4
1. Monoidal categories
1.1. The denition of a monoidal category. A good way of think
ing about category theory (which will be especially useful throughout
these notes) is that category theory is a renement (or categorica
tion) of ordinary algebra. In ot
4
1. Monoidal categories
1.1. The denition of a monoidal category. A good way of think
ing about category theory (which will be especially useful throughout
these notes) is that category theory is a renement (or categorica
tion) of ordinary algebra. In ot
86
1.45. Tensor categories with nitely many simple objects. FrobeniusPerron dimensions. Let A be a Z+ -ring with Z+ -basis I .
Denition 1.45.1. We will say that A is transitive if for any X, Z I
there exist Y1 , Y2 I such that X Y1 and Y2 X involve Z with
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2.11. Main Theorem.
Exercise 2.11.1. Show that for any M M the object Hom(M, M )
with the multiplication dened above is an algebra (in particular, dene
the unit morphism!).
Theorem 2.11.2. Let M be a module category over C , and assume
that M M satise
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1.51. The distinguished invertible object. Let C be a nite tensor
category with classes of simple objects labeled by a set I . Since duals
to projective objects are projective, we can dene a map D : I I
such that Pi = PD(i) . It is clear that D2 (i) =
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1.4. Monoidal functors, equivalence of monoidal categories.
As we have explained, monoidal categories are a categorication of
monoids. Now we pass to categorication of morphisms between monoids,
namely monoidal functors.
Denition 1.4.1. Let (C , , 1, a
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1.25. The Quantum Group Uq (sl2 ). Let us consider the Lie algebra
sl2 . Recall that there is a basis h, e, f sl2 such that [h, e] = 2e, [h, f ] =
2f , [e, f ] = h. This motivates the following denition.
Denition 1.25.1. Let q k , q = 1. The quantum gr
34
1.13. Exactness of the tensor product.
Proposition 1.13.1. (see [BaKi, 2.1.8]) Let C be a multitensor cate
gory. Then the bifunctor : C C C is exact in both factors (i.e.,
biexact).
Proof. The proposition follows from the fact that by Proposition 1.10.
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1.37. Quantum traces. Let C be a rigid monoidal category, V be an
object in C , and a Hom(V, V ). Dene the left quantum trace
(1.37.1)
TrL (a) := evV (a IdV ) coevV End(1).
V
Similarly, if a Hom(V,
trace
(1.37.2)
V ) then we can dene the right quantum
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2.7. First properties of exact module categories.
Lemma 2.7.1. Let M be an exact module category over nite multitensor category C . Then the category M has enough projective objects.
Proof. Let P0 denote the projective cover of the unit object in C .
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1.32. The Andruskiewitsch-Schneider conjecture. It is easy to
see that any Hopf algebra generated by grouplike and skew-primitive
elements is automatically pointed.
On the other hand, there exist pointed Hopf algebras which are not
generated by groupli
14
1.4. Monoidal functors, equivalence of monoidal categories.
As we have explained, monoidal categories are a categorication of
monoids. Now we pass to categorication of morphisms between monoids,
namely monoidal functors.
Denition 1.4.1. Let (C , , 1, a
25
1.9. The MacLane coherence theorem. In a monoidal category,
one can form n-fold tensor products of any ordered sequence of objects
X1 , ., Xn . Namely, such a product can be attached to any parenthe
sizing of the expression X1 . Xn , and such products
34
1.13. Exactness of the tensor product.
Proposition 1.13.1. (see [BaKi, 2.1.8]) Let C be a multitensor cate
gory. Then the bifunctor : C C C is exact in both factors (i.e.,
biexact).
Proof. The proposition follows from the fact that by Proposition 1.10.
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1.21. Bialgebras. Let C be a nite monoidal category, and (F, J ) :
C Vec be a ber functor. Consider the algebra H := End(F ). This
algebra has two additional structures: the comultiplication : H
H H and the counit : H k . Namely, the comultiplication