76 1.37. Quantum traces. Let C be a rigid monoidal category, V be an object in C, and a ∈ Hom(V, V ∗∗). Define the left quantum trace (1.37.1) TrL (a) := evV ∗ ◦ (a ⊗ IdV ∗ ) ◦ coevV ∈ End(1).V Similarly, if a ∈ Hom(V, ∗∗V ) then we can define the right quantum trace (1.37.2) TrR (a) := ev∗∗V ◦ (Id∗V ⊗ a) ◦ coev∗V ∈ End(1).V In a tensor category over k, TrL(a) and TrR(a) can be regarded as elements of k. When no confusion is possible, we will denote TrLV by TrV . The following proposition shows that usual linear algebra formulas hold for the quantum trace. Proposition 1.37.1. If a ∈ Hom(V, V ∗∗), b ∈ Hom(W, W ∗∗) then (1) TrVL (a) = T rVR ∗ (a∗); (2) TrVL ⊕W (a ⊕ b) = TrLW (b) (in additive categories); V (a) + TrL (3) TrVL ⊗W (a ⊗ b) = TrL (a)TrL (b);V W (4) If c ∈ Hom(V, V ) then TrL (ac) = TrL (c∗∗a), TrR (ac) = TrR(∗∗ca).V V V V Similar equalities to (2),(3) also hold for right quantum traces. Exercise 1.37.2. Prove Proposition 1.37.1. If C is a multitensor category, it is useful to generalize Proposi-tion 1.37.1(2) as follows. Proposition 1.37.3. If a Hom(V, V ∗∗) and W V such that ∈ ⊂ a(W ) ⊂ W ∗∗ then TrL = TrL V/W (a). That is, Tr is addi-V (a) W (a) + TrL tive on exact sequences. The same statement holds for right quantum traces. Exercise 1.37.4. Prove Proposition 1.37.3. 1.38. Pivotal categories and dimensions. Definition 1.38.1. Let C be a rigid monoidal category. A pivotal structure on C is an isomorphism of monoidal functors a : Id ∼?∗∗.−→That is, a pivotal structure is a collection of morphisms aX : X ∼−→X∗∗ natural in X and satisfying aX⊗Y = aX ⊗ aY for all objects X, Y in C. Definition 1.38.2. A rigid monoidal category C equipped with a piv-otal structure is said to be pivotal. Exercise 1.38.3. (1) If a is a pivotal structure then aV ∗ = (aV )∗−1 . Hence, aV ∗∗ = a∗∗ V .77 (2) Let C = Rep(H), where H is a finite dimensional Hopf alge-bra. Show that pivotal structures on C bijectively correspond to group-like elements of H such that gxg−1 = S2(x) for all x ∈ H. Let a be a pivotal structure on a rigid monoidal category C. Definition 1.38.4. The dimension of an object X with respect to a is dima(X) := Tr(aX ) ∈ End(1). Thus, in a tensor category over k, dimensions are elements of k. Also, it follows from Exercise 1.38.3 that dima(V ) = dima(V ∗∗). Proposition 1.38.5. If C is a tensor category, then the function X �→dima(X) is a character of the Grothendieck ring Gr(C). Proof. Proposition 1.37.3 implies that dima is additive on exact se-quences, which means that it gives rise to a well-defined linear map from Gr(C) to k. The fact that this map is a character follows from the obvious fact that dima(1) = 1 and Proposition 1.37.1(3). � Corollary 1.38.6. Dimensions of objects in a pivotal finite tensor cat-egory are algebraic integers in k. 12 Proof. This follows from the fact that a character of any ring that is finitely generated as a Z-module takes values in algebraic integers. � 1.39. Spherical categories. Definition 1.39.1. A pivotal structure a on a tensor category C is spherical if dima(V ) = dima(V ∗) for any object V in C. A tensor category is spherical if it is equipped with a spherical structure. Since dima is additive on exact sequences, it suffices to require the property dima(V ) = dima(V ∗) only for simple objects V . Theorem 1.39.2. Let C be a spherical category and V be an object of Then for any x ∈ Hom(V, V ) one has T rL (aV x) = TrR (xa−1).C. V VV Proof. We first note that TrR (a−1) = dima(X∗) for any object X by X X Proposition 1.37.1(1) and Exercise 1.38.3(1). Now let us prove the proposition in the special case when V is semisimple. Thus V = ⊕i Yi ⊗Vi, where Vi are vector spaces and Yi are simple objects. Then x = 12If k has positive characteristic, by an algebraic integer in k we mean an element of a finite subfield of k.� � � 78 ⊕i xi ⊗IdVi with xi ∈ Endk(Yi) and a = ⊕ IdYi ⊗aVi (by the functoriality of a). Hence TrLV (ax) = Tr(xi) dim(Vi), TrRV (xa−1) = Tr(xi) dim(Vi ∗). This implies the result for a semisimple V . Consider now the general case. Then V has the coradical filtration (1.39.1) 0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn = V (such that Vi+1/Vi is a maximal semisimple subobject in V/Vi). This filtration is preserved by x and by a (i.e., a : Vi Vi ∗∗). Since traces →are additive on exact sequences by Proposition 1.37.3, this implies that the general case of the required statement follows from the semisimple case. � Exercise 1.39.3. (i) Let Aut⊗(IdC) be the group of isomorphism classes of monoidal automorphisms of a monoidal category C. Show that the set of isomorphism classes of pivotal structures on C is a torsor over Aut⊗(IdC), and the set of isomorphism classes of spherical structures on C is a torsor over the subgroup Aut⊗(IdC)2 in Aut⊗(IdC ) of elements which act by ±1 on simple objects. 1.40. Semisimple multitensor categories. In this section we will more closely consider semisimple multitensor categories which have some important additional properties compared to the general case. 1.41. Isomorphism between V ∗∗ and V . Proposition 1.41.1. Let C be a semisimple multitensor category and let V be an object in C. = V ∗. = V ∗∗.Then ∗V ∼Hence, V ∼Proof. We may assume that V is simple. We claim that the unique simple object X such that Hom(1, V ⊗X) =�0 is V ∗. Indeed, Hom(1, V ⊗ X) ∼Hom(∗X, V ) which is non-zero if = and only if ∗X ∼= V ∗.= V , i.e., X ∼Similarly, the unique simple object X such that Hom(V ⊗ X, 1) =� 0 is ∗V . But since C is semisimple, dimk Hom(1, V ⊗ X) = dimk Hom(V ⊗ X, 1), which implies the result. Remark 1.41.2. As noted in Remark 1.27.2, the result of Proposi-tion 1.41.1 is false for non-semisimple categories. Remark 1.41.3. Proposition 1.41.1 gives rise to the following ques-tion.�� � � � �79 Question 1.41.4. Does any semisimple tensor category admit a piv-otal structure? A spherical structure? This is the case for all known examples. The general answer is un-known to us at the moment of writing (even for ground fields of char-acteristic zero). Proposition