenius algebra, but we shall explicitly state the structure maps.The green Frobenius algebra is :=:=:= 1 xgxg x gxg − ⊗ x + gx ⊗ g − g ⊗ gx + x ⊗
1= 1 xgxg ⊗ x + gx ⊗ g − g ⊗ gx + x ⊗ x ⊗ x + gx ⊗ gxg ⊗ x + x ⊗ g − ⊗ gx + gx ⊗ gx ⊗ x + x ⊗ gx and the red Frobenius algebra :=:= = 1 ⊗ x + x ⊗ g − g ⊗ gx − gx ⊗ xgxg x gxg − − xgxg x gxg
00 000 00 − xg − gx − g C Additional Background Material
In this section we provide additional definitions and basic properties to flesh out the backgroundmaterial of Section 2.
C.1 Categories with duals
Proposition C.1.
In a monoidal category C suppose that A has two right duals ( B , d , e ) and ( B , d , e ) ; then there exists an isomorphism f : B ∼ = B , satisfying the equations shown below. fB B := d B e B d B A B f = d A B B e B f A = B e A Proof.
Define f as shown above; the required equations follow immediately. C.2 Monoids and Comonoids
Definition C.2. A monoid in a monoidal category C consists of an object M , a binary multipli-cation µ : M ⊗ M → M and a unit morphism η : I → M obeying the familiar associativity and . Collins & R. Duncan comonoid in C is a monoid in C op , concretely depicted below.= = =A (co)monoid is called (co)commutative if its (co)multiplication is invariant under the exchangemap, as depicted below. = =In this paper we will not assume commutativity or cocommutativity. Definition C.3.
Given a monoid (
M, , ), a point a : I → M is left invertible if there existsa point l : I → M satisfying the left equation below; it is right invertible if there exists r : I → M satisfying the right equation; it is invertible if it is both left and right invertible, in which casethe two inverses coincide. l a = = a r Co-invertibility of co-points α : M → I with respect to a comonoid is defined dually. C.3 Frobenius algebras
In the following lemmas we will assume that we have a given Frobenius algebra (
F, , , , ). Definition C.4.
A Frobenius algebra is called symmetric if its cap (or equivalently its cup) isinvariant under the symmetry. = =
Proof.
See Kock [23].
Lemma C.5.
There is a bijective correspondence between invertible points for the monoid andcoinvertible copoints for the comonoid.Proof.
Let ( · ) be the duality induced by the cup and cap; then u : I → F is invertible iff andonly if u : F → I is coinvertible.80 Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Lemma C.6.
Let u be a coinvertible element of the comonoid. Define β ( u ) := u ¯ β ( u ) := u − Then β ( u ) is a Frobenius form for the monoid ( F, , , ) .Proof. We must show that the equations of 2.7 hold. The first follows from associativity of themonoid. For the second we have: u − u = u − u = u − u = =and similarly for the other side. Note that β ( u ) = β ( v ) implies u = v by the uniqueness ofinverses. Lemma C.7.
Suppose that β is a Frobenius form on ; then we obtain a coinvertible element u : F → I as follows: u := β u − := ¯ β Proof.