� 97 1.51. The distinguished invertible object. Let C be a finite tensor category with classes of simple objects labeled by a set I. Since duals to projective objects are projective, we can define a map D : I I such that Pi ∗ = PD(i). It is clear that D2(i) = i∗∗. → Let 0 be the label for the unit object. Let ρ = D(0). (In other words, ∗Lρ is the socle of P0 = P (1)). We have Hom(Pi ∗, Lj ) = Hom(1, Pi ⊗ Lj ) = Hom(1, ⊕kNi kj∗ Pk). This space has dimension Ni Thus we get ρj∗ . Ni ρj∗ = δD(i),j. Let now Lρ be the corresponding simple object. By Proposition 1.47.2, we have = ⊕kNk = PD(m)∗ .L∗ ρ ⊗ Pm ∼ρmPk ∼Lemma 1.51.1. Lρ is an invertible object. Proof. The last equation implies that the matrix of action of Lρ∗ on projectives is a permutation matrix. Hence, the Frobenius-Perron di-mension of Lρ∗ is 1, and we are done. � Lemma 1.51.2. One has: PD(i) = P∗i ⊗ Lρ; LD(i) = L∗i ⊗ Lρ. Proof. It suffices to prove the first statement. Therefore, our job is to show that dim Hom(Pi ∗, Lj ) = dim Hom(P∗i, Lj ⊗ Lρ∗ ). The left hand side was computed before, it is Ni On the other hand, the right hand ρj∗ . side is N∗i (we use that ρ∗ = ∗ρ for an invertible object ρ). Thesej,ρ∗ numbers are equal by the properies of duality, so we are done. � Corollary 1.51.3. One has: Pi∗∗ = L∗ ρ⊗P∗∗i⊗Lρ; Li∗∗ = L∗ ρ⊗L∗∗i⊗Lρ. Proof. Again, it suffices to prove the first statement. We have Pi∗∗ = Pi ∗∗ = (P∗i ⊗ Lρ)∗ = Lρ ∗ ⊗ P∗∗ i = L∗ ρ ⊗ P∗∗i ⊗ Lρ Definition 1.51.4. Lρ is called the distinguished invertible object of C. We see that for any i, the socle of Pi is Lˆi := L∗ ρ ⊗∗∗ Li = L∗∗ i ⊗ Lρ∗. This implies the following result. Corollary 1.51.5. Any finite dimensional quasi-Hopf algebra H is a Frobenius algebra, i.e. H is isomorphic to H∗ as a left H-module. Proof. It is easy to see that that a Frobenius algebra is a quasi-Frobenius algebra (i.e. a finite dimensional algebra for which projective and in-jective modules coincide), in which the socle of every indecomposable� 98 projective module has the same dimension as its cosocle (i.e., the simple quotient). As follows from the above, these conditions are satisfied for finite dimensional quasi-Hopf algebras (namely, the second condition follows from the fact that Lρ is 1-dimensional). � 1.52. Integrals in quasi-Hopf algebras. Definition 1.52.1. A left integral in an algebra H with a counit ε : H k is an element I ∈ H such that xI = ε(x)I for all x ∈ H.→Similarly, a right integral in H is an element I ∈ H such that Ix = ε(x)I for all x ∈ H. Remark 1.52.2. Let H be the convolution algebra of distributions on a compact Lie group G. This algebra has a counit ε defined by ε(ξ) = ξ(1). Let dg be � a left-invariant Haar measure on G. Then the distribution I(f ) = G f(g)dg is a left integral in H (unique up to scaling). This motivates the terminology. Note that this example makes sense for a finite group G over any field k. In this case, H = k[G], and I = g∈G g is both a left and a right integral. Proposition 1.52.3. Any finite dimensional quasi-Hopf algebra ad-mits a unique nonzero left integral up to scaling and a unique nonzero right integral up to scaling. Proof. It suffices to prove the statement for left integrals (for right integrals the statement is obtained by applying the antipode). A left integral is the same thing as a homomorphism of left modules k H.→Since H is Frobenius, this is the same as a homomorphism k H∗, i.e. →a homomorphism H k. But such homomorphisms are just multiples of the counit. → � Note that the space of left integrals of an algebra H with a counit is a right H-module (indeed, if I is a left integral, then so is Iy for all y ∈ H). Thus, for finite dimensional quasi-Hopf algebras, we obtain a character χ : H k, such that Ix = χ(x)I for all x ∈ H. This→character is called the distinguished character of H (if H is a Hopf algebra, it is commonly called the distinguished grouplike element of H∗, see [Mo]). Proposition 1.52.4. Let H be a finite dimensional quasi-Hopf algebra, and C = Rep(H). Then Lρ coincides with the distinguished character χ. Proof. Let I be a nonzero left integral in H. We have xI = ε(x)I and Ix = χ(x)I. This means that for any V ∈ C, I defines a morphism from V ⊗ χ−1 to V .��99 The element I belongs to the submodule Pi of H, whose socle is the trivial H-module. Thus, Pi ∗ = P (1), and hence by Lemma 1.51.2, i = ρ. Thus, I defines a nonzero (but rank 1) morphism Pρ ⊗χ−1 → Pρ. The image of this morphism, because of rank 1, must be L0 = 1, so 1 is a quotient of Pρ ⊗ χ−1, and hence χ is a quotient of Pρ. Thus, χ = Lρ, and we are done. � Proposition 1.52.5. The following conditions on a finite dimensional quasi-Hopf algebra H are equivalent: (i) H is semisimple; (ii) ε(I) = 0 (where I is a left integral in H); (iii) I2 = 0� ; (iv) I can be normalized to be an idempotent. Proof. (ii) implies (i): If ε(I) = 0 then k = 1 is a direct summand in H as a left H-module. This implies that 1 is projective, hence Rep(H) is semisimple (Corollary 1.13.7). (i) implies (iv): If H is semisimple, the integral is a multiple of the projector to the trivial representation, so the statement is obvious. (iv) implies (iii): obvious. (iii) implies (ii): clear, since I2 = ε(I)I. � Definition 1.52.6. A finite tensor category C is unimodular if Lρ = 1. A finite dimensional quasi-Hopf algebra H is unimodular if Rep(H) is a unimodular category, i.e. if left and right integrals in H coincide. Remark 1.52.7. This terminology is motivated by the notion of a unimodular Lie group, which is a Lie group on which a left invariant Haar measure is also right invariant, and vice versa. Remark 1.52.8. Obviously, every semisimple category is automati-cally unimodular. Exercise 1.52.9. (i) Let H be the Nichols Hopf algebra of dimension 2n+1 (Example 1.24.9). Find the projective covers of simple objects, the distinguished invertible object, and show that H is not unimod-ular. In particular, Sweedler’s finite dimensional Hopf algebra is not unimodular. (ii) Do the same if H