M-traces in (non-unimodular) pivotal categories
aa r X i v : . [ m a t h . R T ] S e p M-TRACES IN (NON-UNIMODULAR) PIVOTAL CATEGORIES
NATHAN GEER, JONATHAN KUJAWA, AND BERTRAND PATUREAU-MIRAND
Abstract.
We generalize the notion of a modified trace (or m-trace) to thesetting of non-unimodular categories. M-traces are known to play an impor-tant role in low-dimensional topology and representation theory, as well asin studying the category itself. Under mild conditions we give existence anduniqueness results for m-traces in pivotal categories. Introduction
Background.
Tensor categories with duals have a notion of the categoricaltrace of a morphism. These traces and the corresponding concept of dimensionare a key tool in applications to low-dimensional topology, representation theory,and other fields. However, it is often the case that categories of interest are notsemi-simple, the categorical traces vanish, and these constructions become trivial.In the past decade it became clear there exist non-trivial replacements for tracefunctions on non-semi-simple ribbon and, more generally, pivotal categories (e.g.see [GKP11, GKP13, GPV13]). We call these modified traces or m-traces , forshort. The study of m-traces leads to new, interesting quantum invariants of linksand 3-manifolds as well as applications in the study of representation theory, Hopfalgebras, Deligne categories, logarithmic conformal field theory, and other fields(e.g. see [CMR16, CG17, DGP17, GPT09, CGP14, CK12, Com14, CH17, BKN12,BCGP16, BBG17, BBG18, AS17, Rup16, Mur17, Len17, Phu16]). However, untilnow the existence and theory of m-traces has been limited to unimodular categories(ie. categories in which the projective cover and injective hull of the unit objectcoincide). The goal of this paper is to generalize m-traces to the non-unimodularsetting.1.2. Statement of main results.
In what follows we highlight the main results.For simplicity’s sake, in the introduction we assume k is an algebraically closedfield and C is a pivotal, k -linear, locally-finite, tensor category. Roughly speaking,this is a category with a tensor product, duals, and the morphism sets are finite-dimensional k -vector spaces. See Section 2 for precise definitions. Such categoriesare ubiquitous. For example, they appear: • as categories of finite-dimensional modules for finite-dimensional pivotal(quasi-)Hopf algebras; • in the study of logarithmic conformal field theories (e.g. see [Gab03]); • as fusion categories of categorical dimension zero (e.g. see [EGNO15]). Date : September 5, 2018.Research of the first author has been partially supported by the NSF grants DMS-1452093 andDMS-1664387.Research of the second author was partially supported by a Simons Foundation CollaborationGrant.
See Section 5 and [GKP13] for specific examples of such categories.Given a fixed pair of objects α and β in C and a right ideal I (a certain kind offull subcategory), we define the notion of a right ( α, β )-trace on I . This m-trace isa family of k -linear functions, { t V : Hom C ( α ⊗ V, β ⊗ V ) → k } V ∈ I , where V runs over all objects of I , and such that certain partial trace and cyclicityproperties hold. See Section 3.2 for a precise definition. In the case when α and β are both the unit object of C , then we recover the unimodular m-traces of [GKP11,GKP13, GPV13].Our first main result is Theorem 4.4 in which we show, given an absolutelyindecomposable object P and objects α and β in C with Hom C ( α, P) = k andHom C (P , β ) = k , there exists a right ( α, β )-trace on a certain (possibly empty)right ideal I βα . Furthermore, if either α or β is the unit object then Theorem 4.5implies this m-trace is unique up to scaling.Because of the generality of these existence and uniqueness results it is difficultto explicitly describe the ideal I βα or the functions t V . However, there is a notablecase where we can say more. If P is assumed to be the projective cover of theunit object, , and if α denotes a simple subobject of P, then our second mainresult shows the above theorem defines a unique, nontrivial, right ( α, )-trace on P roj ( C ), the full subcategory of projective objects of C . This is already interestingand powerful in the context of unimodular categories (i.e. when α ∼ = ). It saysany locally-finite, unimodular, pivotal category C with enough projectives has anm-trace on P roj ( C ). Previously this was only known in special cases, such as when C is the category of representations for a finite group or when C contains a simpleprojective object. For example, see [BBG18, GKP13, GR17].We end the paper with a discussion of how the notion of a right ( α, β )-traceon a category leads to a natural generalization of the notion of a Calabi-Yau cate-gory. Variations on Calabi-Yau categories play an important role in mathematicalphysics, algebraic geometry, integrable systems, the representation theory of finite-dimensional algebras, and the categorification of cluster algebras.If F, G are endofunctors of C , then we say C is an ( F, G )-twisted Calabi-Yaucategory if for all objects U and V there is a vector space isomorphismHom C ( F ( U ) , V ) ∼ = Hom C ( V, G ( U )) ∗ , which is functorial in both U and V . For example, an (Id C , G )-twisted Calabi-Yaustructure on C amounts to saying G is a right Serre functor in the sense of Bondal-Kapranov [BK89]. Just as a Calabi-Yau category is a categorical generalization ofthe notion of a symmetric Frobenius algebra, an ( F, G )-twisted Calabi-Yau categorygeneralizes the notion of a Frobenius extension of k as defined by Morita [Mor65].Our main theorem applied to P roj ( C ) can be reformulated as saying there isa twisted Calabi-Yau structure on P roj ( C ) for any pivotal, k -linear, locally-finitetensor category C . As a consequence, P roj ( C ) admits a right Serre functor for anysuch category. See Theorem 6.2.1.3. Future applications.
One motivation for the development of m-traces istheir use in constructing invariants in low-dimensional topology. In particular,in forthcoming work the second two authors with Costantino and Turaev definegeneralized Kuperberg and Turaev-Viro invariants from certain unimodular pivotal -TRACES IN (NON-UNIMODULAR) PIVOTAL CATEGORIES 3 tensor categories [CGPT18]. One of the main ingredients in this construction isthe existence of a non-degenerate m-trace. This is one motivation for Theorem 5.5which says, under mild conditions, a right m-trace is always non-degenerate. Thiswork is still within the context of unimodular categories. An interesting future lineof research is to use the m-traces of this paper to construct generalized Kuperbergand Turaev-Viro invariants from non-unimodular pivotal tensor categories.1.4.
Related work.
While working on this paper we learned A. Fontalvo Orozcoand A. M. Gainutdinov were defining a notion of module trace which is relatedto our m-traces, see [FG]. However, they use different techniques and their workgeneralizes the relation between the theory of integrals in a Hopf algebra H withthe modified trace on the projective ideal P roj ( H - Mod) as established in [BBG18]in the unimodular case.Shimizu recently introduced the notion of integrals for finite tensor categories[Shi14, Shi17]. It would be interesting to generalize the results of [BBG18] andOrozco-Gainutdinov to the categorical setting by relating the m-traces introducedhere to the integrals of Shimizu.2. Preliminaries
Pivotal categories.
We recall the definition of a pivotal tensor category, seefor instance, [BW99]. A tensor category C is a category equipped with a covariantbifunctor ⊗ : C × C → C called the tensor product, an associativity constraint,a unit object , and left and right unit constraints such that the Triangle andPentagon Axioms hold. When the associativity constraint and the left and rightunit constraints are all identities we say that C is a strict tensor category. ByMac Lane’s coherence theorem for pivotal tensor categories, every such category isequivalent (as a pivotal tensor category) to a strict one (e.g. see [NS07, Theorem2.2]). To simplify the exposition, we formulate further definitions only for stricttensor categories; the interested reader will easily extend them to arbitrary ten-sor categories. In what follows we adopt the convention that f g will denote thecomposition of morphisms f ◦ g .A strict tensor category C has a left duality if for each object V of C there isan object V ∗ of C and morphisms ←− coev V : → V ⊗ V ∗ and ←− ev V : V ∗ ⊗ V → (2.1)such that(Id V ⊗ ←− ev V )( ←− coev V ⊗ Id V ) = Id V and ( ←− ev V ⊗ Id V ∗ )(Id V ∗ ⊗ ←− coev V ) = Id V ∗ . A left duality determines for every morphism f : V → W in C the dual (ortranspose) morphism f ∗ : W ∗ → V ∗ by f ∗ = ( ←− ev W ⊗ Id V ∗ )(Id W ∗ ⊗ f ⊗ Id V ∗ )(Id W ∗ ⊗ ←− coev V ) , and determines for any objects V, W of C , an isomorphism γ V,W : W ∗ ⊗ V ∗ → ( V ⊗ W ) ∗ by γ V,W = ( ←− ev W ⊗ Id ( V ⊗ W ) ∗ )(Id W ∗ ⊗ ←− ev V ⊗ Id W ⊗ Id ( V ⊗ W ) ∗ )(Id W ∗ ⊗ Id V ∗ ⊗ ←− coev V ⊗ W ) . Similarly, C has a right duality if for each object V of C there is an object V • of C and morphisms −→ coev V : → V • ⊗ V and −→ ev V : V ⊗ V • → (2.2) NATHAN GEER, JONATHAN KUJAWA, AND BERTRAND PATUREAU-MIRAND such that(Id V • ⊗ −→ ev V )( −→ coev V ⊗ Id V • ) = Id V • and ( −→ ev V ⊗ Id V )(Id V ⊗ −→ coev V ) = Id V . The right duality determines for every morphism f : V → W in C the dual mor-phism f • : W • → V • by f • = (Id V • ⊗ −→ ev W )(Id V • ⊗ f ⊗ Id W • )( −→ coev V ⊗ Id W • ) , and determines for any objects V, W , an isomorphism γ ′ V,W : W • ⊗ V • → ( V ⊗ W ) • by γ ′ V,W = (Id ( V ⊗ W ) • ⊗ −→ ev V )(Id ( V ⊗ W ) • ⊗ Id V ⊗ −→ ev W ⊗ Id V • )( −→ coev V ⊗ W ⊗ Id W • ⊗ Id V • ) . A pivotal category is a tensor category with left duality { ←− coev V , ←− ev V } V ∈ C andright duality { −→ coev V , −→ ev V } V ∈ C which are compatible in the sense that V ∗ = V • , f ∗ = f • , and γ V,W = γ ′ V,W for all
V, W, f as above. Every pivotal category hasnatural tensor isomorphisms φ = { φ V = ( −→ ev V ⊗ Id V ∗∗ )(Id V ⊗ ←− coev V ∗ ) : V → V ∗∗ } V ∈ C . (2.3)We remind the reader of the well-known diagrammatic calculus for pivotal tensorcategories, see for example [Kas95, Chapter XIV] or [GPV13]. For brevity’s sakewe choose to not use it here. Nevertheless, many of the calculations in this paperare most easily understood when done diagrammatically.2.2. Tensor k -categories. Let k be a commutative ring. A tensor k -category isa tensor category C which is enriched over the category of k -modules. That is, C is additive, the hom-sets of C are left k -modules, and the composition and tensorproduct of morphisms are k -bilinear.An object V of a tensor k -category C is absolutely irreducible (or absolutelysimple ) if End C ( V ) is a free k -module of rank one; that is, if the k -homomorphism k → End C ( X ) , k k Id X is an isomorphism. We identify End C ( V ) and k via thismap. We always assume the unit object, , is absolutely irreducible.We call an object V of C absolutely indecomposable ifEnd C ( V ) / J(End C ( V )) ∼ = k . Here J(End C ( V )) denotes the Jacobson radical of the endomorphism ring End C ( V ).We say an absolutely indecomposable object is end-nilpotent if the Jacobson radicalof its endomorphism algebra is nilpotent.2.3. Projective and injective objects.
Let C be a category. Recall that anobject P of C is projective if the functor Hom C ( P, − ) : C → Set preserves epimor-phisms, that is, if for any epimorphism p : X → Y and any morphism f : P → Y in C , there exists a morphism g : P → X in C such that f = pg . We denoteby P roj ( C ) the class of projective objects of C . An object of C is injective if itis projective in the opposite category C op . In other words, an object Q of C isinjective if for any monomorphism i : X → Y and any morphism f : X → Q in C ,there exists a morphism g : Y → Q in C such that f = gi .When C is pivotal the projective and injective objects coincide (e.g. see [GPV13,Lemma 17]). Thus in this case P roj ( C ) is also the class of injective objects of C .The projective cover of an object is unique up to non-unique isomorphism, if itexists. We say C has enough projectives if every object in C has a projective cover. -TRACES IN (NON-UNIMODULAR) PIVOTAL CATEGORIES 5 We call C locally-finite if, for every pair of objects X, Y in C , Hom C ( X, Y )has a finite length composition series as a k -module. If C is a locally-finite tensorcategory, then, for example by [Kra15], an indecomposable projective object P hasa unique simple quotient (which we call the head of P ), a unique simple subobject(which we call the socle of P ), and End C ( P ) is end-nilpotent. By definition, C is unimodular if the socle of the projective cover of is isomorphic to .2.4. Invertible objects.
We call an object X in C invertible if ←− ev X : X ∗ ⊗ X → and ←− coev X : → X ⊗ X ∗ are isomorphisms. For example, is always an invertibleobject and in a finite tensor category the socle of the projective cover of is alwaysan invertible object (see [EGNO15, Section 6.4]).3. Right ( α, β ) -Traces Ideals.
Let C be a pivotal k -category.A right partial trace (with respect to W ) is the map tr Wr : Hom C ( V ⊗ W, X ⊗ W ) → Hom C ( V, X ) defined, for g ∈ Hom C ( V ⊗ W, X ⊗ W ), bytr Wr ( g ) = (Id X ⊗ −→ ev W )( g ⊗ Id W ∗ )(Id V ⊗ ←− coev W )Similarly, a left partial trace (with respect to W ) is the map tr Wl : Hom C ( W ⊗ V, W ⊗ X ) → Hom C ( V, X ) defined bytr Wl ( h ) = ( ←− ev W ⊗ Id X )(Id W ∗ ⊗ h )( −→ coev W ⊗ Id V ) . By a right (resp. left) ideal of C we mean a full subcategory, I , of C such that:(1) Closed under tensor products: If V is an object of I and W is anyobject of C , then V ⊗ W (resp. W ⊗ V ) is an object of I .(2) Closed under retracts: If V is an object of I , W is any object of C , andthere exists morphisms f : W → V , g : V → W such that gf = Id W , then W is an object of I .An ideal of C is a full subcategory of C which is both a right and left ideal. Forexample, the full subcategory whose objects are the class of projective objects, P roj ( C ), is an ideal by [GPV13, Lemma 17].3.2. Traces.
Let α and β be objects of C and I a right ideal in C . A right ( α , β )-trace on I (or m-trace for short) is a family of k -linear functions, { t V : Hom C ( α ⊗ V, β ⊗ V ) → k } V ∈ I , where V runs over all objects of I , and such that the following two conditions hold:(1) Partial trace property. If U ∈ I and W ∈ C , then for any f ∈ Hom C ( α ⊗ U ⊗ W, β ⊗ U ⊗ W ) we have t U ⊗ W ( f ) = t U (cid:0) tr Wr ( f ) (cid:1) (2) ( α , β )-Cyclicity. If U, V ∈ I , then for any morphisms f : α ⊗ V → β ⊗ U and g : U → V in C we have t V ((Id β ⊗ g ) f ) = t U ( f (Id α ⊗ g )) . Similarly, a left ( α , β )-trace on a left ideal I is a family of linear functions, { t V : Hom C ( V ⊗ α, V ⊗ β ) → k } V ∈ I , NATHAN GEER, JONATHAN KUJAWA, AND BERTRAND PATUREAU-MIRAND which satisfies the obvious left partial trace property and the left ( α, β )-cyclicityproperty. An ( α , β )-trace on an ideal I is a left ( α , β )-trace on I which is also aright ( α , β )-trace. Remark 3.1.
When α = β = k , a right (resp. left) ( α , β )-trace is a right (resp.left) modified trace as defined in [GPV13] . The dual trace.
As we now explain, there is a natural notion of the dual ofa right m-trace. If I is a full subcategory of C , define its dual I ∗ to be the fullsubcategory with objects I ∗ = { V ∈ Obj( C ) : V ∗ ∈ I } . It is straightforward tocheck if I is a right (resp. left) ideal then I ∗ is a left (resp. right) ideal. If t is aright ( α, β )-trace t on I , then given V in I ∗ define t ∗ V : Hom C ( V ⊗ β ∗ , V ⊗ α ∗ ) → k by t ∗ V ( f ) = t V ∗ ((Id V ∗ ⊗ φ β − ) f ∗ (Id V ∗ ⊗ φ α ))where φ is the pivotal structure. In light of the following result we call t ∗ the dual of t . Lemma 3.2.
Let t be a right ( α, β ) -trace t on a right ideal, I . Then t ∗ is a left ( β ∗ , α ∗ ) -trace on the left ideal I ∗ .Proof. This follows easily from the observing the left partial trace of the dual mor-phism is the dual of the right partial trace. (cid:3)
One can analogously define the dual of a left m-trace on a left ideal I andobtain a right m-trace on I ∗ . Furthermore, a straightforward check verifies thatthe dualizing a right or left m-trace twice yields the original right or left m-trace.3.4. Related traces.
We next explain how to construct new m-traces from old.Assume α , β , α , and β are a fixed list of objects in C and that we have a fixedmorphism h : α ∗ ⊗ β → α ∗ ⊗ β . For any object V in C , the morphism h inducesa k -linear map h ∗ : Hom C ( α ⊗ V, β ⊗ V ) → Hom C ( α ⊗ V, β ⊗ V )given by f ( −→ ev α ⊗ Id β ⊗ Id V )(Id α ⊗ h ⊗ Id V )(Id α ⊗ Id α ∗ ⊗ f )(Id α ⊗ −→ coev α ⊗ Id V ) . Lemma 3.3.
Let t be a right ( α , β ) -trace on a right ideal I . Assume we havea fixed morphism h ∈ Hom C ( α ∗ ⊗ β , α ∗ ⊗ β ) . Then the family of k -linear maps h ∗ t , { ( h ∗ t ) V : Hom C ( α ⊗ V, β ⊗ V ) } V ∈ I , defined by ( h ∗ t ) V ( f ) = t ( h ∗ ( f )) , is a right ( α , β ) -trace on I .Furthermore, if h ′ ∈ Hom C ( α ∗ ⊗ β , α ∗ ⊗ β ) then h ′∗ ( h ∗ t ) = ( h ′ h ) ∗ t .Proof. The is a straightforward verification using the definition of h ∗ and h ∗ t . (cid:3) Similarly, a morphism h : β ⊗ α ∗ → β ⊗ α ∗ induces a k -linear map h ∗ : Hom C ( V ⊗ α , V ⊗ β ) → Hom C ( V ⊗ α , V ⊗ β )and if t is a left ( α , β )-trace on an ideal I , then we can analogously define a left( α , β )-trace h ∗ t . -TRACES IN (NON-UNIMODULAR) PIVOTAL CATEGORIES 7 Using the obvious morphisms as h in the previous lemma along with Lemma 3.2yields the following result. Proposition 3.4.
Let I be a right ideal of C . Then there are canonical bijectionsbetween the following families: (1) the right ( α, β ) -traces on I , (2) the right ( β ∗ ⊗ α, ) -traces on I , (3) the right ( , α ∗ ⊗ β ) -traces on I , (4) the left ( β ∗ , α ∗ ) -traces on I ∗ , (5) the left ( α ⊗ β ∗ , ) -traces on I ∗ , (6) the left ( , β ⊗ α ∗ ) -traces on I ∗ . Existence of right and left ( α , β )-traces Trace tuples.
Let C be a pivotal k -category. In this section we require k tobe an integral domain. To simplify exposition, in this section we only work withright m-traces and right ideals. However, the interested reader can easily formulatethat analogous left versions of the definitions and statements. Definition 4.1.
Let P , α and β be objects of C . Let η : α → P and ǫ : P → β benonzero morphisms. We say (P , α, β, η, ǫ ) is a trace tuple if the following conditionshold. (1) The object P is absolutely indecomposable and end-nilpotent. (2) The left k -modules Hom C ( α, P) and Hom C (P , β ) are free and generated by η and ǫ , respectively. Let (P , α, β, η, ǫ ) be a trace tuple. Consider the following classes of objects: I α = { V : there exists σ V : P ⊗ V → α ⊗ V such that σ V ( η ⊗ Id V ) = Id α ⊗ V } ,I β = { V : there exists τ V : β ⊗ V → P ⊗ V such that ( ǫ ⊗ Id V ) τ V = Id β ⊗ V } ,I βα = I α ∩ I β . For each of these, we abuse notation by using the same name for the full subcategoryof C consisting of objects isomorphic to an object in the given class. The followinglemma is a straightforward check using the definitions. Lemma 4.2. If (P , α, β, η, ǫ ) is a trace tuple, then I α , I β and I βα are right ideals. We set the following notation. When P is absolutely indecomposable write f f i for the canonical quotient map End C (P) → k . For a trace tuple (P , α, β, η, ǫ )and morphisms g ∈ Hom C ( α, P) and h ∈ Hom C (P , β ), let h g i η , h h i ǫ ∈ k be definedby g = h g i η η and h = h h i ǫ ǫ. Lemma 4.3.
Let (P , α, β, η, ǫ ) be a trace tuple. For any f ∈ End C (P) the followingstatements hold: (1) ǫf = h f i ǫ , (2) f η = h f i η , (3) h f i = h ǫf i ǫ = h f η i η .Proof. Since P is absolutely indecomposable, we have f = h f i Id P + n for h f i ∈ k and n ∈ J (End C ( P )). The first statement then follows once we prove ǫn = 0. SinceHom C (P , β ) is a free left k -module generated by ǫ we have ǫn = λǫ for some λ ∈ k . NATHAN GEER, JONATHAN KUJAWA, AND BERTRAND PATUREAU-MIRAND
But since P is end-nilpotent, n is nilpotent and n k = 0 for some k >
0. But then0 = ǫn k = λ k ǫ and, hence, λ k = 0. Since we are assuming k is an integral domain, itfollows that λ = 0. This proves the first statement, the second follows analogously.The first two parts of the lemma immediately imply the third statement. (cid:3) Existence of m-traces.Theorem 4.4.
Let (P , α, β, η, ǫ ) be a trace tuple. Then there exists a right ( α , β )-trace on I βα defined for V ∈ I βα and f ∈ Hom C ( α ⊗ V, β ⊗ V ) by t V ( f ) = (cid:10) tr Vr ( τ V f ) (cid:11) η = (cid:10) tr Vr ( f σ V ) (cid:11) ǫ where σ V : P ⊗ V → α ⊗ V and τ V : β ⊗ V → P ⊗ V are any morphisms satisfying σ V ( η ⊗ Id V ) = Id α ⊗ V and ( ǫ ⊗ Id V ) τ V = Id β ⊗ V .Proof. First, we note t V is k -linear and the morphisms σ V and τ V exist because V is assumed to lie in I βα . Next, let σ V and τ V be any such morphisms. Then, (cid:10) tr Vr (cid:0) τ V f (cid:1)(cid:11) η = (cid:10) tr Vr (cid:0) τ V f [Id α ⊗ V ] (cid:1)(cid:11) η = (cid:10) tr Vr (cid:0) τ V f [ σ V ( η ⊗ Id V )] (cid:1)(cid:11) η = (cid:10) tr Vr (cid:0) τ V f σ V (cid:1) η (cid:11) η = (cid:10) ǫ tr Vr (cid:0) τ V f σ V (cid:1)(cid:11) ǫ = (cid:10) tr Vr (cid:0) [ ǫ ⊗ Id V ] τ V f σ V (cid:1)(cid:11) ǫ = (cid:10) tr Vr (cid:0) f σ V (cid:1)(cid:11) ǫ , where the fourth equality comes from Lemma 4.3 part (3). Thus, t V ( f ) is indepen-dent of the choice of σ V or τ V .Next we show this family of functions satisfies the partial trace property. Let U ∈ I βα , W ∈ Ob ( C ) and f ∈ Hom C ( α ⊗ U ⊗ W, β ⊗ U ⊗ W ). Since U ∈ I β thereexists τ U : β ⊗ U → P ⊗ U such that ( ǫ ⊗ Id U ) τ U = Id β ⊗ U . Choose τ U ⊗ W to beequal to τ U ⊗ Id W then ( ǫ ⊗ Id U ⊗ W ) τ U ⊗ W = Id β ⊗ U ⊗ W . Therefore, we can use τ U ⊗ W to define t U ⊗ W and we see t U ⊗ W ( f ) = (cid:10) tr U ⊗ Wr (cid:0) τ U ⊗ W f (cid:1)(cid:11) η = (cid:10) tr U ⊗ Wr (cid:0) ( τ U ⊗ Id W ) f (cid:1)(cid:11) η = (cid:10) tr Ur (cid:0) τ U (tr Wr f ) (cid:1)(cid:11) η = t U (cid:0) tr Wr ( f ) (cid:1) . To prove the ( α , β )-cyclicity property, let f : α ⊗ V → β ⊗ U and g : U → V .Then, t V ((Id β ⊗ g ) f ) = (cid:10) tr Vr (cid:0) (Id β ⊗ g ) f σ V (cid:1)(cid:11) ǫ = (cid:10) tr Vr (cid:0) f σ V (Id β ⊗ g ) (cid:1)(cid:11) ǫ = (cid:10) tr Vr (cid:0) ( ǫ ⊗ Id V ) τ V f σ V (Id β ⊗ g ) (cid:1)(cid:11) ǫ = (cid:10) tr Vr (cid:0) τ V f (Id β ⊗ g ) (cid:1)(cid:11) η , where the first equality comes from the definition of the trace, the second from theproperties of the pivotal structure, the third from the definition of I β , and the lastfrom Lemma 4.3 part (3). (cid:3) -TRACES IN (NON-UNIMODULAR) PIVOTAL CATEGORIES 9 Uniqueness.
It is of particular interest when one or both of α and β are theunit object. In this case we have the following uniqueness result. Theorem 4.5.
Suppose (P , α, β, η, ǫ ) is a trace tuple with P ∈ I βα . (1) If β = , then the right ( α, ) -trace on I α is unique up to a scalar. Specifi-cally, if t ′ is a right ( α, ) -trace on I α and t is the right ( α, ) -trace definedby Theorem 4.4, then t ′ = t ′ P ( η ⊗ ǫ ) t . Moreover, t P ( η ⊗ ǫ ) = 1 . (2) If α = , then the right ( , β ) -trace on I β is unique up to a scalar. Specifi-cally, if t ′ is a right ( , β ) -trace on I β and t is the right ( , β ) -trace definedby Theorem 4.4, then t ′ = t ′ P ( ǫ ⊗ η ) t . Moreover, t P ( ǫ ⊗ η ) = 1 .Proof. Let t ′ be right ( α, )-trace on I α . If f ∈ Hom C ( α ⊗ V, V ) then t ′ V (cid:0) f (cid:1) = t ′ V (cid:0) ( ǫ ⊗ Id V ) τ V f ) = t ′ P ⊗ V (( τ V f )(Id α ⊗ ǫ ⊗ Id V ) (cid:1) = t ′ P (cid:0) tr Vr (( τ V f )(Id α ⊗ ǫ )) (cid:1) = t ′ P (cid:0)(cid:10) tr Vr ( τ V f ) (cid:11) η η (Id α ⊗ ǫ ) (cid:1) = t V ( f ) t ′ P (cid:0) η ⊗ ǫ (cid:1) . The first equality comes from the fact that ( ǫ ⊗ Id V ) τ V = Id V by definition of I ; the second from strictness and ( α, )-cyclicity, the third from the partial traceproperty and the last two from the definitions of the η -bracket and the trace t ,respectively.Finally, using the same properties along with Lemma 4.3 yields t P ( η ⊗ ǫ ) = (cid:10) tr P r (cid:0) τ P ( η ⊗ ǫ ) (cid:1)(cid:11) η = h (Id P ⊗ ǫ ) τ P η i η = h ( ǫ ⊗ ǫ ) τ P i ǫ = h ǫ i ǫ = 1 . The proof of the second statement is entirely analogous. (cid:3)
For short, when β = we say a right (resp. left) ( α, )-trace on a right (resp.left) ideal I is a right (resp. left) α -trace on I .4.4. A handy lemma.
The following lemma will be useful in what follows.
Lemma 4.6.
Let t be the right trace associated to a trace tuple (P , α, β, η, ǫ ) as inTheorem 4.4 and let V be an object in I βα . Then: (1) For any f ∈ Hom C (P ⊗ V, β ⊗ V ) , one has t V ( f ( η ⊗ Id V )) = (cid:10) tr Vr ( f ) (cid:11) ǫ . (2) For any g ∈ Hom C ( α ⊗ V, P ⊗ V ) , one has t V (( ǫ ⊗ Id V ) g ) = (cid:10) tr Vr ( g ) (cid:11) η . (3) For any h ∈ Hom C (P ⊗ V, P ⊗ V ) , t V (( ǫ ⊗ Id V ) h ( η ⊗ Id V )) = (cid:10) tr Vr ( h ) (cid:11) .Proof. From definitions and Lemma 4.3, we have t V ( f ( η ⊗ Id V )) = (cid:10) tr Vr ( τ V f ( η ⊗ Id V )) (cid:11) η = (cid:10) tr Vr ( τ V f ) η (cid:11) η = (cid:10) ǫ tr Vr ( τ V f ) (cid:11) ǫ = (cid:10) tr Vr ( f ) (cid:11) ǫ . Similarly, t V (( ǫ ⊗ Id V ) g ) = (cid:10) tr Vr (( ǫ ⊗ Id V ) gσ V ) (cid:11) ǫ = (cid:10) ǫ tr Vr ( gσ V ) (cid:11) ǫ = (cid:10) tr Vr ( gσ V ) η (cid:11) η = (cid:10) tr Vr ( g ) (cid:11) η . The third statement is proven in a similar fashion. (cid:3)
Left and right compatibility.
If the reader formulates the theory for leftideals and m-traces, then the result is compatible with the right version, as we nextexplain.
Definition 4.7.
Let t l be a left ( α, β ) -trace on I l and t r be a right ( α, β ) -trace on I r . We say that t l and t r are compatible if for any ( V, W ) ∈ I l × I r , and for any f ∈ Hom C ( V ⊗ α ⊗ W, V ⊗ β ⊗ W ) , t rW (tr Vl ( f )) = t lV (tr Wr ( f )) . Proposition 4.8.
The left and right trace associated to a trace tuple (P , α, β, η, ǫ ) are compatible.Proof. Since V is in the left ideal I α and W is in the right ideal I β , there exists σ V : V ⊗ P → V ⊗ α and τ W : β ⊗ W → P ⊗ W such that f = (Id V ⊗ ǫ ⊗ Id W )(Id V ⊗ τ W ) f ( σ V ⊗ Id W )(Id V ⊗ η ⊗ Id W ) . Then by Lemma 4.6 we have t lV (tr Wr ( f )) = (cid:10) tr Vl (tr Wr ((Id V ⊗ τ W ) f ( σ V ⊗ Id W ))) (cid:11) = t rW (tr Vl ( f )) . (cid:3) Examples
In addition to the known examples of unimodular m-traces in the literature (e.g.[GKP11, GKP13, BBG18, GR17]), we have the following non-unimodular m-traces.5.1.
The toy example.
Let S be an absolutely irreducible object in C and setP = α = β = S , and let ǫ : P → β and η : α → P be the identity maps. Then(P , α, β, η, ǫ ) is a trace tuple and I βα = C . Also, the ( α, β )-trace of Theorem 4.4 isgiven by t V ( f ) = h tr Vr ( f ) i for all f ∈ Hom C ( S ⊗ V, S ⊗ V ).5.2. Quantized enveloping algebras.
In this subsection let k = C ( q ), where q isan indeterminate. We follow standard conventions without elaboration. The readermay consult [Jan03, Jan96, CP94] for further details. Let g be a complex semisimpleLie algebra and let h ⊆ b ⊆ g be a fixed choice of Cartan and Borel subalgebras,respectively. Let U q ( h ) ⊆ U q ( b ) ⊆ U q ( g ) be the corresponding quantized envelopingalgebras over k .We order elements of the weight lattice, Λ, using the usual dominance orderdetermined by our choice of b . If L is a finite-dimensional simple U q ( g )-module thenthere is a unique maximal nonzero weight space. Let λ ∈ Λ be the highest weightand let L λ denote the corresponding 1-dimensional λ -weight space. Similarly, L hasa unique lowest nonzero weight space, L α . By restriction we can view L as a U q ( b )-module, and then L λ is the simple socle, L α is the simple head, and L is cyclicallygenerated by L α . In particular, L is an absolutely indecomposable U q ( b )-module.In short, the previous paragraph shows the canonical projection and inclusionmaps ε : L → L α and η : L λ → L make ( L, L λ , L α , ǫ, η ) into a trace tuple inthe category of finite-dimensional U q ( b )-modules (which is known to be a pivotal k -tensor category). A similar example holds in the non-quantum case as well.Now let U ζ ( g ) be the restricted specialization of the quantized enveloping algebraat ζ ∈ C , a primitive, odd ℓ th root of unity. We assume ℓ is greater than theCoxeter number for g and is not divisible by 3 if g has a direct summand of type G . For a dominant integral λ ∈ Λ, let H ζ ( λ ), V ζ ( λ ), and T ζ ( λ ) denote the induced, -TRACES IN (NON-UNIMODULAR) PIVOTAL CATEGORIES 11 Weyl, and tilting U ζ ( g )-modules of highest weight λ . Then H ζ ( λ ) and V ζ ( λ ) areabsolutely irreducible and T ζ ( λ ) is absolutely indecomposable. Furthermore, sinceHom U ζ ( g ) ( V ζ ( λ ) , T ( λ )) = C and Hom U ζ ( g ) ( T ζ ( λ ) , H ζ ( λ )) = C , there are maps η : V ζ ( λ ) → T ( λ ) and ǫ : T ζ ( λ ) → H ζ ( λ ) which make (cid:16) T ζ ( λ ) , H ζ ( λ ) , V ζ ( λ ) , ǫ, η (cid:17) intoa trace tuple. A parallel example exists for semisimple algebraic groups over analgebraically closed field.5.3. Projective objects.
In this section k is assumed to be an algebraically closedfield and C is a locally-finite, pivotal, k -tensor category. In particular, in C everysimple object is absolutely simple by Schur’s Lemma and every indecomposableobject is absolutely indecomposable and end-nilpotent by Fitting’s Lemma. Asremarked in Subsection 1.2 such categories include a wide range of examples. Thissubsection implies these examples all admit unique nontrivial right m-traces. Lemma 5.1. If P is an indecomposable projective object in C , then there are uniqueabsolutely irreducible objects α and β and morphisms ǫ : P → β and η : α → P suchthat (P , α, β, η, ǫ ) is a trace tuple.Proof. If P is an indecomposable projective, then it is absolutely indecomposable,end-nilpotent, and has an irreducible head β and irreducible socle α by Section 2.3.Set ǫ : P → β and η : α → P to be the canonical projection and inclusion,respectively. Then (P , α, β, η, ǫ ) is a trace tuple. (cid:3)
Recall P roj ( C ) denotes the ideal of projective objects in C . Lemma 5.2.
Let (P , α, β, η, ǫ ) be an arbitrary trace tuple. Then, P roj ( C ) ⊆ I βα .In particular, if C contains a projective object, then I βα is nonempty.Proof. Let Q be a projective object in C . Then the morphism ǫ ⊗ Id Q : P ⊗ Q → β ⊗ Q is an epimorphism. Since Q is projective and P roj ( C ) is an ideal, it follows β ⊗ Q is projective, and so the morphism ǫ ⊗ Id Q splits. Therefore, Q is an object of I β . Similarly, η ⊗ Id Q : α ⊗ Q → P ⊗ Q is a monomorphism and α ⊗ Q is projective(hence injective), so the morphism η ⊗ Id Q again splits and Q is an object of I α .Taken together this shows Q ∈ I βα (cid:3) Lemma 5.3.
Given any trace tuple (P , α, β, η, ǫ ) where P is projective and either α or β is invertible. Then, I βα = P roj ( C ) .Proof. We do only the case when β is invertible as the other case is similar. Let V ∈ I βα . Then by definition ǫ ⊗ Id V : P ⊗ V → β ⊗ V splits. But P is in theideal P roj ( C ), so β ⊗ V is in P roj ( C ) and, hence, β ∗ ⊗ β ⊗ V ∼ = V is an object of P roj ( C ). The reverse inclusion is given by the previous lemma. (cid:3) Note, if S is a absolutely irreducible, projective object, then the toy example ofSubsection 5.1 shows the previous result could fail if there are no assumptions on α and β .The following theorem summarizes the outcome of the previous lemmas. Theorem 5.4. If P is an indecomposable projective object C , then there are uniqueabsolutely irreducible objects α and β and morphisms ǫ : P → β and η : α → P suchthat (P , α, β, η, ǫ ) is a trace tuple and P roj ( C ) ⊆ I βα . Moreover, if either α or β isinvertible, then P roj ( C ) = I βα . The next result demonstrates the m-trace defined by the previous result com-bined with Theorem 4.4 is nontrivial. Specifically, given Q ∈ P roj ( C ), one canchoose V (since it is arbitrary) so that Hom C ( α ⊗ Q, V ) is nontrivial. Conse-quently, for any Q ∈ P roj ( C ) the next theorem shows both Hom C ( α ⊗ Q, β ⊗ Q )and t Q are nonzero. Theorem 5.5.
Let (P , α, β, η, ǫ ) be the trace tuple given by an absolutely indecom-posable projective P as in the previous theorem. Let t be the right trace given byTheorem 4.4. Then for any Q ∈ P roj ( C ) ⊆ I βα and V ∈ C the map Hom C ( V, β ⊗ Q ) × Hom C ( α ⊗ Q, V ) → k given by ( g, f ) t Q ( gf ) is a non-degenerate pairing.Proof. From Lemma 5.2 P roj ( C ) ⊆ I βα so the function exists. We next show itsright kernel is trivial (the proof for the left kernel is similar). If f ∈ Hom C ( α ⊗ Q, V )is not zero, then f ′ = ( f ⊗ Id Q ∗ )(Id α ⊗ ←− coev Q ) ∈ Hom C ( α, V ⊗ Q ∗ ) is a non zeromap from α to the projective object V ⊗ Q ∗ . Since projective covers (hence injectiveenvelopes) are unique, P is the unique indecomposable projective object with α asa subobject, the map f ′ factors through an indecomposable summand of V ⊗ Q ∗ which is isomorphic to P. That is, there are morphisms ι : P → V ⊗ Q ∗ and p : V ⊗ Q ∗ → P such that pι = Id P and f ′ = ιη .Let g ∈ Hom C ( V, β ⊗ Q ) be given by g = ( ǫ ⊗ Id Q )( p ⊗ Id Q )(Id V ⊗ −→ coev Q ).Then gf = ( ǫ ⊗ Id Q ) f ′′ where f ′′ ∈ Hom C ( α ⊗ Q, P ⊗ Q ) is given by f ′′ = ( p ⊗ Id Q )(Id V ⊗ −→ coev Q ) f . The first of the following equalities holds by Lemma 4.6: t Q ( gf ) = h tr Qr ( f ′′ ) i η = h pf ′ i η = h pιη i η = h η i η = 1 . (cid:3) Combining Theorems 5.4 and 4.5 with the previous result immediately yields thefollowing corollary. Also, there exists an analogous unique, nontrivial left m-traceon P roj ( C ). Corollary 5.6.
Let k be an algebraically closed field and C be a locally-finite,pivotal, k -tensor category which has enough projectives. Let P be the projectivecover of and let α be the socle of P . This data determines a unique (up toscalar), nontrivial right α -trace on P roj ( C ) . Ambidextrous objects.
In earlier work the authors introduced the notionof a right ambidextrous object and the associated right m-trace. We now explainhow that construction is a special case of the one introduced here. Let C be aribbon category, S be an absolutely irreducible object, and let ǫ = −→ ev : S ⊗ S ∗ → and η = ←− coev: → S ⊗ S ∗ . Let S ⊗ S ∗ = ⊕ i W i be the decomposition of S intoindecomposable objects. Then S is right ambi in the sense of [GKP13] if and onlyif there is an i such that the restriction of ǫ and η to P := W i makes (P , , , ǫ, η )into a trace tuple. In which case I equals the ideal generated by S and the m-tracedefined here agrees with the one defined therein.6. Twisted Calabi-Yau Categories
In this section we continue to assume k is a field and C is a k -linear category. -TRACES IN (NON-UNIMODULAR) PIVOTAL CATEGORIES 13 Twisted Calabi-Yau Categories.
Next we introduce the notion of a twistedCalabi-Yau category.
Definition 6.1.
Let
F, G : C → C be fixed endofunctors of a category, C . Then C is an ( F, G )-twisted Calabi-Yau category if it is equipped with a family of k -linearmaps { t U : Hom C ( F ( U ) , G ( U )) → k } U ∈ C such that the following properties hold: (1) Non-degeneracy.
For any objects
U, V in C , the pairing Hom C ( V, G ( U )) × Hom C ( F ( U ) , V ) → k given by ( g, f ) t U ( gf ) is non-degenerate. (2) Cyclicity.
For any objects
U, V in C and any morphisms f : F ( V ) → G ( U ) and g : U → V in C , we have t V ( G ( g ) f ) = t U ( f F ( g )) . The non-degeneracy condition provides a canonical vector space isomorphism,Hom C ( F ( U ) , V ) ∼ = Hom C ( V, G ( U )) ∗ , which is functorial in both U and V .This notion generalizes existing constructions. For example, a (Id C , Id C )-twistedCalabi-Yau category is nothing but a Calabi-Yau category. If a category C is a(Id C , G )-twisted Calabi-Yau category, then G is a right Serre functor in the senseof Bondal-Kapranov [BK89].In the special case when C is a category with a single object, ∗ , then beingan ( F, G )-twisted Calabi-Yau category is equivalent to having a k -linear map t :End C ( ∗ ) → k which satisfies t ( g ( a ) b ) = t ( af ( b )) for fixed algebra endomorphisms f, g : End C ( ∗ ) → End C ( ∗ ) along with the requirement the induced pairing ( a, b ) t ( ab ) be nondegenerate. In this way, it generalizes the well known fact that a Calabi-Yau structure on a category with a single object is equivalent to the notion of asymmetric Frobenius algebra. As a special case, if g is the identity endomorphism,then we exactly have End C ( ∗ ) is a Frobenius extension of k in the sense of Morita[Mor65] (or see [PS16] for a modern treatment).6.2. A twisted Calabi-Yau structure on P roj ( C ) . In this section we assume k is an algebraically closed field and C is a locally-finite, pivotal, k -tensor categorywith enough projectives.If X is an fixed object of C , then we write F X for the endofunctor X ⊗ − . If Pis an indecomposable projective object in C , then there is the corresponding tracetuple (P , α, β, η, ǫ ) and right ( α, β )-trace, t , on P roj ( C ) given by Theorem 5.4.Combining this with Theorem 5.5 yields the following result. Theorem 6.2.
Let P be an indecomposable projective object in C . The corre-sponding trace tuple (P , α, β, η, ǫ ) and right ( α, β ) -trace, t , makes P roj ( C ) into an ( F α , F β ) -twisted Calabi-Yau category. As an application, if we take P to be the injective hull of and β is the simplehead of P , then P roj ( C ) is an (Id C , F β )-twisted Calabi-Yau category and, hence, F β is a right Serre functor on P roj ( C ). We also have the following special caseof the previous theorem. Recently Gainutdinov-Runkel [GR17] obtained the sameresult under the assumption C is finite and factorisable. Corollary 6.3.
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