aa r X i v : . [ m a t h . QA ] D ec ∗ -STRUCTURES ON MODULE-ALGEBRAS MATTHEW TUCKER-SIMMONS
Abstract.
This chapter lays out a framework for discussing ∗ -structures onmodule-algebras over a Hopf ∗ -algebra H . We define a complex conjugationfunctor V V , which is an involution on the module category HMod , anddiscuss its interaction with natural constructions such as direct sums, duality,Hom, and tensor products. We define ∗ -structures first at the level of modules.We say that V is a ∗ -module if there is an isomorphism ∗ : V → V in HMod which is involutive in an appropriate sense. Then we define ∗ -structures onalgebras in HMod by requiring compatibility with multiplication. We showthat a ∗ -structure on a module lifts uniquely to the tensor algebra, and weprove that the tensor algebra has a universal mapping properly for morphismsof ∗ -modules. We also discuss inner products and adjoints in this framework.Finally, we discuss the interaction between ∗ -structures, R -matrices, and braid-ings. Background and notation
Background.
My motivation for writing this document was to understandwhat is the appropriate notion of a ∗ -algebra in the category of modules over aHopf ∗ -algebra. Although this is not very deep, it is perhaps a little tricky tophrase things properly. Since ∗ -structures are antilinear, one is forced to workwith antilinear maps of complex vector spaces. This is aesthetically displeasingsince one is then required to move outside the category of vector spaces and linearmaps to deal with what is almost a linear phenomenon. The framework of complexconjugate linear algebra allows one to phrase everything in terms of linear mapsrather than antilinear ones. It is then relatively straightforward to generalize thenotions to modules (and module-algebras) over a Hopf ∗ -algebra.The structure of this document is as follows. In the remainder of Section 1 weset notation. In Section 2 we give the necessary background on Hopf ∗ -algebrasand discuss some properties of their module categories. In Section 3 we discussantilinear maps, define complex conjugation as an endofunctor on the category ofcomplex vector spaces, and show that this functor is natural with respect to manycommon operations on linear spaces. In Section 4 we extend the notions fromSection 3 to modules over a Hopf ∗ -algebra, and in Section 5 we examine whatthese notions mean for module-algebras. In Section 6 we define ∗ -structures onmodules, and in Section 7 we extend this to module-algebras. In Section 8 we treatinner products and adjoints of linear maps in our framework. Finally, in Section 9we discuss the interaction between ∗ -structures, R -matrices, and braidings.1.2. Notation.
All vector spaces are over C . We denote the category of vectorspaces over C with linear maps by C Vect , and we denote the full subcategory of
The research reported here was supported in part by National Science Foundation grant DMS-1066368. finite-dimensional vector spaces by C Vect f . The linear dual of a vector space V will be denoted V ∗ . In order to avoid a proliferation of ∗ ’s, of which there are quiteenough already, we denote the transpose (or dual map) of a linear map T : V → W by T tr : W ∗ → V ∗ . The vector space of all linear maps from V to W will bedenoted Hom( V, W ). An undecorated Hom will always refer to linear maps, notmodule maps.For the rest of this document, H will denote a Hopf algebra over C with coproduct∆, counit ε , and antipode S . We use the Sweedler notation∆( a ) = a (1) ⊗ a (2) for the coproduct of H , with implied summation. When we refer to H -modules,we mean left modules for the underlying algebra structure of H . We denote thecategory of left H -modules with H -module maps by HMod , and we denote thefull subcategory of finite-dimensional modules by
HMod f . The vector space ofall module morphisms from V to W will be denoted by Hom H ( V, W ). The vectorspace Hom(
V, W ) of all linear maps from V to W has the structure of an H -module,but as we will see in Section 2 there are two choices for the action when H is aHopf ∗ -algebra and one must distinguish between them.For any category C , we write X ∈ C to mean that X is an object of C .2. Hopf ∗ -algebras and their modules In this section we recall some necessary definitions and facts about Hopf ∗ -algebras. One important (and not immediately obvious) fact is that the antipodeof a Hopf ∗ -algebra is invertible. This has the consequence that some linear spaces–namely the linear dual Hom( V, C ) of a module V and the space Hom( V, W ) of linearmaps between two modules–carry two natural actions of H . We assume throughoutthe rest of this document that H is a Hopf ∗ -algebra (see Definition 2.2), althoughwe note that in § § ∗ -algebras are discussed, for example, in Section 1.2.7 of [KS97] and inSection 4.1.F of [CP95]. Tensor products and duals of modules, as well as themodule structure on Hom-spaces, can be found in Section 4.1.C of [CP95]. Notation 2.1.
The symbol ⊲ will generally indicate the action of a Hopf algebraon a module, so that a ⊲ v means the action of a on v . When we consider more thanone action of H on the same vector space, we will distinguish one of the actions byusing the symbol ◮ rather than ⊲ .2.1. Hopf ∗ -algebras.Definition 2.2. A ∗ -structure on H is an antilinear map ∗ : H → H such that(a) ∗ is involutive: ( a ∗ ) ∗ = a for all a ∈ H .(b) ∗ reverses products: ( ab ) ∗ = b ∗ a ∗ for all a, b ∈ H .(c) ∆ is a ∗ -homomorphism: ∆( a ∗ ) = a ∗ (1) ⊗ a ∗ (2) for all a ∈ H .If H is equipped with a ∗ -structure then we say that H is a Hopf ∗ -algebra .It follows from Definition 2.2 that the counit is also a ∗ -homomorphism, i.e. that ε ( a ∗ ) = ε ( a ) for a ∈ H . Another consequence of the definition is that(2.1) ∗ ◦ S ◦ ∗ ◦ S = id H , which is shown in Proposition 10, Section 1.2.7 of [KS97]. The proof proceeds byshowing that ∗ ◦ S ◦ ∗ is an antipode for H op (which a priori is only a bialgebra), -STRUCTURES ON MODULE-ALGEBRAS 3 and hence S is invertible with S − = ∗ ◦ S ◦ ∗ by Proposition 6, Section 1.2.4 of thesame reference. Remark 2.3.
The fact that the antipode is necessarily invertible shows that notevery Hopf algebra over C can be given a ∗ -structure. In [Tak71], a so-called freeHopf algebra H ( C ) is constructed for any coalgebra C , and those coalgebras forwhich the antipode of H ( C ) is invertible are classified. In particular, if C = M n ( C ) ∗ is the linear dual of a matrix algebra with n >
1, then the antipode of H ( C ) is notinvertible, and hence H ( C ) cannot be equipped with a ∗ -structure.However, there are also examples of Hopf algebras over C with bijective antipodewhich cannot be endowed with a ∗ -structure. It is shown in Example 10, Section1.2.7 of [KS97] that if g is a complex Lie algebra, then ∗ -structures on the universalenveloping algebra U ( g ) correspond bijectively to real forms of g . It is shown inExample 1, Section 1.7.1 of [Vin94] that the complex Lie algebra spanned by X, Y ,and Z with relations [ X, Y ] = Y, [ X, Z ] = 2 iZ, [ Y, Z ] = 0has no real form, and hence its enveloping algebra has no ∗ -structures.2.2. Tensor products of modules.
We recall that the coproduct of H allows usto form the tensor product of modules. For V, W ∈ HMod we endow V ⊗ W withthe tensor product action(2.2) a ⊲ ( v ⊗ w ) = ( a (1) ⊲ v ) ⊗ ( a (2) ⊲ w ) . This gives
HMod the structure of a monoidal category in which the monoidal unitis the ground field C , which is an H -module via the counit ε .2.3. Duals of modules.
For any vector space V we can form the dual vectorspace Hom( V, C ). If V is a left module for a complex algebra A , then Hom( V, C ) isnaturally a right A -module, or equivalently a left A op -module. For a Hopf algebra H , the antipode can be viewed as an algebra homomorphism S : H → H op . Hencefor V ∈ HMod , the linear space Hom( V, C ) becomes a left H -module via the action(2.3) ( a ⊲ f )( v ) = f ( S ( a ) ⊲ v )for all a ∈ H , f ∈ Hom( V, C ), and v ∈ V .We have seen in § S is invertible since H is a Hopf ∗ -algebra.Thus S − gives a second algebra homomorphism H → H op , and corresponding tothis is a second left H -module structure on Hom( V, C ) given by(2.4) ( a ◮ f )( v ) = f ( S − ( a ) ⊲ v )for all a ∈ H , f ∈ Hom( V, C ), and v ∈ V . As this action is less well-known thanthe standard dual action (2.3), we briefly verify that it is an action. Linearity in a and f is clear, so we just need to check compatibility with multiplication in H .For a, b ∈ H and f ∈ Hom( V, C ) we have( a ◮ ( b ◮ f ))( v ) = ( b ◮ f )( S − ( a ) ⊲ v )= f ( S − ( b ) ⊲ ( S − ( a ) ⊲ v ))= f (( S − ( b ) S − ( a )) ⊲ v )= f ( S − ( ab ) ⊲ v )= ( ab ◮ f )( v ) , MATTHEW TUCKER-SIMMONS so a ◮ ( b ◮ f ) = ab ◮ f . Definition 2.4.
For V ∈ HMod , we define V ∗ = Hom( V, C ) with the action ⊲ of H given by (2.3), and we refer to this as the left dual of V . We define ∗ V = Hom( V, C )with the action ◮ of H given by (2.4), and we refer to this as the right dual of V . Remark 2.5.
Lemma 2.6 below explains how to remember which dual is the leftand which one is the right. The left dual, V ∗ , goes to the left of V in the evaluationpairing, while the right dual, ∗ V , goes to the right. In both cases, the superscript ∗ is adjacent to the ⊗ symbol; one can think of the ∗ as “eating” the vector. Lemma 2.6.
For V ∈ HMod , we have(a) The evaluation map ev V : V ∗ ⊗ V → C given by ϕ ⊗ v ϕ ( v ) is a morphismof H -modules.(b) The evaluation map V ev : V ⊗ ∗ V → C given by v ⊗ ϕ ϕ ( v ) is a morphismof H -modules.Proof. (a) For f ∈ V ∗ , v ∈ V , a ∈ H we haveev V ( a ⊲ ( f ⊗ v )) = ( a (1) ⊲ f )( a (2) ⊲ v )= f ( S ( a (1) ) a (2) ⊲ v )= ε ( a ) f ( v )= a ⊲ (ev V ( f ⊗ v )) . (b) Similar to (a). (cid:3) Proposition 2.7.
Let V ∈ HMod , and for v ∈ V let δ v : Hom( V, C ) → C be theevaluation map δ v ( f ) = f ( v ) .(a) The linear map V → ∗ ( V ∗ ) given by v δ v is a morphism of H -modules, andit is an isomorphism for V ∈ HMod f .(b) The linear map V → ( ∗ V ) ∗ given by v δ v is a morphism of H -modules, andit is an isomorphism for V ∈ HMod f .Proof. (a) On the one hand, for a ∈ H , v ∈ V and f ∈ V ∗ we have δ a⊲v ( f ) = f ( a ⊲ v ) , while on the other hand we have( a ◮ δ v )( f ) = δ v ( S − ( a ) ⊲ f )= ( S − ( a ) ⊲ f )( v )= f ( S ( S − ( a )) ⊲ v )= f ( a ⊲ v ) , and hence δ a⊲v = a ◮ δ v . If V is finite-dimensional then the dimensions of V and ∗ ( V ∗ ) coincide. Since v δ v is clearly injective, it is also surjective, andhence is an isomorphism.(b) Similar to (a). (cid:3) In some Hopf algebras, e.g. quasitriangular ones, the square of the antipode isan inner automorphism; see Proposition 5, Section 8.1.3 of [KS97]. In that casethe left and right duals coincide, and then for finite-dimensional modules the thesecond dual is naturally isomorphic to the original module: -STRUCTURES ON MODULE-ALGEBRAS 5
Proposition 2.8.
Suppose that there is an invertible element u of H such that S ( a ) = uau − for all a ∈ A . Then:(a) au ⊲ f = ua ◮ f for all f ∈ Hom( V, C ) and a ∈ H . In particular u ⊲ f = u ◮ f .(b) The linear map ∗ V → V ∗ given by f u ⊲ f is an isomorphism of moduleswhose inverse is f u − ⊲ f .(c) The linear map V → V ∗∗ given by v δ u⊲v is a morphism of modules, and itis an isomorphism if V ∈ HMod f , where δ is as defined in Proposition 2.7.Proof. (a) First note that for a ∈ H we have S ( au ) = S − ( S ( au )) = S − ( ua ) , so for any v ∈ V we have( au ⊲ f )( v ) = f ( S ( au ) ⊲ v )= f ( S − ( ua ) ⊲ v )= ( ua ◮ f )( v ) , so au ⊲ f = ua ◮ f .(b) We compute u ⊲ ( a ◮ f ) = u ◮ ( a ◮ f ) = ua ◮ f = au ⊲ f = a ⊲ ( u ⊲ f ) , using (a) for the first and third equalities. Hence f u ⊲ f is a module map ∗ V → V ∗ .(c) For a ∈ A and v ∈ V we have( a ⊲ δ u⊲v )( f ) = δ u⊲v ( S ( a ) ⊲ f )= ( S ( a ) ⊲ f )( u ⊲ v )= f ( S ( a ) u ⊲ v )= f ( ua ⊲ v )= δ u⊲ ( a⊲v ) ( f ) , so we see that a ⊲ δ u⊲v = δ u⊲ ( a⊲v ) , i.e. v δ u⊲v is a module map. The map isinjective, and hence bijective if V is finite-dimensional since the dimensions of V and V ∗∗ coincide. (cid:3) Remark 2.9.
The existence of left and right duals makes
HMod f into a rigid or autonomous category; see, for example, Section 2.1 of [BK01] for a precise definition.If in addition S is inner then part (c) of Proposition 2.8 implies that HMod f isa pivotal category in the sense of Definition 5.1 of [FY92].2.4. Module structures on
Hom(
V, W ) . The two actions of H on Hom( V, C )give rise to two actions of H on the space Hom( V, W ) for
V, W ∈ HMod . Thestandard action is given by(2.5) a ⊲ T = a (1) T S ( a (2) )for T ∈ Hom(
V, W ) and a ∈ H . It is not difficult to check directly that (2.5) definesan action of H . However, the definition can be better motivated by viewing thespace of linear maps as(2.6) Hom( V, W ) ≃ W ⊗ V ∗ , MATTHEW TUCKER-SIMMONS where a simple tensor w ⊗ ϕ ∈ W ⊗ V ∗ acts as a linear map on a vector v ∈ V by( w ⊗ ϕ )( v ) = wϕ ( v ) . Note that we have written the scalar on the right here. Then it is straightforwardto see that [ a ⊲ ( w ⊗ ϕ )]( v ) = [( a (1) ⊲ w ) ⊗ ( a (2) ⊲ ϕ )]( v )= ( a (1) ⊲ w )( a (2) ⊲ ϕ )( v )= a (1) ⊲ [ wϕ ( S ( a (2) ) ⊲ v )]= [ a (1) ( w ⊗ ϕ ) S ( a (2) )]( v ) , which explains (2.5). Remark 2.10.
We should note that the decomposition (2.6) holds only when atleast one of V and W is finite-dimensional. In general there is an injective linearmap W ⊗ V ∗ → Hom(
V, W ) whose range consists of all finite-rank operators. Thepoint here is that we only use the correspondence (2.6) heuristically; the action(2.5) is well-defined no matter whether V or W is finite-dimensional. The sameremarks apply to the decomposition (2.8) and the action (2.9) below as well.In order to get the tensor product action to agree with (2.5) it was essential thatwe used V ∗ rather than ∗ V in the decomposition (2.6). Also note that the orderof the two factors in this decomposition was arranged so that evaluation of linearmaps on vectors is given by(2.7) id W ⊗ ev V : W ⊗ V ∗ ⊗ V → W, which itself is a morphism of modules. We prove this in general (i.e. for moduleswhich are not necessarily finite-dimensional) in Proposition 2.15.This tells us how we can get another canonical action of H on Hom( V, W ); weexchange ∗ V for V ∗ and swap the order of the factors. Thus we write(2.8) Hom( V, W ) ≃ ∗ V ⊗ W, with the action of a simple tensor ϕ ⊗ w ∈ ∗ V ⊗ W on v ∈ V defined by( ϕ ⊗ w )( v ) = ϕ ( v ) w. Remark 2.11.
If we were bolder we would write v on the left of ϕ ⊗ w in thepreceding display, but as the overwhelming convention is to write functions to theleft of their arguments, we seem to be stuck with the current clunky formulation.In any case, the decomposition (2.8) allows us to define another action of H onHom( V, W ), which we denote using the symbol ◮ . For a ∈ H , ϕ ⊗ w ∈ ∗ V ⊗ W and v ∈ V , the tensor product action gives[ a ◮ ( ϕ ⊗ w )]( v ) = [( a (1) ◮ ϕ ) ⊗ ( a (2) ⊲ w )]( v )= ( a (1) ◮ ϕ )( v ) a (2) ⊲ w = a (2) ⊲ [ ϕ ( S − ( a (1) ) ⊲ v ) w ]= [ a (2) ( ϕ ⊗ w ) S − ( a (1) )]( v ) . Hence for T ∈ Hom(
V, W ) the action ◮ of H is given by(2.9) a ◮ T = a (2) T S − ( a (1) ) . -STRUCTURES ON MODULE-ALGEBRAS 7 Definition 2.12.
For
V, W ∈ HMod , we define Hom ℓ ( V, W ) to be the linear spaceHom(
V, W ) with the action ⊲ given by (2.5), and we call this the left Hom -space . Wedefine Hom r ( V, W ) to be the linear space Hom(
V, W ) with the action ◮ given by(2.9), and we call this the right Hom -space . When V = W , we denote Hom ℓ ( V, V )and Hom r ( V, V ) by End ℓ ( V ) and End r ( V ), respectively.We now formalize the statements made in Remark 2.10: Proposition 2.13.
Let
V, W ∈ HMod .(a) The map W ⊗ V ∗ → Hom ℓ ( V, W ) given by ( w ⊗ ϕ )( v ) = wϕ ( v ) for v ∈ V , w ∈ W , and ϕ ∈ V ∗ is an injective module map. It is an isomorphism if V and W are finite-dimensional.(b) The map ∗ V ⊗ W → Hom r ( V, W ) given by ( ϕ ⊗ w )( v ) = ϕ ( v ) w for v ∈ V , w ∈ W , and ϕ ∈ V ∗ is an injective module map. It is an isomorphism if V and W are finite-dimensional.Proof. The proofs that these are module maps are exactly the displayed com-putations immediately preceding Remark 2.10 and Definition 2.12, respectively.Injectivity is straightforward, and then these maps are isomorphisms for finite-dimensional modules just by a dimension count. (cid:3)
Remark 2.14.
As with duals, there is a simple way to to remember which is theleft Hom-space and which is the right. The left Hom-space Hom ℓ ( V, W ) goes to theleft of V in the evaluation map, and it is constructed using the left dual of V . Theright Hom-space Hom r ( V, W ) goes to the right of V in the evaluation map, and itis constructed using the right dual of V . This is encapsulated in: Lemma 2.15.
For
V, W ∈ HMod , we have(a) The evaluation map
Hom ℓ ( V, W ) ⊗ V → W given by T ⊗ v T ( v ) is a mor-phism of H -modules.(b) The evaluation map V ⊗ Hom r ( V, W ) → W given by v ⊗ T T ( v ) is amorphism of H -modules.Proof. We do part (a) only; part (b) is similar. For finite-dimensional modules theresult follows from Equation (2.7). We now give a direct proof which works forarbitrary modules. Denoting the evaluation map by ev, for a ∈ H we haveev ( a ⊲ ( T ⊗ v )) = ev (cid:0) ( a (1) ⊲ T ) ⊗ ( a (2) ⊲ v ) (cid:1) = a (1) ⊲ T ( S ( a (2) ) a (3) ⊲ v )= a (1) ⊲ T ( ε ( a (2) ) v )= a ⊲ ( T v )= a ⊲ ev( T ⊗ v ) . (cid:3) An element v of an H -module V is called invariant if a ⊲ v = ε ( a ) v for all a ∈ H .The submodule of invariant elements is denoted by V H . For the standard action(2.5) of H on Hom( V, W ), it is well-known that the invariant elements are preciselythe H -module maps. It is therefore natural to ask what the invariants are for theaction (2.9). It turns out that they are the same: Proposition 2.16.
For
V, W ∈ HMod we have
Hom ℓ ( V, W ) H = Hom H ( V, W ) = Hom r ( V, W ) H . MATTHEW TUCKER-SIMMONS
Proof.
First suppose that T is a module map. Then for a ∈ H we have a ⊲ T = a (1) T S ( a (2) ) = a (1) S ( a (2) ) T = ε ( a ) T, and similarly we see that a ◮ T = a (2) T S − ( a (1) ) = a (2) S − ( a (1) ) T = ε ( a ) T ;the last equality follows since S ( a (2) S − ( a (1) )) = a (1) S ( a (2) ) = ε ( a ).Now suppose that T ∈ Hom ℓ ( V, W ) H . Using part (a) of Lemma 2.15, for v ∈ V we have a ⊲ ( T ( v )) = ( a (1) ⊲ T )( a (2) ⊲ v )= ε ( a (1) ) T ( a (2) ⊲ v )= T ( a ⊲ v ) . Similarly, if T ∈ Hom r ( V, W ) H , we have using part (b) of Lemma 2.15 that a ⊲ ( T ( v )) = ( a (2) ◮ T )( a (1) ⊲ v )= ε ( a (2) ) T ( a (1) ⊲ v )= T ( a ⊲ v ) . (cid:3) Antilinear maps and the conjugation functor on C Vect
While our main goal is to describe complex conjugates of modules for Hopf ∗ -algebras, these notions also make sense for vector spaces over C . Actually vectorspaces are just the special case of modules over the Hopf ∗ -algebra C itself, butanyway . . . Here we set out the basics of complex conjugate linear algebra before movingon to study modules in Section 4. We begin with some elementary remarks onantilinear maps. Then we introduce the complex conjugation functor, our maintechnical tool in what follows. We show that the conjugation functor allows anti-linear maps to be interpreted naturally as linear ones. Finally we discuss severalnatural constructions in C Vect and examine their interactions with conjugation.We emphasize the functorial nature of the constructions throughout.3.1.
Elementary remarks on antilinear maps.
When doing linear algebra over C , one often encounters the notion of an antilinear map between vector spaces. Atrivial example is the map λ λ of C to itself. A less trivial example is the map T T ∗ , where T is a linear map between complex inner product spaces. Anotherexample is a complex inner product itself, which is antilinear in the first variable.While these maps are generally not difficult to deal with on their own, theframework of complex conjugate linear algebra allows us to understand antilinearmaps while working within C Vect , at the cost of some added complexity (punintended).
Definition 3.1.
Let
V, W ∈ C Vect . We say that a function T : V → W is antilinear if T ( αu + βv ) = αT ( u ) + βT ( v )for α, β ∈ C and u, v ∈ V . If T is bijective then we say that T is an anti-isomorphism ; an anti-isomorphism of V with itself will be called an anti-automorphism .We denote the collection of antilinear maps from V to W by cHom( V, W ). -STRUCTURES ON MODULE-ALGEBRAS 9 Lemma 3.2. (a) For
V, W ∈ C Vect , the set cHom(
V, W ) is a subspace of thevector space of all functions from V to W .(b) The composition of two antilinear maps is linear. The composition of an anti-linear map and a linear map, in either order, is antilinear.Proof. Straightforward. (cid:3)
Remark 3.3.
While the proof of Lemma 3.2 is trivial, there is some nontrivialcontent. While it is clear how to define addition in cHom(
V, W ), there is some choicein the definition of scalar multiplication. If we defined instead ( λT )( v ) = λT ( v ),this would give us a vector space structure different from that defined in Lemma3.2. The key observation is that there is a natural vector space structure on the setof all (not necessarily linear) functions from V to W ; viewing cHom( V, W ) insidethis space allows us to choose naturally between the two options.We would like to be able to interpret cHom(
V, W ) as a Hom-set in C Vect . Apriori this is not possible due to the banal fact that antilinear maps are not linear.In § The conjugation functor.Definition 3.4.
For V ∈ C Vect , we define the complex conjugate vector space of V or just complex conjugate of V to be the complex vector space V consisting offormal symbols c V ( v ) = v for v ∈ V with addition and scalar multiplication givenby(3.1) v + w = v + w, λ · v = λv for v, w ∈ V, λ ∈ C , respectively. Equivalently, we can define the operations in V by declaring the map c V : V → V given by v v = c V ( v ) to be an antilinearbijection.For a linear map T : V → W , we define the complex conjugate of T to be themap T : V → W given by T ( v ) = T ( v ) for v ∈ V , i.e. T = c W ◦ T ◦ c − V . In otherwords, T is the unique map making the diagram(3.2) V T −−−−→ W c V y y c W V −−−−→ T W commute. It follows from Lemma 3.2 that T is linear. Remark 3.5.
The map c V is an isomorphism of the underlying real vector spacesof V and V . Although V and V are isomorphic as complex vector spaces sincetheir dimensions are the same, there is in general no natural isomorphism. Theexception is the ground field C , as we will see below in Lemma 3.10.We emphasize that T is the unique linear map making (3.2) commute; thisfollows immediately from the fact that c V and c W are bijective.Note also that the map T T is itself an antilinear map since λT = c W ◦ ( λT ) ◦ c − V = λ · c W ◦ T ◦ c − V = λ · T .
Notation 3.6.
We generally use the notation v rather than the more cumbersome c V ( v ) except in the following two situations when confusion may arise from doingso.The first possible source of confusion is when V = C , where λ may refer eitherto the element c C ( λ ) ∈ C or to the complex conjugate element λ ∈ C . We show inLemma 3.10 that C is naturally isomorphic to C and that c C ( λ ) is identified with λ . The second possible source of confusion occurs when discussing linear maps. Wehave seen that we can define the complex conjugate of a linear map T : V → W ;the symbol T can then refer either to the conjugate map T : V → W or to theelement c Hom(
V,W ) ( T ) of Hom( V, W ). We show in Proposition 4.10 that there is anatural way to identify T with c Hom(
V,W ) ( T ).Nevertheless, we will explicitly say what we mean whenever the notation maycause confusion. Lemma 3.7.
Complex conjugation is a functor from the category C Vect to itself.That is, for
V, W, X ∈ C Vect and linear maps U : V → W and T : W → X wehave(a) id V = id V .(b) T ◦ U = T ◦ U .Proof. (a) According to (3.2) we have V VV V id V c V c V id V which proves the claim.(b) Using (3.2) twice we have V W XV W X Uc V Tc W c X U T so we see that T ◦ U : V → X is a linear map which fulfils the uniquenesscriterion discussed in Remark 3.5, and hence T ◦ U = T ◦ U . (cid:3) The next proposition allows us to view antilinear maps as linear ones, as men-tioned in Remark 3.3. In order to state the result properly, note that the functortaking a pair of vector spaces (
V, W ) to cHom(
V, W ) is contravariant in V andcovariant in W : given linear maps U : V ′ → V and R : W → W ′ , the mapcHom( V, W ) → cHom( V ′ , W ′ ) is given by T R ◦ T ◦ U , i.e.cHom( U, R ) : (cid:16) V T −→ W (cid:17) (cid:16) V ′ U −→ V T −→ W R −→ W ′ (cid:17) . -STRUCTURES ON MODULE-ALGEBRAS 11 Similarly the functor taking (
V, W ) to Hom(
V , W ) is contravariant in V and covari-ant in W : for linear maps U : V ′ → V and R : W → W ′ , the map Hom( V , W ) → Hom( V ′ , W ′ ) is given byHom( U , R ) : (cid:16) V T −→ W (cid:17) (cid:18) V ′ U −→ V T −→ W R −→ W ′ (cid:19) . Proposition 3.8.
For
V, W ∈ C Vect , the map Ψ V W : cHom(
V, W ) → Hom(
V , W ) given by T T ◦ c − V is a linear isomorphism. If T is an antilinear isomorphismthen Ψ V W ( T ) is a linear isomorphism. The collection of isomorphisms (Ψ V W ) is a natural transformation in the sense that for linear maps U : V ′ → V and R : W → W ′ , the following diagram commutes: (3.3) cHom( V, W ) cHom( U,R ) −−−−−−−→ cHom( V ′ , W ′ ) Ψ V W y y Ψ V ′ W ′ Hom(
V , W ) −−−−−−−→ Hom(
U,R ) Hom( V ′ , W ) Remark 3.9.
The definition of Ψ
V W is best captured by a diagram. For anantilinear map T : V → W , the linear map Ψ V W ( T ) is the unique map making thefollowing diagram commute: V WV Tc V Ψ V W ( T ) In fancy terms, this proposition states that for a fixed V ∈ C Vect , the pair(
V , (Ψ V − )) represents the covariant functor cHom( V, − ). Proof of Proposition 3.8.
The image of Ψ
V W lands in Hom(
V , W ) since the compo-sition of two antilinear maps is linear. The inverse of Ψ
V W is given by T T ◦ c V ,so indeed Ψ V W is an isomorphism. Since c V is bijective, we see that Ψ V W ( T ) isbijective if and only if T is. The naturality statement (3.3) follows from commuta-tivity of the diagram V ′ V W W ′ V ′ V Uc V ′ Tc V RU Ψ V W ( T ) (cid:3) Further properties of the conjugation functor.
In this subsection weexamine how the conjugation functor interacts with some natural constructions in C Vect . We show also that conjugation is an autoequivalence of C Vect and that itssquare is naturally isomorphic to the identity functor. We begin by showing that C is naturally isomorphic to its own conjugate. Our main tool in this subsectionwill be Proposition 3.8. Lemma 3.10.
Let ( λ λ ) ∈ cHom( C , C ) be the usual complex conjugation mapfrom C to itself. Then γ = Ψ CC ( λ λ ) ∈ Hom( C , C ) is a linear isomorphism.Explicitly, γ is given by γ ( c C ( λ )) = λ .Proof. Straightforward. (cid:3)
The analogue of Lemma 3.10 holds for any complex vector space with an anti-linear automorphism. Of course, any vector space has such an anti-automorphismif one fixes a basis; the point is that the anti-automorphism of C is canonical. Lemma 3.11.
The map σ V : V → V given by v v is a linear isomorphism.Given a linear map T : V → W , the following diagram commutes: (3.4) V T −−−−→ W σ V y y σ W V −−−−→ T W Proof.
Note that σ V = ( c V ◦ c V ) − = Ψ V V ( c − V ); since both c V and c V are antilinearisomorphisms, σ V is a linear isomorphism. Commutativity of (3.4) follows fromcommutativity of the diagram V c V −−−−→ V c V −−−−→ V T y T y T y W −−−−→ c W W −−−−→ c W W (cid:3) Proposition 3.12.
The conjugation functor is an autoequivalence of C Vect whosequasi-inverse is itself. The conjugation functor is exact.Proof.
Lemma 3.11 shows that the square of the conjugation functor is naturallyisomorphic to the identity functor of C Vect . Exactness holds because any equiva-lence of abelian categories is exact. (cid:3)
Lemma 3.13.
Let
V = ( V j ) j ∈ J be any family of vector spaces. Then the map π V : M j ∈ J V j → M j ∈ J V j given by π V (cid:16) ( v j ) j ∈ J (cid:17) = ( v j ) j ∈ J is a linear isomorphism. If W = ( W j ) j ∈ J is another family of vector spaces indexedby J and if ( T j : V j → W j ) j ∈ J is a family of linear maps, then the diagram (3.5) L j ∈ J V j T V −−−−→ L j ∈ J W jπ V y y π W L j ∈ J V j −−−−→ T V L j ∈ J W j commutes, where T V = ⊕ j ∈ J T j and T V = ⊕ j ∈ J T j . -STRUCTURES ON MODULE-ALGEBRAS 13 Proof.
Note that π V = Ψ ⊕ j V j , ⊕ j V j ( ⊕ j c V j ). Since ⊕ j c V j is an antilinear isomor-phism, it follows from Proposition 3.8 that π V is a linear isomorphism. Commuta-tivity of (3.5) is straightforward. (cid:3) Remark 3.14.
We may well ask about the interaction of complex conjugationwith several other natural constructions in C Vect , namely duals, tensor products,and Hom. We defer discussion of these constructions to later sections becauseof additional complications which are introduced when working with H -modulesrather than vector spaces.4. Complex conjugation of modules
In this section H is a fixed Hopf ∗ -algebra. We define complex conjugation of H -modules and define the concept of an anti-module map. We show how complexconjugation interacts with tensor products and duals of modules and we examinethe complex conjugates of Hom-spaces also. We discuss how the results of Section3 extend to H -modules.4.1. The conjugation functor on HMod. If H is a Hopf algebra over C then HMod is a subcategory of C Vect . It is therefore natural to ask whether thecomplex conjugation functor restricts to an endofunctor on
HMod . This boilsdown to two questions: first, whether we can define a module structure on V for V ∈ HMod ; and second, whether complex conjugates of module maps are againmodule maps. The fact that H is a Hopf ∗ -algebra allows us to answer thesequestions affirmatively. Definition 4.1.
Given any module V ∈ HMod , we define its complex conjugatemodule to be the complex conjugate vector space V with action given by(4.1) a ⊲ v = S ( a ) ∗ ⊲ v, or equivalently a ⊲ c V ( v ) = c V ( S ( a ) ∗ ⊲ v )for a ∈ H and v ∈ V . Remark 4.2.
It is straightforward to check that (4.1) defines an action of H on V ; for the sake of completeness we carry out this computation in the proof ofProposition 4.3. It is interesting to note that, except for interchanging the orderof the antipode and the ∗ -operation, this is essentially the only the choice for amodule structure on V : the ∗ is necessary in order to keep the operation of a on v linear in a , but since the ∗ reverses products, the S is required also to make V intoa left module.If we instead switched the order of the S and the ∗ in (4.1), we would have ananalogous concept. More precisely, we would define e V to be V as a complex vectorspace, with the H -action given by a ⊲ ˜ v = ^ S ( a ∗ ) ⊲ v for a ∈ A and v ∈ V . Certainlyall of the following theory could be developed in that framework, and while it doesnot appear that there is a natural transformation which directly connects these twocomplex conjugation functors, there are some relationships between them.First, we claim that ( e V ) ∗ ≃ ∗ V . Indeed, note that the underlying complex vectorspace of each of these modules is just Hom( V , C ). Then for f ∈ Hom(
V , C ) and v ∈ V , we have( a ⊲ f )(˜ v ) = f ( S ( a ) ⊲ ˜ v ) = f ( ^ S ( S ( a ) ∗ ) ⊲ v ) = f ( ^ a ∗ ⊲ v ) , while on the other hand we have( a ◮ f )( v ) = f ( S − ( a ) ⊲ v ) = f ( a ∗ ⊲ v ) , so the identity map on the underlying vector space Hom( V , C ) in fact is an isomor-phism of modules ( e V ) ∗ ≃ ∗ V .Also we can consider the modules e V and e V . For v ∈ V and a ∈ A , in the formermodule we have a ⊲ e v = ^ S ( a ∗ ) ⊲ v = ^ S ( a ∗ ) ∗ ⊲ v = ^ S − ( a ) ⊲ v, while in the latter we get a ⊲ ˜ v = S ( a ) ∗ ⊲ ˜ v = ^ S ( a ) ⊲ v. Thus the two actions are related by the automorphism S of H . Hence if S is aninner automorphism, the two modules are isomorphic, while if S is an inner auto-morphism then both modules are isomorphic to V (and hence the two conjugationfunctors are quasi-inverse to one another). As mentioned above, if H is quasitrian-gular then in fact S is inner. In general, however, this is not the case, although insome situations S is “almost inner” in a certain sense, and this may allow one to saymore about the relationship between the modules e V and e V . For finite-dimensionalHopf algebras the result on S is due to Radford [Rad76], extending a result ofLarson; for various generalizations to different classes of infinite-dimensional Hopfalgebras one can see [BBT07], [BB09], and references therein. Proposition 4.3.
Let
V, W ∈ HMod .(a) Equation (4.1) defines an action of H on V , i.e. V ∈ HMod .(b) If T : V → W is a module map, then T : V → W is also a module map.(c) Complex conjugation is an endofunctor of HMod .Proof. (a) It is clear that the action is linear in H and in v . We just check that itis compatible with multiplication in H . For a, b ∈ H and v ∈ V we have a ⊲ ( b ⊲ v ) = a ⊲ S ( b ) ∗ ⊲ v = S ( a ) ∗ ⊲ ( S ( b ) ∗ ⊲ v )= ( S ( a ) ∗ S ( b ) ∗ ) ⊲ v = ( S ( b ) S ( a )) ∗ ⊲ v = S ( ab ) ∗ ⊲ v = ( ab ) ⊲ v. (b) For v ∈ V and a ∈ H we have T ( a ⊲ v ) = T ( S ( a ) ∗ ⊲ v )= T ( S ( a ) ∗ ⊲ v )= S ( a ) ∗ ⊲ T ( v )= a ⊲ T ( v )= a ⊲ ( T ( v ))Thus T is a module map. -STRUCTURES ON MODULE-ALGEBRAS 15 (c) We know already from Lemma 3.7 that complex conjugation takes identitiesto identities and preserves composition. By part (b) it takes module maps tomodule maps, so it defines a functor from HMod to itself. (cid:3)
Antimodule maps.
In Section 3 we defined antilinear maps and introducedcomplex conjugation of vector spaces as a way to turn antilinear maps into linearones. Here we do the opposite: having defined complex conjugation of modules,we use this to motivate the definition of antimodule maps. We then show thatcomplex conjugation of modules turns antimodule maps into module maps, as onewould hope. This will prove useful later on when we show how the results of Section3 extend to the category of H -modules. Definition 4.4.
Let
V, W ∈ HMod . We say that a function T : V → W is an antimodule map if T is antilinear and satisfies T ( a ⊲ v ) = S ( a ) ∗ ⊲ T ( v )for all a ∈ H and v ∈ V . We denote the collection of antimodule maps from V to W by cHom H ( V, W ).Of course, the prototype for the definition of an antimodule map is c V : V → V .The analogue of Lemma 3.2 (b) holds: Lemma 4.5.
The composition of two antimodule maps is a module map. The com-position of an antimodule map and a module map, in either order, is an antimodulemap.Proof.
Straightforward. (cid:3)
Proposition 4.6.
Let
V, W ∈ HMod . Then the linear isomorphism Ψ V W fromProposition 3.8 restricts to an isomorphism Ψ V W : cHom H ( V, W ) → Hom H ( V , W ) . The analogue of the naturality statement (3.3) holds with respect to module maps S : V ′ → V and R : W → W ′ .Proof. Since c V is an antimodule map, Lemma 4.5 implies that Ψ V W carries anti-module maps to module maps. The inverse of Ψ
V W is given by T T ◦ c V . Thenaturality statement is immediate from (3.3). (cid:3) Proposition 4.7.
Let V and W be objects in HMod and let
V = ( V j ) j ∈ J be afamily of objects in HMod .(a) The map γ : C → C from Lemma 3.10 is a module isomorphism.(b) The map σ V : V → V from Lemma 3.11 is a module isomorphism.(c) The map π V : L j ∈ J V j → L j ∈ J V j from Lemma 3.13 is a module isomorphism.Proof. We have already shown that γ, σ V , and π V are linear isomorphisms in Section3, so it is only left to show that they are morphisms of modules. Since these mapsare all constructed from antilinear maps using the natural isomorphisms Ψ, ourstrategy is to show that these antilinear maps are actually anti module maps; thenthe result will follow from Proposition 4.6.(a) For a ∈ H and λ ∈ C we have a ⊲ λ = ε ( a ) λ = ε ( S ( a )) · λ = ε ( S ( a ) ∗ ) λ = S ( a ) ∗ ⊲ λ, so λ λ is an antimodule map. Hence γ = Ψ CC ( λ λ ) is a module map. (b) Since c V is an antimodule map, σ V = Ψ V V ( c − V ) is a module map.(c) Each c V j is an antimodule map, so ⊕ j c V j is as well. Hence π V = Ψ ⊕ j V j , ⊕ j V j ( ⊕ j c V j )is a module map. (cid:3) Conjugation of tensor products.
In this subsection we show that complexconjugation reverses the order of tensor products of modules.
Proposition 4.8.
Given any modules
V, W ∈ HMod , the map ρ V W : V ⊗ W → W ⊗ V given by v ⊗ w w ⊗ v is an isomorphism of H -modules. Given modulemaps S : W → Y and T : X → Z , the diagram (4.2) W ⊗ X S ⊗ T −−−−→ Y ⊗ Z ρ WX y y ρ Y Z X ⊗ W −−−−→ T ⊗ S Z ⊗ Y commutes. Furthermore, for any U, V, W in HMod , the diagram (4.3) U ⊗ V ⊗ W ρ U,V ⊗ W −−−−−→ V ⊗ W ⊗ U ρ U ⊗ V,W y y ρ V W ⊗ id U W ⊗ U ⊗ V −−−−−−−→ id W ⊗ ρ UV W ⊗ V ⊗ U commutes.Proof. Our goal is to show that η : v ⊗ w w ⊗ v is an antimodule map; thenit will follow from Proposition 4.6 that ρ V W = Ψ V ⊗ W,W ⊗ V ( η ) is a module map.Noting that ∆( S ( a ) ∗ ) = S ( a (2) ) ∗ ⊗ S ( a (1) ) ∗ , we have η ( a ⊲ v ⊗ w ) = η ( a (1) ⊲ v ⊗ a (2) ⊲ w )= a (2) ⊲ w ⊗ a (1) ⊲ v = ( S ( a (2) ) ∗ ⊲ w ) ⊗ ( S ( a (1) ) ∗ ⊲ v )= S ( a ) ∗ ⊲ η ( v ⊗ w ) , so η is an antimodule map, as claimed. It is clear that ρ V W is a linear isomorphismand that (4.2) and (4.3) commute. (cid:3)
Conjugation of duals.
It is natural to ask whether (left or right) dualizationcommutes with complex conjugation of modules. It turns out that the situation isslightly subtler than that. In fact, complex conjugation intertwines left and rightduals, i.e. V ∗ ≃ ∗ V . This can be seen as follows. Consider the evaluation pairingev V : V ∗ ⊗ V → C . Taking the complex conjugate of this map, composing with theisomorphism γ : C → C , and precomposing with the isomorphism ρ − V ∗ V of V ⊗ V ∗ with V ∗ ⊗ V gives γ ◦ ev V ◦ ρ − V ∗ V : V ⊗ V ∗ → C ;thus V ∗ plays the role of the right dual of V . We formalize this as follows: Proposition 4.9.
Let
V, W ∈ HMod .(a) The map b V : V ∗ → ∗ V given by b V ( f )( v ) = f ( v ) is a bijective antimodule map. -STRUCTURES ON MODULE-ALGEBRAS 17 (b) The map β V : V ∗ → ∗ V given by β V = Ψ V ∗ , ∗ V ( b V ) is an isomorphism of modules.(c) For any module map T : V → W , the following diagram commutes: (4.4) W ∗ T tr −−−−→ V ∗ β W y y β V ∗ W −−−−→ T tr ∗ V (d) The collection of maps ( β V ) is a natural equivalence of the (contravariant) end-ofunctors of HMod given by left dual followed by conjugation, and conjugationfollowed by right dual, respectively.Proof. (a) Note first that b V ( f ) = ( λ λ ) ◦ f ◦ c − V , where λ λ is the complexconjugation map of C to itself. This shows both that b V ( f ) is a linear functionalon V and that b V itself is an antilinear map. The inverse of b V is given by g ( λ λ ) ◦ g ◦ c V , so b V is bijective. Thus we just need to show that b V isan antimodule map. For f ∈ V ∗ , v ∈ V , and a ∈ H we have( a ◮ b V ( f ))( v ) = b V ( f )( S − ( a ) ⊲ v )= b V ( f )( a ∗ ⊲ v )= f ( a ∗ ⊲ v )= ( S − ( a ∗ ) ⊲ f )( v )= b V ( S ( a ) ∗ ⊲ f )( v ) , so a ◮ b V ( f ) = b V ( S ( a ) ∗ ⊲ f ). Hence b V is an antimodule map.(b) Immediate from (a) together with Proposition 4.6.(c) Straightforward.(d) Immediate from (b) and (c). (cid:3) Conjugation of
Hom ’s.
The fact that conjugation switches left and right du-als implies also that conjugation switches left and right Hom-spaces. Heuristically,we have Hom ℓ ( V, W ) ≃ W ⊗ V ∗ ≃ V ∗ ⊗ W ≃ ∗ V ⊗ W ≃ Hom r ( V , W ) , where the first isomorphism comes from the decomposition (2.6), the second fromProposition 4.8, the third from Proposition 4.9, and the last from (2.8). As we notedin Remark 2.10, these tensor product decompositions of the Hom-spaces are validonly when at least one of V and W is finite-dimensional. However, the followingresults are valid for all modules since they refer only to actions on Hom-spaces andnot to the tensor product decompositions. Proposition 4.10.
Let
V, W ∈ HMod .(a) The map b V : Hom ℓ ( V, W ) → Hom r ( V , W ) given by b V ( T ) = T is a bijective antimodule map. (b) The map β V : Hom ℓ ( V, W ) → Hom r ( V , W ) given by β V = Ψ Hom ℓ ( V,W ) , Hom r ( V ,W ) ( b V ) is an isomorphism of modules. On elements β V is given by β V ( c Hom ℓ ( V,W ) ( T )) = T .
Proof. (a) Note first that b V ( T ) = c W ◦ T ◦ c − V . This shows both that b V ( T ) is alinear map V → W and that b V itself is an antimodule map. The inverse of b V is given by U c − W ◦ U ◦ c V , so b V is bijective. Thus we just need to show that b V is an antimodule map. For T ∈ Hom(
V, W ), v ∈ V , and a ∈ H we have( a ◮ T )( v ) = a (2) ⊲ ( T ( S − ( a (1) ) ⊲ v ))= a (2) ⊲ ( T ( a ∗ (1) ⊲ v ))= a (2) ⊲ T ( a ∗ (1) ⊲ v )= S ( a (2) ) ∗ ⊲ T ( a ∗ (1) ⊲ v )= ( S ( a ) ∗ ⊲ T )( v )= ( S ( a ) ∗ ⊲ T )( v ) , so we see that a ◮ b V ( T ) = a ◮ T = S ( a ) ∗ ⊲ T = b V ( S ( a ) ∗ ⊲ T )and hence b V is an antimodule map.(b) Follows immediately from (a) together with Proposition 4.6. (cid:3) Remark 4.11.
Part (b) of Proposition 4.10 realizes the identification of the linearmap T with the abstract element c Hom(
V,W ) ( T ) that we promised in the discussionimmediately preceding Lemma 3.7.5. Complex conjugation of algebras
In this section we look at algebras in the category
HMod and we show that thecomplex conjugate of an algebra is again an algebra. We examine tensor algebras ofmodules in this light and show that for a module V , both T ( V ) and T ( V ) satisfy auniversal property with respect to lifting of antimodule maps from V into algebrasin HMod .For an introduction to module-algebras over a Hopf algebra one can see Section1.3.3 of [KS97] or Section 4.1.C of [CP95]; a much deeper treatment can be foundin Chapter 4 of [Mon93].5.1.
The complex conjugate of an algebra.
Recall that an H -module algebra is a C -algebra ( A, m, u ), where A ∈ HMod and m : A ⊗ A → A and u : C → A aremorphisms in HMod . We will generally just say that A is an algebra in HMod and refer to the maps m and u explicitly only when necessary; we write the productas m ( x ⊗ y ) = xy and the unit as u (1 C ) = 1 A . In terms of elements, the fact that m and u are module maps means that for a ∈ H and x, y ∈ A we have(5.1) a ⊲ ( xy ) = ( a (1) ⊲ x )( a (2) ⊲ y ) and a ⊲ A = ε ( a )1 A . If A and B are algebras in HMod , then we say that f : A → B is a morphismof module-algebras or a module-algebra morphism if f is simultaneously a modulemap and an algebra homomorphism. -STRUCTURES ON MODULE-ALGEBRAS 19 Proposition 5.1. (a) If ( A, m, u ) is an algebra in HMod then ( A, m A , u A ) is analgebra in HMod with the structure maps given by u A = u ◦ γ − , m A = m ◦ ρ − AA , where γ : C → C and ρ AA : A ⊗ A → A ⊗ A are the isomorphisms fromProposition 4.7 ( a ) and Proposition 4.8, respectively.(b) If f : A → B is a morphism of module-algebras, then f : A → B is also amorphism of module-algebras.(c) Complex conjugation is an endofunctor of the category of H -module algebraswith module-algebra morphisms. Remark 5.2.
Unwinding the definition of the multiplication in A , we see that for a, b ∈ A we have a · b = m A ( a ⊗ b ) = m ( b ⊗ a ) = m ( b ⊗ a ) = ba ;this implies that 1 A · a = a · A = a and similarly a · A = a . To summarize, in the conjugate algebra A we have(5.2) a · b = ba and 1 A = 1 A . Proof of Proposition 5.1. (a) By Proposition 4.3, both m and u are module maps.Then m A and u A are both module maps, as they are compositions of modulemaps. We just need to verify that m A and u A satisfy the associativity and unitlaws. But this is clear from (5.2).(b) We already know that f is a module morphism by Proposition 4.3, so we justneed to check that it is an algebra homomorphism. For a, b ∈ A we have f ( a · b ) = f ( ba )= f ( ba )= f ( b ) f ( a )= f ( a ) · f ( b )= f ( a ) · f ( b ) , so f is multiplicative. We have also f (1 A ) = f (1 A ) = f (1 A ) = 1 B = 1 B , so f is unital, and hence is an algebra homomorphism.(c) Follows immediately from (a) and (b) since we already know that conjugationrespects identity maps and composition. (cid:3) We now investigate some properties of morphisms of module-algebras and thecorresponding conjugate-linear notions. For this we need the following:
Definition 5.3. If A and B are algebras in HMod then we say that a function f : A → B is an antimodule-algebra morphism if f is simultaneously an antimodulemap, f ( ab ) = f ( b ) f ( a ) for all a, b ∈ A , and f (1 A ) = 1 B . Lemma 5.4.
Let A and B be algebras in HMod . (a) The composition of two antimodule-algebra morphisms is a module-algebra mor-phism. The composition of an antimodule-algebra morphism and a module-algebra morphism, in either order, is an antimodule-algebra morphism.(b) The map c A : A → A is an antimodule-algebra morphism.(c) If f : A → B is an antimodule map, then f is an antimodule-algebra morphismif and only if Ψ AB ( f ) : A → B is a module-algebra morphism.Proof. (a) Straightforward.(b) We have c A (1 A ) = 1 A = 1 A , and for a, b ∈ A we have c A ( ab ) = ab = ba = c A ( b ) c A ( a ) . (c) Since Ψ AB ( f ) = f ◦ c − A , the result follows immediately from parts (b) and (c). (cid:3) Corollary 5.5.
The morphism γ : C → C from Lemma 3.10 is an isomorphism ofmodule-algebras.Proof. Since γ = Ψ CC ( λ λ ), this follows from Lemma 5.4 (c) and the observationthat λ λ is an antimodule-algebra morphism. (cid:3) Corollary 5.6. If A is an algebra in HMod then the map σ A : A → A introducedin Lemma 3.11 is an isomorphism of module-algebras.Proof. We know that σ A is a linear isomorphism from Lemma 3.11 and that it isa module map by Proposition 4.7. Since σ A = c − A ◦ c − A , we have that σ A is amorphism of module-algebras from parts (a) and (b) of Lemma 5.4. (cid:3) The complex conjugate of the tensor algebra.
For V in HMod , thetensor algebra of V is defined to be T ( V ) = ∞ M n =0 V ⊗ n , where by definition V ⊗ = C (with H -action given by the counit), and all of thetensor products are taken over C . We denote by ι V the canonical injection of V into T ( V ), and note that ι V is a module map. We denote homogeneous elementsof T ( V ) by v . . . v n rather than v ⊗ · · · ⊗ v n in order to save space.Our goal is to show that there is a natural isomorphism of module-algebrasbetween T ( V ) and T ( V ). In order to do so, we begin with a discussion on theuniversal mapping property of the tensor algebra. Although the following result iswell-known, we include the formal statement and proof for the sake of completeness. Lemma 5.7.
Let V ∈ HMod and let T ( V ) be defined as above.(a) T ( V ) is an algebra in HMod with multiplication given by concatenation oftensors and the unit C → T ( V ) given by the isomorphism C ≃ V ⊗ .(b) For any algebra A in HMod and any module map f : V → A , there is a uniquemorphism of module-algebras ˜ f : T ( V ) → A such that ˜ f ◦ ι V = f .Proof. (a) The fact that T ( V ) is an associative algebra over C is straightforward.The crux of the matter is to show that the multiplication and unit are mapsof H -modules. For this proof only we denote the multiplication map by m : -STRUCTURES ON MODULE-ALGEBRAS 21 T ( V ) ⊗ T ( V ) → T ( V ). For v , . . . , v m , w , . . . , w n ∈ V , denote v = v . . . v m and w = w . . . w n . Then for a ∈ H we have m ( a ⊲ ( v ⊗ w )) = m (( a (1) ⊲ v ) ⊗ ( a (2) ⊲ w ))= ( a (1) ⊲ v ) . . . ( a ( m ) ⊲ v m )( a ( m +1) ⊲ w ) . . . ( a ( m + n ) ⊲ w n )= a ⊲ ( v . . . v m w . . . w n )= a ⊲ m ( v ⊗ w ) , so we see that all we really used was coassociativity of the comultiplication.The fact that the unit is a module map is immediate, as the module action on C ≃ V ⊗ is just given by the counit of H .(b) Fix n ≥
0. If n = 0, define f : V ⊗ → A by f (1) = 1 A . If n >
0, the map Q nj =1 V → A given by ( v , . . . , v n ) f ( v ) . . . f ( v n )is C -multilinear and hence induces a unique map f n : V ⊗ n → A given by f n ( v . . . v n ) = f ( v ) . . . f ( v n );recall that we are omitting ⊗ signs inside T ( V ). We claim that f n is a map of H -modules. For a ∈ A , we have f n ( a ⊲ ( v . . . v n )) = f n (( a (1) ⊲ v ) . . . ( a ( n ) ⊲ v n ))= f ( a (1) ⊲ v ) . . . f ( a ( n ) ⊲ v n )= ( a (1) ⊲ f ( v )) . . . ( a ( n ) ⊲ f ( v n ))= a ⊲ ( f ( v ) . . . f ( v n ))= a ⊲ f n ( v . . . v n ) , which verifies the claim. Finally, we define ˜ f to be the direct sum of themaps f n for n ≥
0. It is immediate from the definition that ˜ f is an algebrahomomorphism. It is also a map of H -modules since it is a direct sum ofsuch maps. Clearly ˜ f ◦ ι V = f , and ˜ f is unique with this property since ι ( V )generates T ( V ) as an algebra. (cid:3) We will use similar ideas to show that T ( V ) and T ( V ) are isomorphic; bothof these algebras are equipped with antimodule maps from V and they satisfy auniversal mapping property with respect to algebras with antimodule maps com-ing from V . Then the usual abstract nonsense will show that there is a uniqueisomorphism between T ( V ) and T ( V ).We define maps θ V : V → T ( V ) and ϑ V : V → T ( V ) by(5.3) θ V = ι V ◦ c V and ϑ V = c T ( V ) ◦ ι V , where ι V and ι V are the embeddings of V and V into T ( V ) and T ( V ), respectively,as defined in the discussion at the beginning of the subsection. Both θ V and ϑ V are antimodule maps by Lemma 4.5. Proposition 5.8.
Let V ∈ HMod and let θ V and ϑ V be as in (5.3) . Let A be analgebra in HMod and let f : V → A be an antimodule map.(a) There is a unique module-algebra morphism ˆ f : T ( V ) → A such that ˆ f ◦ θ V = f .(b) There is a unique module-algebra morphism ˇ f : T ( V ) → A such that ˇ f ◦ ϑ V = f . Proof. (a) By Proposition 4.6, Ψ
V A ( f ) : V → A is a module map. Then theuniversal property of the tensor algebra implies that there is a unique module-algebra morphism ˆ f : T ( V ) → A such that ˆ f ◦ ι V = Ψ V A ( f ). Then we haveˆ f ◦ θ V = ˆ f ◦ ι V ◦ c V = Ψ V A ( f ) ◦ c V = f. This is encapsulated in the commutativity of the diagram AT ( V ) V V ˆ f fθ V c V ι V Ψ V A ( f ) The lower left triangle commutes by definition of θ V , the right-hand trianglecommutes by definition of Ψ V A ( f ), and the large triangle commutes by the uni-versal property of the tensor algebra. Hence the upper left triangle commutesas well, and this shows that ˆ f has the desired property.To see that ˆ f is unique, note that θ V ( V ) generates T ( V ) as an algebra, andthe equation ˆ f ◦ θ V = f uniquely specifies ˆ f on the generators. Since ˆ f is analgebra homomorphism, this implies that it is uniquely specified on the algebra T ( V ).(b) By Lemma 4.5, the function c A ◦ f : V → A is a module map. The universalproperty of T ( V ) then gives a unique extension to a module-algebra morphism f : T ( V ) → A such that f ◦ ι V = c A ◦ f . Then f : T ( V ) → A is amorphism of module-algebras by Proposition 5.1 (b). We define ˇ f : T ( V ) → A by ˇ f = σ A ◦ f . Since σ A is a module-algebra morphism by Corollary 5.6, thenˇ f is a module-algebra morphism by Lemma 5.4 (a). We haveˇ f ◦ ϑ V = σ A ◦ f ◦ c T ( V ) ◦ ι V m = σ A ◦ c A ◦ f ◦ ι V = c − A ◦ c A ◦ f = f. This is encapsulated in the commutativity of the diagram T ( V ) AV AT ( V ) A f c T ( V ) c − A c A ι V ϑ V c A ◦ ff σ A ˇ f f -STRUCTURES ON MODULE-ALGEBRAS 23 The large rectangle commutes by (3.2). For the five triangles starting onthe left-hand side of the diagram and going clockwise, the reasons they com-mute are, respectively: by definition of ϑ V ; by the universal property of T ( V )(i.e. since f lifts c A ◦ f ); trivially; by definition of σ A ; and by definition ofˇ f . Then the final interior triangle commutes because everything else does, andthis shows that ˇ f has the desired property.The proof of uniqueness from part (a) carries over here almost verbatim. (cid:3) Since T ( V ) and T ( V ) satisfy the same universal property, they are isomorphic: Corollary 5.9.
There is a unique isomorphism of module-algebras κ V : T ( V ) → T ( V ) such that κ V ( v ) = v for all v ∈ V . On simple tensors κ V is given by (5.4) κ V ( v . . . v n ) = v n . . . v , and hence the restriction of κ V to V ⊗ n gives an isomorphism V ⊗ n ≃ V ⊗ n .Proof. By Proposition 5.8 (a) applied to the antimodule map ϑ V : V → T ( V ), thereis a unique map of module-algebras κ V : T ( V ) → T ( V ) such that κ V ◦ θ V = ϑ V ,i.e. such that κ V ( v ) = v .Applying part (b) of Proposition 5.8 to the antimodule map θ V : V → T ( V ) givesa unique map of module-algebras ψ V : T ( V ) → T ( V ) such that ψ V ◦ ϑ V = θ V .According to part (v) of Proposition 5.8, there is a unique morphism of module-algebras ˇ ϑ V : T ( V ) → T ( V ) such that ˇ ϑ V ◦ ϑ V = ϑ V ; it is clear that ˇ ϑ V = id T ( V ) .But on the other hand, we have( κ V ◦ ψ V ) ◦ ϑ V = κ V ◦ θ V = ϑ V , so we must have κ V ◦ ψ V = id T ( V ) by uniqueness. Similarly we have ψ V ◦ κ V =id T ( V ) , so both maps are isomorphisms of module-algebras.Finally, the formula (5.4) follows immediately from the description of multipli-cation in the conjugate of an algebra given in Remark 5.2. (cid:3) We now use Proposition 5.8 to show that T ( V ) has a universal property allowingus to lift antimodule morphisms from V to antimodule-algebra morphisms: Proposition 5.10.
Let V ∈ HMod , let A be an algebra in HMod , and let f : V → A be an antimodule map. Then there is a unique antimodule-algebra morphism f ♯ : T ( V ) → A such that f ♯ ◦ ι V = f .Proof. By part (b) of Proposition 5.8 implies that there is a unique homorphismof module-algebras ˇ f : T ( V ) → A such that ˇ f ◦ ϑ V = f . Then define f ♯ by f ♯ = ˇ f ◦ c T ( V ) , i.e. so that the top right triangle of the diagram T ( V ) T ( V ) V A c T ( V ) f ♯ ˇ fι V f commutes; recall that ϑ V = c T ( V ) ◦ ι V . The square commutes by the definingproperty of ˇ f , so the lower left triangle commutes as well, i.e. we have f ♯ ◦ ι V = f . Then f ♯ is an antimodule-algebra morphism by Lemma 5.4, and it is unique sinceit is uniquely determined on the generators ι V ( V ) of T ( V ). (cid:3) ∗ -structures on modules We now turn to the main theme of this chapter, which is ∗ -structures. Althoughour goal is to formulate the correct notion of a ∗ -algebra in the category HMod ,it is useful to consider modules first and then extend to algebras afterward. InSection 6.1 we begin by defining ∗ -modules and ∗ -morphisms. We show in Propo-sition 6.8 that ∗ -modules are the “fixed points up to homotopy” of the conjugationfunctor in the sense that the conjugation functor is naturally isomorphic to the iden-tity functor when restricted to the subcategory of HMod consisting of ∗ -modules.We then show that modules of the form V ⊗ V and V ⊗ V carry natural ∗ -structures;these ∗ -modules will be used later in Section 8 to formulate the notion of an innerproduct in our framework. We also discuss ∗ -submodules and quotients.6.1. The category HMod ∗ . The usual definition of a ∗ -structure on, say, a com-plex algebra A is an antilinear map ∗ : A → A such that ( a ∗ ) ∗ = a and ( ab ) ∗ = b ∗ a ∗ .Simply omitting the last condition then gives a reasonable definition of a ∗ -structureon a vector space. Translating this into our framework of complex conjugate mod-ules using the map Ψ V V leads to the following definition:
Definition 6.1.
We say that a module V ∈ HMod is a ∗ -module if V is equippedwith a module map ∗ : V → V , such that ∗ ◦ ∗ = σ V V : V → V . We refer to themap ∗ as a ∗ -structure on V .If W ∈ HMod is also a ∗ -module, then we say that a linear map T : V → W isa ∗ -morphism or a ∗ -map if T ◦ ∗ = ∗ ◦ T , i.e. if the following diagram commutes:(6.1) V ∗ −−−−→ V T y y T W −−−−→ ∗ W If T is also a module map then we may refer to it as a ∗ -module morphism or ∗ -module map . If v ∈ V and v ∗ = v then we will say that v is a self-adjoint element or just that v is self-adjoint . Remark 6.2.
It might seem more natural for a ∗ -structure on V to be a maporiginating from V rather than from V . However, since a ∗ -structure in the usualformulation is an antilinear map from V to V , Proposition 4.6 tells us that Definition6.1 is appropriate.It follows immediately from the requirement ∗ ◦ ∗ = σ V that the map ∗ itselfmust be an isomorphism of modules, hence we do not require it in the definition. Lemma 6.3. (a) For any ∗ -module V , id V is a morphism of ∗ -modules.(b) The composition of two ∗ -module morphisms is a ∗ -module morphism.Proof. Straightforward. (cid:3)
Definition 6.4.
The preceding lemma tells us that the collection of all ∗ -modulesin HMod together with all ∗ -module maps forms a subcategory of HMod . Wedenote this category by
HMod ∗ . -STRUCTURES ON MODULE-ALGEBRAS 25 Notation 6.5.
We will mostly not need to work with more than one ∗ -structureon any given module at a time, so we generally employ the notation ∗ for all ∗ -structures we consider. If necessary we will decorate the ∗ ’s with labels.If V ∈ HMod ∗ , we write v ∗ rather than ∗ ( v ). In this notation, the requirementthat ∗ ◦ ∗ = σ V means that (( v ) ∗ ) ∗ = v for all v ∈ V . It is quite cumbersometo carry around the ∗ in the superscript, and henceforth we will just write ∗ forthe morphism V → V as well as for its complex conjugate V → V . Thus thecondition of involutivity becomes the slightly more palatable (( v ) ∗ ) ∗ = v ; omittingparentheses gives the almost satisfactory v ∗∗ = v .We now show that out definition of a ∗ -structure is equivalent to the usualdefinition in terms of an involutive antilinear (antimodule) map: Proposition 6.6.
Let
V, W ∈ HMod .(a) If † : v v † is an antimodule map V → V , then ∗ = Ψ V V ( † ) : V → V is a ∗ -structure in the sense of Definition 6.1 if and only if ( v † ) † = v for all v ∈ V .(b) Let † : V → V and † : W → W be involutive antimodule maps with correspond-ing ∗ -structures as in (a). Then a linear map T : V → W is a ∗ -map if andonly if T ( v † ) = ( T v ) † for all v ∈ V .Proof. (a) From Proposition 4.6 we know that ∗ is a module map. For v ∈ V wehave v ∗ = Ψ V V ( † )( v ) = v † , so v ∗ = v † . Hence v ∗∗ = ( v † ) ∗ = ( v † ) † , so we see that v ∗∗ = σ V ( v ) = v if and only if ( v † ) † = v .(b) On the one hand we have T ( v ∗ ) = T ( v † ) , while on the other hand we have (cid:0) T ( v ) (cid:1) ∗ = (cid:0) T v (cid:1) ∗ = ( T v ) † , so we see that the diagram (6.1) commutes if and only if T ( v † ) = ( T v ) † . (cid:3) Remark 6.7.
The upshot of Proposition 6.6 is that we can work with involutiveantimodule maps rather than directly with Definition 6.1. This is convenient, espe-cially in the proofs below, where this approach allows us to avoid the complicationof carrying ’s around all over the place.
Proposition 6.8.
Let
V, W ∈ HMod ∗ .(a) The module V is a ∗ -module with the ∗ -structure ∗ : V → V .(b) If T : V → W is a morphism of ∗ -modules then T : V → W is a morphism of ∗ -modules with respect to the ∗ -structures defined in (a) .(c) Complex conjugation is an endofunctor of HMod ∗ , and moreover the mor-phisms ∗ : V → V give a natural isomorphism of the complex conjugationfunctor with the identity functor of HMod ∗ .Proof. (a) The complex conjugate morphism ∗ : V → V is a module morphism byProposition 4.3. Then we have (trivially and unenlighteningly) ∗ ◦ ∗ = ∗ ◦ ∗ = σ V = σ V , so ∗ is a ∗ -structure on V .(b) Follows immediately from taking the complex conjugate of the diagram (6.1).(c) Since each morphism ∗ : V → V is an isomorphism, the statement is exactlythe commutativity of the diagram (6.1). (cid:3) Building up the category HMod ∗ . Although we have defined the category
HMod ∗ , we have not exhibited any modules in it. In this subsection we show that C is a ∗ -module and that we can in fact construct a large class of ∗ -modules withoutknowing anything about the specific Hopf ∗ -algebra H that we are working with.We also discuss direct sums and tensor products of ∗ -modules. Lemma 6.9.
The map γ : C → C is a ∗ -structure on C .Proof. Follows immediately from Proposition 6.6. (cid:3)
Proposition 6.10.
Let V ∈ HMod .(a) The module V e = V ⊗ V is a ∗ -module with ∗ -structure given by ( v ⊗ w ) ∗ = w ⊗ v for v, w ∈ V .(b) The module e V = V ⊗ V is a ∗ -module with ∗ -structure given by ( v ⊗ w ) ∗ = w ⊗ v for v, w ∈ V .Proof. We show the proof for (a) only; (b) is similar. Let us define a map † : V e → V e by ( v ⊗ w ) † = w ⊗ v. It is clear that ( x † ) † = x for all x ∈ V e . Moreover we claim that † is an antimodulemap. For v, w ∈ V and a ∈ H we have( a ⊲ ( v ⊗ w )) † = [( a (1) ⊲ v ) ⊗ ( a (2) ⊲ w )] † = [( S ( a (1) ) ∗ ⊲ v ) ⊗ ( a (2) ⊲ w )] † = ( a (2) ⊲ w ) ⊗ ( S ( a (1) ) ∗ ⊲ v )= ( S ( a (2) ) ∗ ⊲ w ) ⊗ ( S ( a (1) ) ∗ ⊲ v )= S ( a ) ∗ ⊲ ( w ⊗ v )= S ( a ) ∗ ⊲ ( v ⊗ w ) † , which verifies the claim. Then Proposition 6.6 implies that ∗ = Ψ V e V e ( † ) is a ∗ -structure. (cid:3) Proposition 6.11.
For a family ( V j ) j ∈ J in HMod ∗ with ∗ -structures ∗ : V j → V j ,the direct sum L j ∈ J V j is a ∗ -module with the ∗ -structure given by (6.2) (( v j ) j ∈ J ) ∗ = ( v j ∗ ) j ∈ J . Proof.
For each j ∈ J , define a map † : V j → V j by † = Ψ − V j V j ( ∗ ). Each † is anantimodule map by Proposition 4.6, and ( v j † ) † = v j for all v j ∈ V j by Proposition6.6. -STRUCTURES ON MODULE-ALGEBRAS 27 Then we define † : L j ∈ J V j → L j ∈ J V j to be the direct sum of the maps † : V j → V j and observe that † is an involutive antimodule map since all of itsdirect summands are such. Finally, the map (6.2) is given by ∗ = Ψ ⊕ j V j , ⊕ j V j ( † ) , so it is a ∗ -structure by Proposition 6.6. (cid:3) Proposition 6.12.
For V ∈ HMod ∗ and n a positive integer, the module V ⊗ n isa ∗ -module with the ∗ -structure given by (6.3) ( v ⊗ · · · ⊗ v n ) ∗ = v n ∗ ⊗ · · · ⊗ v ∗ . Proof.
Let † = Ψ − V V ( ∗ ) : V → V be the involutive antimodule map correspondingto the given ∗ -structure, and then define another map † : V ⊗ n → V ⊗ n by( v ⊗ · · · ⊗ v n ) † = v † n ⊗ · · · ⊗ v † . This map is involutive by construction, and we claim that it is an antimodule mapas well. For a ∈ H we have[ a ⊲ ( v ⊗ · · · ⊗ v n )] † = [( a (1) ⊲ v ) ⊗ · · · ⊗ ( a ( n ) ⊲ v n )] † = ( a ( n ) ⊲ v n ) † ⊗ · · · ⊗ ( a (1) ⊲ v ) † = ( S ( a ( n ) ) ∗ ⊲ v † n ) ⊗ · · · ⊗ ( S ( a (1) ) ∗ ⊲ v † )= S ( a ) ∗ ⊲ ( v † n ⊗ · · · ⊗ v † )= S ( a ) ∗ ⊲ ( v ⊗ · · · ⊗ v n ) † , so the claim is verified. Finally, the map (6.3) is given by ∗ = Ψ V ⊗ n V ⊗ n ( † ), so it isa ∗ -structure by Proposition 6.6. (cid:3) Remark 6.13.
It seems that in general there is no way to define a ∗ -structureon the tensor product of two arbitrary ∗ -modules V and W . In order to do so wewould need a module map ∗ : V ⊗ W → V ⊗ W ; since V ⊗ W ≃ W ⊗ V , however,the tensor product of the ∗ -structures gives us a map ∗ : V ⊗ W → W ⊗ V . Insome situations there is a braiding on the category HMod (or
HMod f ), and ifthere is some compatibility between the braidings and the ∗ -structures, then morecan be said. We will revisit this issue later in Section 9.It seems also that there is no natural way to define a ∗ -structure on the dualof a ∗ -module. Indeed, if ∗ : V → V is a ∗ -structure, then the dual map is ∗ tr : V ∗ → V ∗ ∼ = ∗ V , and taking the complex conjugate of this gives a map V ∗ → ∗ V , which is not of the correct form to be a ∗ -structure.6.3. ∗ -submodules and quotients. We now prove the unsurprising fact that thequotient of a ∗ -module V by a submodule W inherits a ∗ -structure when W is stableunder the ∗ -operation, and we show also that the kernel and image of a ∗ -modulemorphism are ∗ -submodules. Definition 6.14.
Let V ∈ HMod ∗ and let W ⊆ V be a submodule. We say that W is a ∗ -submodule if W ∗ ⊆ W . (Note that involutivity of the ∗ -structure on V implies that W ∗ = W if W is a ∗ -submodule.) Lemma 6.15.
Let V ∈ HMod ∗ and let W be a ∗ -submodule. Then there is aunique ∗ -structure on the quotient V /W such that the quotient map q : V → V /W is a morphism of ∗ -modules, i.e. so that the following diagram commutes: (6.4) V ∗ −−−−→ V q y y q V /W −−−−→ ∗ V /W
Proof.
Let † : V → V be the involutive antimodule map corresponding to ∗ byProposition 6.6. The fact that W is a ∗ -submodule implies that W † ⊆ W . Then ( v + W ) † = v † + W is a well-defined involutive antimodule map on V /W , so Proposition6.6 (a) implies that the corresponding module map is a ∗ -structure. By definitionwe have q ( v ) † = q ( v † ), so q is a ∗ -map by part (b) of Proposition 6.6. (cid:3) Lemma 6.16.
Let
V, W ∈ HMod ∗ and let T : V → W be a ∗ -module map.(a) ker( T ) is a ∗ -submodule of V .(b) im( T ) is a ∗ -submodule of W .Proof. (a) For v ∈ ker( T ), we have T ( v ∗ ) = (cid:0) T ( v ) (cid:1) ∗ = (cid:0) T v (cid:1) ∗ = 0 , so v ∗ ∈ ker( T ), and hence ker( T ) is a ∗ -submodule of V.(b) For w ∈ im( T ), write w = T ( v ) for some v ∈ V . Then we have w ∗ = (cid:0) T v (cid:1) ∗ = T ( v ∗ ) ∈ im( T ) , so im( T ) is a ∗ -submodule of W . (cid:3) ∗ -structures on algebras In this section we extend to algebras our discussion of ∗ -structures on modulesfrom Section 6. We define ∗ -algebras and their morphisms and show how to con-struct ∗ -algebra structures from involutive antimodule-algebra maps. We discuss ∗ -ideals and quotients, and give a criterion for when an ideal generated by a sub-module of a ∗ -algebra is a ∗ -ideal. We show that ∗ -structures on a module can belifted to a ∗ -structure on its tensor algebra, and further that the the tensor algebrahas a universal mapping property for ∗ -module morphisms. We also make someobservations about ∗ -structures on algebras presented by generators and relations.7.1. Algebras in HMod ∗ .Definition 7.1. Suppose that A is an algebra in HMod and that A is equippedwith a ∗ -structure in the sense of Definition 6.1. Then we say that A is a ∗ -modulealgebra , or just a ∗ -algebra in HMod if ∗ : A → A is an algebra homomorphism,and we refer to ∗ as a ∗ -algebra structure on A .If A and B are ∗ -algebras in HMod , then a ∗ -homomorphism is an algebrahomomorphism f : A → B which is also a map of ∗ -modules. Proposition 7.2.
Suppose that A is an algebra in HMod and † : A → A is anantimodule map. Then ∗ = Ψ AA ( † ) is a ∗ -algebra structure on A if and only if † isan antimodule-algebra homomorphism satisfying ( a † ) † = a for all a ∈ A . -STRUCTURES ON MODULE-ALGEBRAS 29 Proof.
Proposition 6.6 (a) implies that ∗ is a ∗ -module structure on A if and onlyif † is an involutive antimodule map. Then Lemma 5.4 (c) implies that ∗ is amodule-algebra morphism if and only if † is an antimodule-algebra morphism. (cid:3) ∗ -ideals and quotients. Here we show that, as one might expect, the quo-tient of a ∗ -algebra by an ideal inherits a ∗ -structure exactly when the ideal isa ∗ -algebra. This will be useful in § ∗ -structures on algebrasdefined by generators and relations. Definition 7.3.
An ideal (two-sided) I in a ∗ -algebra A in HMod is called a ∗ -ideal if ( I ) ∗ ⊆ I . (Note that this implies that ( I ) ∗ = I because of involutivity.) Lemma 7.4.
Let A be a ∗ -algebra in HMod and suppose that the ideal I is a ∗ -submodule of A in the sense of Definition 6.14. With respect to the ∗ -structureon A/I defined in Lemma 6.15, the quotient map q : A → A/I is a morphism of ∗ -module algebras.Proof. The quotient map is a map of module-algebras by construction, and it is a ∗ -map by Lemma 6.15. (cid:3) Lemma 7.5.
Let A be a ∗ -algebra in HMod and let W ⊆ A be a ∗ -submodule.Then the two-sided ideal J W of A generated by W is a ∗ -submodule of A , and henceis a ∗ -ideal.Proof. The elements of J W are finite sums of terms of the form awb , where a, b ∈ A and w ∈ W . For x ∈ H we have x ⊲ ( awb ) = ( x (1) ⊲ a )( x (2) ⊲ w )( x (3) ⊲ b );since W is a submodule, the middle term x (2) ⊲ w is in W , so x ⊲ ( awb ) is in J W .Hence J W is a submodule.To see that J W is a ∗ -ideal, we need to show that ( awb ) ∗ ∈ J W for all a, b ∈ A and w ∈ W . We have ( awb ) ∗ = ( bw a ) ∗ = b ∗ w ∗ a ∗ ;since W is a ∗ -submodule, the middle term w ∗ is in W , so ( awb ) ∗ is in J W . Thus J W is a ∗ -ideal in A . (cid:3) ∗ -structures on T ( V ) . Here we show that a ∗ -structure on a module V ex-tends uniquely to the tensor algebra T ( V ), and we show that T ( V ) has a universalmapping property for ∗ -algebras equipped with module maps coming from V . Proposition 7.6.
Let V ∈ HMod ∗ and let T ( V ) be the tensor algebra of V .There is a unique ∗ -module algebra structure on T ( V ) such that the inclusion map ι V : V → T ( V ) is a ∗ -module morphism, i.e. such that the following diagramcommutes: (7.1) V ∗ −−−−→ V ι V y y ι V T ( V ) −−−−→ ∗ T ( V ) For an element v ⊗ · · · ⊗ v n ∈ V ⊗ n , we have (7.2) ( v ⊗ · · · ⊗ v n ) ∗ = v n ∗ ⊗ · · · ⊗ v ∗ . Hence the ∗ -structure on T ( V ) coincides with the direct sum of the ∗ -structures onthe modules V ⊗ n coming from Proposition 6.12.Proof. Let † : V → V be the involutive antimodule map corresponding to the given ∗ -structure on V . Now ι V ◦ † is an antimodule map V → T ( V ), so by Proposition5.10 it extends uniquely to an antimodule-algebra morphism, which we also denote † : T ( V ) → T ( V ). Note that †◦ † is a module-algebra endomorphism of T ( V ) whichis the identity on the generators, so it must be the identity map. Hence Proposition6.6 (a) implies that ∗ = Ψ T ( V ) T ( V ) ( † ) is a ∗ -structure on T ( V ).The fact that ι V is a ∗ -map follows from commutativity of the diagram V T ( V ) V T ( V ) V T ( V ) ι V ∗ c − V c − T ( V ) ∗ ι V † † ι V The upper trapezoid commutes by definition of the complex conjugate of ι V . Thetriangle on the right commutes by the definition of ∗ . The lower trapezoid commutesby Proposition 5.10. The triangle on the left commutes by definition of † . Thusthe large rectangle commutes as well, which gives (7.1).The ∗ -structure on T ( V ) is unique with this property since T ( V ) is generatedas an algebra by ι V ( V ), and (7.1) uniquely specifies ∗ on the generators.The equation (7.2) follows from the fact that ∗ is an algebra map together withthe description (5.2) of multiplication in the complex conjugate of an algebra. Wesee that (7.2) is identical to (6.3), so this ∗ -structure on T ( V ) is indeed the directsum of the star structures coming from Proposition 6.12. (cid:3) Remark 7.7.
We now briefly discuss ∗ -structures on algebras given by generatorsand relations. This means the following: we take a module V (the generators)and a submodule W ⊆ T ( V ) (the relations), and form the quotient algebra A = T ( V ) /J W , where J W is the 2-sided ideal generated by W as in Lemma 7.5. If V is a ∗ -module, then T ( V ) is a ∗ -algebra, and we would like to know when this ∗ -structure descends to A . By Lemma 7.4, it is sufficient for J W to be a ∗ -ideal.Then by Lemma 7.5, it is sufficient that W is a ∗ -submodule of T ( V ). Accordingto Lemma 6.16, this happens, for instance, when W is the kernel or image ofa morphism of ∗ -modules. While these observations are more or less trivial inthe abstract, they may be useful later on in specific circumstances when checkingrelations by hand is complicated. Proposition 7.8.
Let V ∈ HMod ∗ , and endow T ( V ) with the corresponding ∗ -structure as described in Proposition 7.6. Let A be a ∗ -algebra in HMod and let f : V → A be a ∗ -module map. Then the morphism ˜ f : T ( V ) → A from Lemma5.7 is in fact a morphism of ∗ -algebras.Proof. We know from Lemma 5.7 that ˜ f is a morphism of module-algebras, so weonly need to show that ˜ f respects the ∗ -structures on T ( V ) and A . We claim that -STRUCTURES ON MODULE-ALGEBRAS 31 the following diagram commutes: T ( V ) T ( V ) V VA A ∗ e f ι V ι V e f ∗ f f ∗ Indeed, the upper trapezoid commutes by Proposition 7.6, the triangle on the rightcommutes by Lemma 5.7, the lower trapezoid commutes since f is a ∗ -map, andthe triangle on the left is just the complex conjugate of the triangle on the right.Thus the large rectangle commutes also, which means that ˜ f is a ∗ -map. (cid:3) Inner products and adjoints
We now turn to a discussion of inner products. In the traditional formulation,an inner product on a complex vector space V is a map ( , ) : V × V → C which isantilinear in the first variable, linear in the second variable, satisfies ( v, w ) = ( w, v )for v, w ∈ V , and which is positive definite. This definition, like that of a ∗ -structure, does not lend itself easily to a module-theoretic approach because of theantilinearity. Inner products allow one to introduce the notion of adjoint lineartransformations; this too is an antilinear concept. In this section we develop thesenotions in our framework.Although so far we have discussed ∗ -structures only on modules, not on arbitrarycomplex vector spaces, the discussion extends to complex vector spaces just bytaking H = C with ∗ -structure given by complex conjugation. In particular, wecan speak of ∗ -vector spaces and morphisms of ∗ -vector spaces.8.1. Inner products.
We now formulate the definition of an inner product in thelanguage of complex conjugate vector spaces and modules:
Definition 8.1.
For V ∈ C Vect we define an inner product on V to be a linearmap h , i : V e = V ⊗ V → C such that(a) With respect to the ∗ -structures on V e and C constructed in Proposition 6.10and Lemma 3.10, respectively, h , i is a morphism of ∗ -vector spaces.(b) h v, v i ≥ v ∈ V , and h v, v i = 0 only if v = 0.If in addition V is an H -module and if h , i is a module map, then we say that h , i is an inner product in HMod ; in this case we will call V together with thisinner product a Hermitian module . If V and W are Hermitian modules with innerproducts h , i V and h , i W , respectively, then we say that a module map T : V → W is an isometry if (cid:10) T u, T v (cid:11) W = h u, v i V for all u, v ∈ V . We say that T is unitary if T is an invertible isometry. Remark 8.2.
Untangling the definitions shows that requiring h , i to be a ∗ -mapmeans just that h v, w i = h w, v i for all v, w ∈ V .As we have done in previous sections, we could make the Hermitian modules intoa category where the morphisms are the isometries. This is somewhat restrictive, as the isometry condition forces all morphisms to be injective, and for our purposeswe don’t need to consider this category separately, so we omit the definition. Notation 8.3.
As we did in Definition 8.1, we will frequently abuse notation bywriting h ( w, x ) rather than h ( w ⊗ x ) when h : W ⊗ X → Y is a linear map. Thisis nothing more than the canonical identification of bilinear maps W × X → Y with linear ones W ⊗ X → Y , and we will make this identification without furthercomment from now on.We now show that Definition 8.1 is equivalent to the usual definition of aninner product, and we formulate the appropriate notion of H -invariance for thetraditional incarnation ( , ) : V × V → C of the inner product in the case when V is an H -module: Proposition 8.4.
Let V ∈ C Vect and let h , i : V ⊗ V → C be a linear map.Define a function ( , ) : V × V → C by ( v, w ) = h v, w i for v, w ∈ V . Then(a) h , i is an inner product in the sense of Definition 8.1 if and only if ( , ) satisfiesthe usual definition of an inner product given at the beginning of § V ∈ HMod , then h , i is a module map if and only if ( , ) is H -invariant inthe sense that ( a ⊲ v, w ) = ( v, a ∗ ⊲ w ) for all v, w ∈ V and a ∈ H .Proof. (a) Since v v is antilinear, it is clear that ( v, w ) is antilinear in v , andlinearity in w is also clear. The formulations of positive-definiteness for h , i and ( , ) are also clearly equivalent. We claim now that h , i is a ∗ -map, i.e. thatthe diagram V ⊗ V h , i −−−−→ C ∗ y y γ V ⊗ V h , i −−−−→ C commutes, if and only if ( , ) satisfies conjugate-symmetry. For v, w ∈ V ,chasing v ⊗ w right and then down in the diagram gives γ ( c C ( h v, w i )) = h v, w i def = ( v, w ) . On the other hand, chasing v ⊗ w down and then right gives (cid:10) v ⊗ w ∗ (cid:11) = h w, v i def = ( w, v ) , so our claim is verified.(b) First, observe that for H -modules W and X , a linear map h : W ⊗ X → C (i.e. a bilinear form) is a module map if and only if h ( a ⊲ w, x ) = h ( w, S ( a ) ⊲ x )for all w ∈ W and x ∈ X . Then for v, w ∈ V and a ∈ H we have( a ⊲ v, w ) = h a ⊲ v, w i = h S ( a ) ∗ ⊲ v, w i , while on the other hand we have( v, a ∗ ⊲ w ) = h v, a ∗ ⊲ w i . Thus ( a ⊲ v, w ) = ( v, a ∗ ⊲ w ) for all a ∈ H if and only if h S ( a ) ∗ ⊲ v, w i = h v, a ∗ ⊲ w i for all a ∈ H . Replacing a with S ( a ) ∗ and using (2.1), the latterequation becomes h a ⊲ v, w i = h v, S ( a ) ⊲ w i . Then our first observation con-cludes the proof. (cid:3) -STRUCTURES ON MODULE-ALGEBRAS 33 Proposition 8.5.
Suppose that V ∈ HMod is a Hermitian module. Then µ V : V → V ∗ given by µ V ( v ) = h v, −i is an injective module map, and µ V is an isomor-phism if V is finite-dimensional. If W ∈ HMod is another Hermitian module andthe module map T : V → W is an isometry, then the following diagram commutes: (8.1) V T −−−−→ W µ V y y µ W V ∗ ←−−−− T tr W ∗ Remark 8.6.
We caution the reader that µ V is not intrinsic to V ; it is only definedrelative to a fixed inner product. Proof of Proposition 8.5.
Since µ V = Ψ V V ∗ ( v
7→ h v, −i ), we need to show that v
7→ h v, −i is an antimodule map. For a ∈ H and v, w ∈ V we have h a ⊲ v, w i = h S ( a ) ∗ ⊲ v, w i = h v, S ( S ( a ) ∗ ) ⊲ w i = ( S ( a ) ∗ ⊲ h v, −i )( w ) , where for the second equality we used the fact that h , i is a module map, and forthe third we used the definition (2.3) of the H -action on V ∗ . Hence µ V is a modulemap by Proposition 4.6. Injectivity follows immediately from the fact that h , i ispositive-definite, and then µ V must be an isomorphism when V is finite-dimensionalbecause the dimensions of V and V ∗ coincide.Finally, for any u, v ∈ V , we have[ T tr µ W T ( u )]( v ) = [ µ W ( T u )](
T v )= (cid:10) T u, T v (cid:11) = h u, v i = [ µ V ( u )]( v ) , so T tr µ W T = µ V , and hence (8.1) commutes. (cid:3) Remark 8.7.
Many authors define an inner product to be linear in the first variableand antilinear in the second. While this makes no difference for vector spaces,it matters for modules: if we were to translate this alternate definition into ourframework, an inner product would become a positive-definite ∗ -module map h , i : V ⊗ V → C . With this definition, Proposition 8.5 would instead give an isomorphism V ≃ ∗ V . All of the subsequent theory would carry over, mutatis mutandis , but weomit it here for the sake of brevity.8.2. Some remarks on positivity.
We have seen in the proof of Proposition8.4 that the conjugate-symmetry condition ( v, w ) = ( w, v ) is encapsulated in therequirement that h , i respects the ∗ -structures on V e = V ⊗ V and C . Thus we canexpress conjugate-symmetry entirely as a condition on maps in the category ratherthan as a condition involving elements. This raises the question of whether thepositivity criterion for an inner product can also be phrased in such a manner. Itappears that the answer is no, although this may be a failure of imagination ratherthan an insight into necessity. One way to avoid this question entirely would be to include the notion of “positive cone” as part of the data when discussing innerproducts. We will begin with an example and then give a general definition.The fact that h , i is a ∗ -map implies that any self-adjoint element of V e is mappedto R (the self-adjoint part of C ). Positivity is a further restriction: it requires thatthe particular self-adjoint elements in V e of the form v ⊗ v are mapped into thenon-negative real numbers R + . If we denote by ( V e ) + the R + -span of the vectors v ⊗ v , then it is not difficult to convince oneself that ( V e ) + spans the whole self-adjoint part of V e and that ( V e ) + ∩ − ( V e ) + = { } . The non-negative reals R + share the same properties in C . Then the inner product is a ∗ -map which whichtakes ( V e ) + into R + . This motivates the following definition.Consider the category HMod ∗ + of pairs ( V, V + ) where V is a ∗ -module and V + , the positive cone of V , is a subset consisting of self-adjoint elements of V which is closed under addition and multiplication by R + . We also require that V + ∩ ( − V + ) = { } and that V + − V + = V sa , the real subspace of self-adjointelements of V . A morphism T : ( V, V + ) → ( W, W + ) in HMod ∗ + is a ∗ -modulemap T : V → W such that T ( V + ) ⊆ W + . Then there is a functor from HMod to HMod ∗ + given on objects by V V e = V ⊗ V and on morphisms by T T ⊗ T ,and we define the positive cone of V e to be the R + -span of vectors of the form v ⊗ v for v ∈ V , as above. The pair ( C , R + ) is an object of HMod ∗ + , and then an innerproduct on V would be defined to be a morphism h , i in HMod ∗ + from V e to C ,with positive cones as above.The necessity to define the positive cone and to include that as part of the datamanifests itself in other situations where there may be more than one relevantnotion of positivity in a single vector space. For example, in a ∗ -algebra A (inthe usual sense, not in the sense presented here, although the discussion could beformulated in our setting as well) one can define the positive cone A + to be the R + -span of elements of the form a ∗ a . Then we can say that a linear functional f ∈ A ∗ is positive if f ( A + ) ⊆ R + , and finally we can define another positive conein A by declaring A ++ to be the collection of self-adjoint elements of A which aremapped to R + by all positive functionals (and then check that this forms a cone).Certainly A + ⊆ A ++ , and there is equality if, for instance, A is a C ∗ -algebra, butin general these notions will not coincide.Although this would give an even more structural approach to the study of innerproducts, for our purposes we don’t need to make things quite so abstract, so weconfine this construction to this subsection. We caution also that this approach hasnot been carefully considered by the author, and some tweaking of the conditionsabove may be necessary to develop the approach rigorously.8.3. Adjoints of linear operators.
Now we show how to use our notion of innerproducts to frame adjoints of linear maps in our language. We begin with somemotivational discussion on the usual formulation of adjoints. For this we restrictto finite-dimensional modules.
Remark 8.8.
When V and W are finite-dimensional inner product spaces with(ordinary, sesquilinear) inner products ( , ) V and ( , ) W , respectively, one definesthe adjoint of a linear transformation T : V → W to be the unique linear map T † : W → V satisfying(8.2) (cid:0) T † w, v (cid:1) V = ( w, T v ) W for all v ∈ V and w ∈ W . Then one shows that the following properties hold: -STRUCTURES ON MODULE-ALGEBRAS 35 (a) The map Hom( V, W ) → Hom(
W, V ) given by T T † is antilinear.(b) Taking the adjoint again gives ( T † ) † = T in Hom( V, W ).(c) If X is another finite-dimensional inner product space, then for any linear map U : W → X , we have ( U T ) † = T † U † .Rather than repeating the construction of the adjoint operator from scratch, we usethe correspondence from Proposition 8.4 between sesquilinear inner products andinner products in our sense in order to transport the traditional notion of adjointgiven above into our setting. Notation 8.9.
We make the following notational convention: for a Hermitianmodule (or just complex vector space) V with inner product h , i V in the sense ofDefinition 8.1, we will always denote by ( , ) V the corresponding positive-definite,conjugate-symmetric, sesquilinear form given by( v, w ) V = h v, w i V for v, w ∈ V , as discussed in Proposition 8.4. With this convention in mind, trans-lating the defining property (8.2) of the adjoint transformation into our languagegives(8.3) D T † w, v E V = h w, T v i W for v ∈ V and w ∈ W .In Proposition 8.10 we explore some properties of the adjoint map T † . Part (a)shows that the adjoint of a module map is a module map. Part (b), while somewhattechnical in its statement, is really just formalizing the notion that the adjoint of alinear map is its conjugate transpose. Proposition 8.10.
Let
V, W ∈ HMod f be finite-dimensional Hermitian modules,let T : V → W be a linear map, and let T † : W → V be the adjoint of T . Then:(a) T is a module map if and only if T † is a module map.(b) T † coincides with the composition W σ − W −−−−→ W µ W −−−−→ W ∗ T tr −−−−→ V ∗ µ − V −−−−→ V σ V −−−−→ V Proof. (a) Since ( T † ) † = T , we only need to do one direction of the proof. Assum-ing that T is a module map, we need to show that T † ( a ⊲ w ) = a ⊲ ( T † w ) for w ∈ W and a ∈ H . For any v ∈ V , we have (cid:0) T † ( a ⊲ w ) , v (cid:1) V = ( a ⊲ w, T v ) W = ( w, a ∗ ⊲ ( T v )) W = ( w, T ( a ∗ ⊲ v )) W = ( T w, a ∗ ⊲ v ) V = ( a ⊲ ( T w ) , v ) V , using (8.2), Proposition 8.4 (b), and the fact that T is a module map. Sincethis holds for all v ∈ V , we conclude that T † ( a ⊲ w ) = a ⊲ ( T † w ), so T † is amodule map.(b) Recall that c W ∗ is the canonical antimodule map from W ∗ to its conjugate.For w ∈ W , we have µ W ( σ − W ( w )) = c W ∗ ( h w, −i W ) . Applying T tr to this gives T tr ( c W ∗ ( h w, −i W )) = c V ∗ ( h w, T ( − ) i W )= c V ∗ (cid:16)D T † w, − E V (cid:17) . Finally, applying σ V µ − V now gives σ V µ − V (cid:16) c V ∗ (cid:16)D T † w, − E V (cid:17)(cid:17) = σ V ( T † w )= T † w, which proves our claim. (cid:3) We now turn to the module-theoretic properties of the map T T † . Recallfrom Definition 2.12 that we have two actions of H on each of the linear spacesHom( V, W ) and Hom(
W, V ) which we called left and right . We know that takingthe adjoint is an antilinear map Hom(
V, W ) → Hom(
W, V ), but a priori it is notclear whether this should be an antimodule map for the left or right actions on thetwo Hom-spaces.It turns out that we need to use the left action for both of them. The point hereis that since we are dealing with finite-dimensional modules, we have isomorphisms V ≃ V ∗ and Hom ℓ ( V, W ) ≃ W ⊗ V ∗ from Propositions 8.5 and 2.13, respectively.Combining these gives an isomorphism Hom ℓ ( V, W ) ≃ W ⊗ V , and similarly wehave Hom ℓ ( W, V ) ≃ V ⊗ W . As in Remark 8.7, we can see that this is a result ofour convention that inner products are maps defined on V ⊗ V ; if we had chosen theopposite convention then we would have Hom r instead of Hom ℓ in the following: Proposition 8.11.
Let
V, W ∈ HMod f be finite-dimensional Hermitian modules.(a) The map Hom ℓ ( V, W ) → Hom ℓ ( W, V ) given by T T † is an anti-isomorphismof modules.(b) The map ∗ : Hom ℓ ( V, W ) → Hom ℓ ( W, V ) associated to T T † by Proposition4.6 is an isomorphism of modules.Proof. (a) We need to show that ( a ⊲ T ) † = S ( a ) ∗ ⊲ T † for T ∈ Hom ℓ ( V, W ) and a ∈ H , where the action of H on Hom ℓ ( V, W ) is given by (2.5). We will do thisby showing that (cid:0) ( a ⊲ T ) † w, v (cid:1) V = (cid:0) ( S ( a ) ∗ ⊲ T † ) w, v (cid:1) V for all v ∈ V , w ∈ W , and a ∈ H . Then the result will follow from nondegener-acy of the sesquilinear form. For the sake of readability, we omit the ⊲ symbolsfor the H -actions on V and W in the following. Beginning with the right-handside, we have (cid:0) ( S ( a ) ∗ ⊲ T † ) w, v (cid:1) V = (cid:0) S ( a (2) ) ∗ T † S ( S ( a (1) ) ∗ ) w, v (cid:1) V = (cid:16) T † a ∗ (1) w, S ( a (2) ) v (cid:17) V = (cid:0) w, a (1) T S ( a (2) ) v (cid:1) W = ( w, ( a ⊲ T ) v ) W = (cid:0) ( a ⊲ T ) † w, v (cid:1) V , where we used the property (8.2) of the adjoint, as well as the invariance of( , ) V and ( , ) W under the H -action described in part (b) of Proposition 8.4. -STRUCTURES ON MODULE-ALGEBRAS 37 (b) Follows immediately from (a) together with Proposition 4.6. (cid:3) Remark 8.12.
Although the notation is suggestive, the map ∗ : Hom ℓ ( V, W ) → Hom ℓ ( W, V ) is not a ∗ -structure in the sense of Definition 6.1 because it is not ofthe form ∗ : Y → Y for a module Y except when V = W . We address this casenow. Since we have not yet shown that End ℓ ( V ) is an H -module algebra, we takecare of this detail first. Proposition 8.13.
For V ∈ HMod , the algebra
End ℓ ( V ) is an algebra in HMod .Proof.
We need to show that the conditions (5.1) hold. For x, y ∈ End ℓ ( V ) and a ∈ H we have ( a (1) ⊲ x )( a (2) ⊲ y ) = a (1) xS ( a (2) ) a (3) yS ( a (4) )= a (1) xε ( a (2) ) yS ( a (3) )= a (1) xyS ( ε ( a (2) ) a (3) )= a (1) xyS ( a (2) )= a ⊲ ( xy ) , so multiplication in End ℓ ( V ) is a module map. For the unit map, we write 1 = id V ;we have a ⊲ a (1) S ( a (2) ) = ε ( a )1 , so the unit is also a module map, which concludes the proof. (cid:3) Proposition 8.14.
Let V ∈ HMod f be a finite-dimensional Hermitian module.Then the map ∗ : End ℓ ( V ) → End ℓ ( V ) from Proposition 8.11 makes End ℓ ( V ) intoa ∗ -algebra in HMod .Proof.
We know already from Proposition 8.13 that End ℓ ( V ) is an H -module alge-bra. We know also from Proposition 8.11 that T T † is an anti-isomorphism ofmodules, and according to the properties (b) and (c) listed in Remark 8.8, we seethat this map is actually an involutive antimodule-algebra morphism. Thus ∗ is a ∗ -structure by Proposition 7.2. (cid:3) Remark 8.15.
As in the discussion preceding Proposition 8.11, note that End ℓ ( V ) ≃ V ⊗ V , which we denoted by e V in § ∗ -structure on End ℓ ( V )from Proposition 8.14 corresponds through this isomorphism with the ∗ -structureon e V defined in Proposition 6.10.8.4. Inner products, ∗ -structures, and bilinear forms. Let V ∈ HMod f . Inthis subsection we explore the relationship between inner products, ∗ -structures,and bilinear forms on V .By definition, a ∗ -structure on V is an isomorphism ∗ : V → V with a certainextra property (namely involutivity). If V is also a Hermitian module, then we candefine a bilinear form h : V ⊗ V → C by h ( v ∗ , w ) = h v, w i . In other words, wedefine the bilinear form so that the following diagram commutes:(8.4) V ⊗ V V ⊗ V C ∗ ⊗ id h , i h Since ∗ is an isomorphism and since the inner product is nondegenerate, the bilinearform h will be nondegenerate as well. Note that h is a morphism in HMod . Remark 8.16. A ∗ -structure on V gives an isomorphism V ≃ V . An inner producton V gives an isomorphism V ≃ V ∗ via v
7→ h v, −i ; see Proposition 8.5. Similarly,a nondegenerate bilinear form h : V ⊗ V → C gives an isomorphism V ≃ V ∗ via v h ( v, − ).We saw above that having a ∗ -structure and an inner product allowed us toobtain a bilinear form. This corresponds to composing the corresponding isomor-phisms V ≃ V and V ≃ V ∗ to obtain the isomorphism V ≃ V ∗ associated to thebilinear form. We would now like to ask if there is a “two-out-of-three” type result,i.e. whether having any two of a bilinear form, an inner product, and a ∗ -structure,allows us to obtain the third one.In the discussion preceding this remark, we got the bilinear form for free. Butwe will see that in fact this bilinear form necessarily satisfies some conditions, so itis not the case that, for example, an arbitrary bilinear form plus an inner productwill give a ∗ -structure. Similarly, we cannot expect an arbitrary bilinear form plusa ∗ -structure to give an inner product. Thus a completely general two-out-of-threeresult is not possible. But if we restrict the bilinear forms we consider, we do obtainsome results. Proposition 8.17.
Let V ∈ HMod f and let ∗ : V → V , h , i : V ⊗ V → C , and h : V ⊗ V → C be module maps such that the diagram (8.4) commutes (we do notassume ∗ to be a ∗ -structure nor h , i to be an inner product). Then(a) If h , i is an inner product on V , then ∗ is a ∗ -structure on V if and only if h ( w ∗ , v ∗ ) = h ( v, w ) for all v, w ∈ V .(b) If ∗ is a ∗ -structure on V , then h , i is an inner product on V if and only if h satisfies the conditions h ( w ∗ , v ∗ ) = h ( v, w ) for v, w ∈ V , h ( v ∗ , v ) ≥ for v ∈ V , and h ( v ∗ , v ) = 0 only if v = 0 .Proof. Consider the following diagram:(8.5) V ⊗ V V ⊗ V V ⊗ VV ⊗ V V ⊗ V V ⊗ V C C ρ V V ∗ ⊗ id h , i id ⊗ σ V id ⊗∗ ∗ ⊗ id h , i ρ V V h id ⊗∗∗ ⊗ ∗ hγ Certain parts of this diagram commute automatically. The triangle on the right-hand side commutes since it is exactly (8.4), and the triangle on the left-hand side isthe complex conjugate of that on the right, so it commutes as well. The trapezoid onthe upper left commutes by Proposition 4.8. The small interior triangle commutestrivially.This leaves the triangle on the upper right, the pentagon, and the large rectangle.The triangle on the upper right commutes if and only if ∗ is a ∗ -structure. The -STRUCTURES ON MODULE-ALGEBRAS 39 pentagon commutes if and only if h ( w ∗ , v ∗ ) = h ( v, w ) for all v, w ∈ V . Notingthat the composition across the top of (8.5) is exactly the ∗ -structure on V ⊗ V described in Proposition 6.10, we see that the large rectangle commutes if and onlyif h is a ∗ -map.If h , i is an inner product, it is a ∗ -map, and hence the large rectangle commutes.Then the upper right triangle commutes if and only if the pentagon commutes, i.e. ∗ is a ∗ -structure if and only if h ( w ∗ , v ∗ ) = h ( v, w ) for all v, w ∈ V . This proves (a).On the other hand, if ∗ is a ∗ -structure, then the upper right triangle commutes.Thus the large rectangle commutes if and only if the pentagon commutes, i.e. h , i is a ∗ -map if and only if h ( w ∗ , v ∗ ) = h ( v, w ) for all v, w ∈ V . From (8.4) we have h ( v ∗ , w ) = h v, w i , so we see that h , i satisfies the appropriate positivity requirementif and only if h ( v ∗ , v ) ≥ v ∈ V and h ( v ∗ , v ) = 0 only if v = 0. This proves(b). (cid:3) Remark 8.18.
We now explain the meaning of the condition h ( w ∗ , v ∗ ) = h ( v, w ).We said in the proof of Proposition 8.17 that this condition is equivalent to com-mutativity of the pentagon in the diagram (8.5). If ∗ is a ∗ -structure on V , thenthe composition ( ∗ ⊗ ∗ ) ◦ ρ V V (i.e. the top of the pentagon) is the ∗ -structure on V ⊗ V described in 6.12 (for n = 2). Then the statement that the pentagon com-mutes means exactly that h : V ⊗ V → C is a ∗ -map. We could also formulate thepositivity criterion h ( v ∗ , v ) ≥ § ∗ -structures and braidings In this section we examine the interaction between ∗ -structures, R-matrices,and braidings. We show that the complex conjugate of a braiding on the modulecategory of a Hopf ∗ -algebra is again a braiding. When the braiding comes froman R-matrix, we show how conditions on the R-matrix translate into relationsbetween the braiding and its conjugate, and relations between the braiding and the ∗ -structure on modules. In particular, when the R-matrix is real (see Definition9.10), then the braiding of V with itself is a module map for any ∗ -module V .9.1. Braidings and their complex conjugates.
In this subsection we define abraiding and show that the complex conjugate of a braiding on a category of H -modules gives another braiding. Although the notion of braiding makes sense forany monoidal category, we define it here only for HMod (or for a sub-monoidal cat-egory of
HMod ). This simplifies our presentation because we already understandthe monoidal structure well: it is the tensor product of vector spaces over C . Thuswe can and do suppress the associativity isomorphisms U ⊗ ( V ⊗ W ) ≃ ( U ⊗ V ) ⊗ W . Definition 9.1.
Let C be a subcategory of HMod containing C and such that V ⊗ W ∈ C for all V, W ∈ C (a sub-monoidal category ). A braiding on C is acollection of isomorphisms ψ V W : V ⊗ W → W ⊗ V (in C ) for all V, W ∈ C satisfying the conditions(9.1) ψ U,V ⊗ W = (id V ⊗ ψ UW )( ψ UV ⊗ id W ) , ψ U ⊗ V,W = ( ψ U,W ⊗ id V )(id U ⊗ ψ V W ) , and such that for any modules U, V, W, X and any morphisms f : U → W and g : V → X in C , the following diagram commutes:(9.2) U ⊗ V ψ UV −−−−→ V ⊗ U f ⊗ g y y g ⊗ f W ⊗ X −−−−→ ψ WX X ⊗ W We refer to the property (9.2) as naturality of the braiding. We refer to the relations(9.1) as the hexagon axioms ; the reason for this is that the relevant diagrams wouldeach form a hexagon if we did not suppress the associators.
Remark 9.2.
A fancier way to define a braiding is as follows. One can form theCartesian product category
C × C . Then the tensor product in C defines a functor N : C × C → C , which takes a pair of objects (
U, V ) to their tensor product U ⊗ V ,and a pair of morphisms ( f, g ) to the tensor product morphism f ⊗ g . The factthat the tensor product is associative means that there is a natural isomorphismbetween the functors N ◦ ( N × id) and N ◦ (id × N ) from C × C × C to C . Similarly,there is a functor N op which takes the pairs ( U, V ) and ( f, g ) to V ⊗ U and g ⊗ f ,respectively. Then a braiding can be defined as a natural isomorphism between thefunctors N and N op .It follows from the axioms that the braidings ψ C V and ψ V C are compatible withthe canonical identifications of C ⊗ V and V ⊗ C with V . We refer to § § .
1, assume that C is subcategory of HMod which is equipped with a braiding ψ = ( ψ V W ) V,W ∈C as in Definition 9.1. Suppose furthermore that C is closed undercomplex conjugation, i.e. that V ∈ C for all V ∈ C and f is a morphism in C for allmorphisms f in C . Finally, assume that the isomorphisms ρ V W : V ⊗ W → W ⊗ V and σ V : V → V from Proposition 4.8 and Lemma 3.11, respectively, along withtheir inverses, are in C as well.We now define a new family of isomorphisms obtained from the original braiding ψ by conjugation. As this is somewhat technical, we make the definition in twostages. First, for each X, Y ∈ C , we define an isomorphism ξ XY : X ⊗ Y → Y ⊗ X via the composition(9.3) ξ XY : X ⊗ Y ρ − Y X −−−−→ Y ⊗ X ψ XY −−−−→ X ⊗ Y ρ XY −−−−→ Y ⊗ X. Note that our assumptions ensure that ξ XY is a morphism in C .We have now provided an isomorphism ξ XY between the modules X ⊗ Y and Y ⊗ X for any X and Y . In order to define the conjugated braiding, we use thefact that every module is naturally isomorphic to the conjugate of some module(namely to the conjugate of its conjugate): Definition 9.3.
For any
X, Y ∈ C we define an isomorphism ψ XY (not to beconfused with ψ XY ) from X ⊗ Y to Y ⊗ X via the composition(9.4) ψ XY : X ⊗ Y σ − X ⊗ σ − Y −−−−−−→ X ⊗ Y ξ X Y −−−−→ Y ⊗ X σ Y ⊗ σ X −−−−−→ Y ⊗ X. We refer to the family ψ = ( ψ XY ) X,Y ∈C as the conjugate braiding to ψ . -STRUCTURES ON MODULE-ALGEBRAS 41 As the name suggests, the family ψ is also a braiding on C . Verifying the detailsis a little tedious. We begin by proving the relevant properties of the maps ξ XY before using them to establish the properties of ψ . First we show that the ξ XY ’sare natural with respect to module maps. Lemma 9.4.
Let
U, V, W, X ∈ C and let f : U → W and g : V → X be morphismsin C . Then the following diagram commutes: (9.5) U ⊗ V ξ UV −−−−→ V ⊗ U f ⊗ g y y g ⊗ f W ⊗ X −−−−→ ξ WX X ⊗ W Proof.
This follows from (9.2) together with (4.2) (used twice). (cid:3)
The harder part of showing that ψ is a braiding is checking that the hexagonaxioms (9.1) hold. We now prove a technical lemma, which will assist us in checkingthese relations. Lemma 9.5.
Let
U, V, W ∈ C . The following diagram commutes: (9.6) U ⊗ V ⊗ W V ⊗ W ⊗ UU ⊗ W ⊗ V W ⊗ U ⊗ V W ⊗ V ⊗ U ξ U,V ⊗ W id U ⊗ ρ V W ρ V W ⊗ id U ξ UW ⊗ id V id W ⊗ ξ UV Proof.
Commutativity of (9.6) follows from commutativity of the diagram (9.7).Indeed, expanding all of the ξ ’s in (9.6) according to the definition gives the outeredges of (9.7), so if (9.7) commutes then (9.6) will as well.(9.7) U ⊗ V ⊗ W V ⊗ W ⊗ U U ⊗ V ⊗ W V ⊗ W ⊗ U W ⊗ V ⊗ UV ⊗ U ⊗ WU ⊗ W ⊗ V W ⊗ U ⊗ V U ⊗ W ⊗ V W ⊗ U ⊗ V W ⊗ V ⊗ U W ⊗ U ⊗ V ρ − id ⊗ ρ ψ id ⊗ ψρ ρ ρ ρ ⊗ id ψ ⊗ id ρ ρρ − ⊗ id ψ ⊗ id ρ ⊗ id id ⊗ ρ − id ⊗ ψ id ⊗ ρ We have omitted the subscripts labeling the objects on the ρ ’s and ψ ’s for legibility.This should cause no confusion because there is only one choice of subscripts in eachcase that matches the domain and codomain of each arrow.It is left to prove that the diagram above commutes. We begin at the left-handside and move to the right. The rectangle commutes by (4.3). The trapezoidcommutes by (4.2). The upper triangle commutes since ψ is a braiding. The lowertriangle commutes by (4.3). The quadrilateral commutes by (4.2). Finally, thetriangle on the right-hand side commutes by (4.3). This completes the proof of thelemma. (cid:3) We are now ready to prove our first main result in this section:
Proposition 9.6.
The conjugate braiding ψ is a braiding on C .Proof. First we check the naturality property (9.2). Let
U, V, W, X ∈ C and let f : U → W and g : V → X be morphisms in C . Then we claim that the followingdiagram commutes (where we have removed most of the subscripts on the maps forlegibility):(9.8) U ⊗ V σ − ⊗ σ − −−−−−−→ U ⊗ V ξ U V −−−−→ V ⊗ U σ ⊗ σ −−−−→ V ⊗ U f ⊗ g y f ⊗ g y y g ⊗ f y g ⊗ f W ⊗ X −−−−−−→ σ − ⊗ σ − W ⊗ X −−−−→ ξ W X X ⊗ W −−−−→ σ ⊗ σ X ⊗ W Each of the two outer squares is a tensor product of two diagrams of the form(3.4), so they commute by Lemma 3.11. The central square commutes by Lemma9.4 applied to the modules
U , V , W , X and the morphisms f and g . Noting thatthe compositions across the top and bottom of (9.8) are exactly ψ UV and ψ W X ,respectively, this completes the proof of naturality.For the hexagon axioms, we will verify only the equality(9.9) ψ X,Y ⊗ Z = (id Y ⊗ ψ XZ )( ψ XY ⊗ id Z );the other one is similar. Consider the following diagram: X ⊗ Y ⊗ Z X ⊗ Y ⊗ Z Y ⊗ Z ⊗ X Y ⊗ Z ⊗ XX ⊗ Z ⊗ Y Z ⊗ Y ⊗ XX ⊗ Y ⊗ Z X ⊗ Y ⊗ Z Y ⊗ Z ⊗ XY ⊗ X ⊗ Z Y ⊗ X ⊗ ZY ⊗ X ⊗ Z Y ⊗ X ⊗ Z Y ⊗ Z ⊗ X σ − ⊗ σ − σ − ⊗ σ − ⊗ id ξ id ⊗ ρ σ ⊗ σρ ⊗ id ξ id ⊗ ρ ρ ⊗ idid ⊗ id ⊗ σ − ξ ⊗ id ξ ⊗ id σ ⊗ σ ⊗ σσ ⊗ id ⊗ idid ⊗ id ⊗ σ − σ ⊗ σ ⊗ id id ⊗ ξ σ ⊗ ξσ ⊗ id ⊗ idid ⊗ σ − ⊗ σ − id ⊗ ξ id ⊗ σ ⊗ σ Although this looks rather unpleasant, it is mostly harmless. Almost all of thepieces commute tautologically. The only pieces that are not obvious are the squarein the top middle part of the diagram, and the pentagon immediately below thatsquare. The square commutes by Lemma 9.4, and the pentagon commutes byLemma 9.5.To complete the proof, note that the left-hand side of (9.9) is exactly the com-position along the top row of the diagram, while the right-hand side of (9.9) isthe composition down the left side, across the bottom, and up to the top rightcorner. (cid:3) -STRUCTURES ON MODULE-ALGEBRAS 43
Quasitriangular Hopf ∗ -algebras. Quasitriangular Hopf algebras are im-portant because their module categories are automatically equipped with a braiding.Our goal in § ψ looks like when H is a quasitriangular Hopf ∗ -algebra. We begin by recalling the definition andproperties of quasitriangular Hopf algebras.Recall that a Hopf algebra H is quasitriangular [KS97, Chapter 8] if there is aninvertible element R ∈ H ⊗ H such that R ∆( a ) R − = ∆ op ( a ) def = τ ◦ ∆( a ) , (9.10)and (∆ ⊗ id)( R ) = R R , (id ⊗ ∆)( R ) = R R , (9.11)where τ is the tensor flip. The element R is called a universal R-matrix for H . In(9.11) we are using the so-called leg-numbering notation : if R = P j x j ⊗ y j , then R = P j x j ⊗ y j ⊗ R = P j x j ⊗ ⊗ y j , and R = P j ⊗ x j ⊗ y j . We alsodenote R = τ ( R ) = P j y j ⊗ x j .The important point here is that if H is quasitriangular, then HMod acquires abraiding from the action of the R-matrix. Before describing the braiding, we recordhere for later use some of the consequences of quasitriangularity. The proofs ofthese statements can be found in § Proposition 9.7.
Suppose that H is a quasitriangular Hopf algebra with universalR-matrix R . Then the following hold: R R R = R R R (9.12) ( ε ⊗ id)( R ) = (id ⊗ ε )( R ) = 1(9.13) ( S ⊗ id)( R ) = R − , (id ⊗ S )( R − ) = R , ( S ⊗ S )( R ) = R . (9.14) Furthermore, the element u = m ( S ⊗ id)( R ) is invertible in H with inverse u − = m (id ⊗ S )( R ) (here m denotes the multiplication map of H ), and the element uS ( u ) = S ( u ) u is central in H . The antipode of H is invertible, and we have (9.15) S ( a ) = uau − , S − ( a ) = u − S ( a ) u for all a ∈ H . The relation (9.12) is called the quantum Yang-Baxter equation .If
U, V are any two H -modules, then R acts in U ⊗ V coordinate-wise. Then wehave the following result [KS97, § Proposition 9.8.
Suppose that H is a quasitriangular Hopf algebra with universalR-matrix R . For any U, V ∈ HMod , define ψ UV = τ ◦ R : U ⊗ V → V ⊗ U. Then ψ = ( ψ UV ) U,V ∈ HMod is a braiding on
HMod . Now that the prerequisites are in place, we can begin to explore the effect ofconjugation on the braiding coming from the R-matrix.
Notation 9.9.
We extend the ∗ -structure of H to a ∗ -structure on H ⊗ H coordinate-wise. For a simple tensor a ⊗ b ∈ H ⊗ H , we have ( a ⊗ b ) ∗ = a ∗ ⊗ b ∗ . In particular, for R = P j x j ⊗ y j we have R ∗ = X j x ∗ j ⊗ y ∗ j . The following definition is inspired by [KS97]Definition 2 in § Definition 9.10.
Suppose that H is a quasitriangular Hopf ∗ -algebra with univer-sal R-matrix R . We say that R is real if R ∗ = R . We say that R is inverse real if R ∗ = R − .When the R-matrix is either real or inverse real, we can draw some conclusionsabout the conjugated braiding: Proposition 9.11.
Suppose that H is a quasitriangular Hopf ∗ -algebra with uni-versal R-matrix R . Let ψ be the braiding on HMod as in Proposition 9.8(a) If R is real, then ψ = ψ , i.e. for any U, V ∈ HMod we have ψ UV = ψ UV .(b) If R is inverse real, then ψ = ψ − , i.e. for any U, V ∈ HMod we have ψ UV = ψ − V U .Proof.
Let us write R = x j ⊗ y j with implied summation. First we compute themaps ξ V W that we defined in (9.3). For v ∈ V , w ∈ W we have ξ V W ( v ⊗ w ) = ρ V W (cid:16) ψ W V ( w ⊗ v ) (cid:17) = ρ V W (cid:16) τ ◦ R ( w ⊗ v ) (cid:17) = ρ V W (cid:16) ( y j ⊲ v ) ⊗ ( x j ⊲ w ) (cid:17) = ( x j ⊲ w ) ⊗ ( y j ⊲ v )= ( S ( x j ) ∗ ⊲ w ) ⊗ ( S ( y j ) ∗ ⊲ v )= [( S ⊗ S )( R ) ∗ ] ( w ⊗ v )= R ∗ ( w ⊗ v )= ( τ ◦ R ∗ )( v ⊗ w ) . In other words, we have shown that ξ V W = τ ◦ R ∗ .When R is real, then we have ξ V W = τ ◦ R ∗ = τ ◦ R = ψ V W . Applying naturality of the braiding ψ to the morphisms σ V and σ W (and suppress-ing subscripts for readability), we have ψ V W = ( σ ⊗ σ ) ξ V W ( σ − ⊗ σ − ) = ( σ ⊗ σ ) ψ V W ( σ − ⊗ σ − ) = ψ V W . This establishes (a).When R is inverse real, then our initial computation gives ξ V W = τ ◦ R ∗ = τ ◦ R − = R − ◦ τ = ψ − W V ;as above, using naturality of the braiding ψ , we conclude that ψ V W = ψ − W V . Thisestablishes (b). (cid:3) -STRUCTURES ON MODULE-ALGEBRAS 45
Braidings and ∗ -structures. For this subsection we will assume that H isa quasitriangular Hopf ∗ -algebra, so that HMod is braided. As above, we denotethe braiding coming from the R-matrix by ψ . Recall from Proposition 6.12 thatif V is a ∗ -module, then V ⊗ V is also a ∗ -module, with ∗ -structure given by( u ⊗ v ) ∗ = v ∗ ⊗ u ∗ . It therefore makes sense to ask: under what circumstances isthe braiding map ψ V V a morphism of ∗ -modules? In other words, when does thefollowing diagram commute:(9.16) V ⊗ V ∗ −−−−→ V ⊗ V ψ V V y y ψ V V V ⊗ V −−−−→ ∗ V ⊗ V Note that on the left-hand side of (9.16) we really mean the complex conjugateof the map ψ V V , not the conjugated braiding ψ V V (which wouldn’t make senseanyway).
Proposition 9.12.
Let V ∈ HMod be a ∗ -module, and give V ⊗ V the ∗ -structuredescribed in Proposition 6.12. If the R-matrix R of H is real, then the diagram (9.16) commutes, so ψ V V is a morphism of ∗ -modules.Proof. Again, let us denote the universal R-matrix by R = x j ⊗ y j , with impliedsummation. Let u, v ∈ V . On the one hand, we have ψ V V (( u ⊗ v ) ∗ ) = ψ V V ( v ∗ ⊗ u ∗ )= ( τ ◦ R )( v ∗ ⊗ u ∗ )= ( y j ⊲ u ∗ ) ⊗ ( x j ⊲ v ∗ )= R ( u ∗ ⊗ v ∗ ) . On the other hand, we have (cid:0) ψ V V ( u ⊗ v ) (cid:1) ∗ = (cid:16) ψ V V ( u ⊗ v ) (cid:17) ∗ = (cid:16) ( τ ◦ R )( u ⊗ v ) (cid:17) ∗ = (cid:16) ( y j ⊲ v ) ⊗ ( x j ⊲ u ) (cid:17) ∗ = ( x j ⊲ u ) ∗ ⊗ ( y j ⊲ v ) ∗ = ( S ( x j ) ∗ ⊲ u ∗ ) ⊗ ( S ( y j ) ∗ ⊲ v ∗ )= ( S ⊗ S )( R ) ∗ ( u ∗ ⊗ v ∗ )= R ∗ ( u ∗ ⊗ v ∗ ) . If R is real, then R ∗ = R by definition, so the two expressions agree, and thus(9.16) commutes. Hence ψ V V is a ∗ -map. (cid:3) References [BK01] Bojko Bakalov and Alexander Kirillov Jr.,
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Department of Mathematics, University of California, Berkeley, CA 94720-3840
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