( �44 1.21. Bialgebras. Let C be a finite monoidal category, and (F, J) : C → Vec be a fiber functor. Consider the algebra H := End(F ). This algebra has two additional structures: the comultiplication Δ : H →H ⊗ H and the counit ε : H k. Namely, the comultiplication is →defined by the formula Δ(a) = α−1 Δ(a)),F,F where �Δ(a) ∈ End(F ⊗ F ) is given by �X,Y aX⊗Y JX,Y ,Δ(a)X,Y = J−1 and the counit is defined by the formula ε(a) = a1 ∈ k. Theorem 1.21.1. (i) The algebra H is a coalgebra with comultiplica-tion Δ and counit ε. (ii) The maps Δ and ε are unital algebra homomorphisms. Proof. The coassociativity of Δ follows form axiom (1.4.1) of a monoidal functor. The counit axiom follows from (1.4.3) and (1.4.4). Finally, ob-serve that for all η, ν ∈ End(F ) the images under αF,F of both Δ(η)Δ(ν) and Δ(ην) have components J−1 (ην)X⊗Y JX,Y ; hence, Δ is an algebra X,Y homomorphism (which is obviously unital). The fact that ε is a unital algebra homomorphism is clear. � Definition 1.21.2. An algebra H equipped with a comultiplication Δ and a counit ε satisfying properties (i),(ii) of Theorem 1.21.1 is called a bialgebra. Thus, Theorem 1.21.1 claims that the algebra H = End(F ) has a natural structure of a bialgebra. Now let H be any bialgebra (not necessarily finite dimensional). Then the category Rep(H) of representations (i.e., left modules) of H and its subcategory Rep(H) of finite dimensional representations of H are naturally monoidal categories (and the same applies to right modules). Indeed, one can define the tensor product of two H-modules X, Y to be the usual tensor product of vector spaces X ⊗ Y , with the action of H defined by the formula ρX⊗Y (a) = (ρX ⊗ ρY )(Δ(a)), a ∈ H (where ρX : H End(X), ρY : H End(Y )), the associativity iso-→ →morphism to be the obvious one, and the unit object to be the 1-dimensional space k with the action of H given by the counit, a ε(a).→Moreover, the forgetful functor Forget : Rep(H) Vec is a fiber func-→ tor.� � � 45 Thus we see that one has the following theorem. Theorem 1.21.3. The assignments (C, F ) �→ H = End(F ), H �→(Rep(H), Forget) are mutually inverse bijections between 1) finite abelian k-linear monoidal categories C with a fiber functor F , up to monoidal equivalence and isomorphism of monoidal functors; 2) finite dimensional bialgebras H over k up to isomorphism. Proof. Straightforward from the above. � Theorem 1.21.3 is called the reconstruction theorem for finite dimen-sional bialgebras (as it reconstructs the bialgebra H from the category of its modules using a fiber functor). Exercise 1.21.4. Show that the axioms of a bialgebra are self-dual in the following sense: if H is a finite dimensional bialgebra with multiplication µ : H ⊗ H H, unit i : k H, comultiplication → →Δ : H → H ⊗ H and counit ε : H → k, then H∗ is also a bialgebra, with the multiplication Δ∗, unit ε∗, comultiplication µ∗, and counit i∗. Exercise 1.21.5. (i) Let G be a finite monoid, and C = VecG. Let F : C → Vec be the forgetful functor. Show that H = End(F ) is the bialgebra Fun(G, k) of k-valued functions on G, with comultiplication Δ(f)(x, y) = f(xy) (where we identify H ⊗ H with Fun(G × G, k)), and counit ε(f) = f(1). (ii) Show that Fun(G, k)∗ = k[G], the monoid algebra of G (with basis x ∈ G and product x y = xy), with coproduct Δ(x) = x ⊗x, and · counit ε(x) = 1, x ∈ G. Note that the bialgebra k[G] may be defined for any G (not necessarily finite). Exercise 1.21.6. Let H be a k-algebra, C = H −mod be the category of H-modules, and F : C → Vec be the forgetful functor (we don’t assume finite dimensionality). Assume that C is monoidal, and F is given a monoidal structure J. Show that this endows H with the structure of a bialgebra, such that (F, J) defines a monoidal equivalence C → Rep(H). Note that not only modules, but also comodules over a bialgebra H form a monoidal category. Indeed, for a finite dimensional bialgebra, this is clear, as right (respectively, left) modules over H is the same thing as left (respectively, right) comodules over H∗. In general, if X, Y are, say, right H-comodules, then the right comodule X ⊗ Y is the usual tensor product of X, Y with the coaction map defined as follows: if x ∈ X, y ∈ Y , π(x) = xi ⊗ ai, π(y) = yj ⊗ bj , then πX⊗Y (x ⊗ y) = xi ⊗ yj ⊗ aibj .� � � �� � � �� � � �46 For a bialgebra H, the monoidal category of right H-comodules will be denoted by H − comod, and the subcategory of finite dimensional comodules by H − comod. 1.22. Hopf algebras. Let us now consider the additional structure on the bialgebra H = End(F ) from the previous subsection in the case when the category C has right duals. In this case, one can define a linear map S : H H by the formula→ S(a)X = a∗ X∗ , where we use the natural identification of F (X)∗ with F (X∗). Proposition 1.22.1. (“the antipode axiom”) Let µ : H ⊗ H H and→i : k H be the multiplication and the unit maps of H. Then→ µ (Id ⊗ S) Δ = i ε = µ (S ⊗ Id) Δ ◦ ◦ ◦ ◦ ◦ as maps H H.→ Proof. For any b ∈ End(F ⊗F ) the linear map µ◦(Id⊗S)(α−1 (b))X , X ∈F,F C is given by (1.22.1) coevF (X) bX,X∗ evF (X)F (X) −−−−−→ F (X)⊗F (X)∗⊗F (X) −−−→ F (X)⊗F (X)∗⊗F (X) −−−−→ F (X), where we suppress the identity isomorphisms, the associativity con-straint, and the isomorphism F (X)∗ = F (X∗). Indeed, it suffices to∼check (1.22.1) for b = η ⊗ ν, where η, ν ∈ H, which is straightforward. Now the first equality of the proposition follows from the commuta-tivity of the diagram coevF (X)(1.22.2) F (X) ��F (X) ⊗ F (X)∗ ⊗ F (X) Id JX,X∗ F (coevX )F (X) ��F (X ⊗ X∗) ⊗ F (X) η1 ηX⊗X∗ F (coevX )F (X) ��F (X ⊗ X∗) ⊗ F (X) J−1 Id X,X∗ evF (X)F (X) ��F (X) ⊗ F (X)∗ ⊗ F (X), for any η ∈ End(F ). Namely, the commutativity of the upper and the lower square fol-lows from the fact that upon identification of F (X)∗ with F (X∗), the morphisms evF (X) and coevF