lecture 12 - stanford universitysporadic.stanford.edu/quantum/lecture12.pdf · 12t 13: explicitly t...

The Drinfeld Double (II) Modular Tensor Categories

Lecture 12

Daniel Bump

May 29, 2019

Vi V∗iVj V∗j


Review: convolution theory

Suppose C is a coalgebra and A is an algebra. Then the spaceHom(C,A) of all linear maps C→ A has an algebra structure. Iff , g ∈ Hom(C,A) then f ? g ∈ End(H) is defined to be thecomposition:

C C ⊗ C A⊗ A A .∆ f⊗g µ

Associativity follows from the associativity of µ and thecoassociativity of ∆.

As a special case, if A is a Hopf algebra then A⊗ A∗ ∼= End(A)as a vector space. This is an isomorphism of algebras usingthe convolution product in End(A).


Review: Convolution theory (continued)

The counit ε is the unit in the convolution ring. This is aparaphrase of the definition of ε.

If f ∈ End(A) then the corresponding element of A⊗ A∗ isf (ei)⊗ ei (implied summation). In particular the canonicalelement T = ei ⊗ ei ∈ A⊗ A∗ corresponds the identity mapIA ∈ End(A). Transferring this fact to A⊗ A∗ gives the identity

ε(ei)⊗ ei = 1A⊗A∗ .

The antipode S is the inverse of I in the canonical ring. Again,this is a paraphrase of the definition of S. So transferring thisfact to A⊗ A∗,

T−1 =∑

i

S(ei)⊗ ei.


Review: the canonical element

Let A be a finite-dimensional Hopf algebra with basis ei, and letei be the dual basis of A∗. The canonical elementT = ei ⊗ ei ∈ A⊗ A∗ does not depend on choice of basis.

In Lecture 11 we proved

(∆⊗ 1)(T) = T13T23, (1⊗∆)(T) = T12T13.

Explicitly

T13T23 = ei ⊗ ej ⊗ eiej, T12T13 = eiej ⊗ ei ⊗ ek.

(In Lecture 11 we wrote this for Aop ⊗ A∗ so the second formulaappeared differently.)


The story so far

The Drinfeld double was characterized thus by Drinfeld in his1986 ICM talk.

Let A be a Hopf algebra. Denote by A◦ the algebra A∗ with theopposite comultiplication. It can be shown that there is a uniquequastriangular Hopf algebra (D(A),R) such that (1) D(A)contains A and A◦ as Hopf subalgebras (2) R is the image of thecanonical element of A⊗ A◦ under the embeddingA⊗ A◦ → D(A) and (3) the linear mapping A⊗ A◦ → D(A) givenby a⊗ b→ ab is bijective.

In Lecture 11 we constructed the dual Hopf algebra D(A)∗ bymodifying the comultiplication of A∗ ⊗ Aop, using one of its twocanonical elements to twist. Today We will see that the othercanonical element provides the R-matrix.


Review: D(A)∗

We created D(A)∗ by Drinfeld twisting A∗ ⊗ Aop. Thus as analgebra D(A)∗ = A∗ ⊗ Aop but the comultiplication is

∆F(x) = F∆(x)F−1

whereF = (1A∗ ⊗ S−1ei)⊗ (ei ⊗ 1A),

F−1 = (1A∗ ⊗ ei)⊗ (ei ⊗ 1A).

Then we define D(A) to be the dual of D(A)∗. As a coalgebra itis A⊗ A◦, where A◦ = (Aop)∗ = (A∗)cop. Since thecomultiplication of D(A)∗ differs from the comultiplication inA∗ ⊗ Aop, the multiplication in D(A) is correspondingly modifiedfrom A⊗ Aop.


Subalgebras of D(A) isomorphic to A,A◦

Since the multiplication in D(A) is changed from A⊗ A◦, we willmodify the notation and write a � λ for the elementcorresponding to a ∈ A and λ ∈ Acirc. We will continue to usethe notation λ⊗ a for the element of D(A)∗.

PropositionThe maps A −→ D(A) and A◦ −→ D(A) given by a 7→ a � 1 andλ 7→ 1 � λ are Hopf algebra homomorphisms. Thus D(A)contains isomorphic copies of A and A◦.

Since D(A) and A⊗ A◦ are identified as coalgebras, we haveonly to show that the maps a 7→ a � 1 and λ 7→ 1 � λ respectmultiplication. These are similar and we prove the first.We must show that for ν ⊗ x ∈ D(A)∗ we have

〈(a � 1)(b � 1), ν ⊗ x〉 = 〈ab � 1, ν ⊗ x〉


Proof

Since

F = (1A∗ ⊗ S−1ei)⊗ (ei ⊗ 1A), F−1 = (1A∗ ⊗ ei)⊗ (ei ⊗ 1A),

we have

∆F(ν ⊗ x) = (ν(1) ⊗ S−1(ei)x(1)ej)⊗ (eiν(2)ej ⊗ x(2)).

Thus 〈(a � 1)(b � 1), ν ⊗ x〉 equals

〈a � 1, ν(1) ⊗ S−1(ei)x(1)ej〉〈b � 1, eiν(2)ej ⊗ x(2)〉.

We now remember〈1, x〉 = ε(x).

Also ε(S−1(ei)) = ε(ei)S−1(1) = ε(ei).


Proof (concluded)

From the previous page, 〈(a � 1)(b � 1), ν ⊗ x〉 equals

〈a � 1, ν(1) ⊗ S−1(ei)x(1)ej〉〈b � 1, eiν(2)ej ⊗ x(2)〉.

Thus

〈(a � 1)(b � 1), ν ⊗ x〉 = 〈a, ν(1)〉ε(ei)ε(x(1))ε(ej)〈b, eiν(2)ej〉ε(x(2)).

Since ε(ei)⊗ ei = 1A⊗A∗ we get

〈(a�1)(b�1), ν⊗x〉 = 〈a, ν(1)〉〈b, ν(2)〉ε(x) = 〈ab, ν〉ε(x) = 〈ab�1, ν⊗x〉.

Therefore (a � 1)(b � 1) = (ab � 1) in D(A).

Similarly (1 � λ)(1 � µ) = 1 � λµ.


Multiplication in D(A)

Although the multiplication in D(A) is different from A⊗ A◦, thefollowing formulas can be used.

PropositionWe have

(a � 1)(1 � λ) = a � λ,

(1 � λ)(a � 1) = 〈λ(1), S−1a(1)〉(a(2) � λ(2))〈λ(3), a(3)〉.

Since

∆F(ν ⊗ x) = (ν(1) ⊗ S−1(ei)x(1)ej)⊗ (eiν(2)ej ⊗ x(2)),

we have

〈(a�1)(1�λ), ν⊗x〉 = 〈a�1, ν(1)⊗S−1(ei)x(1)ej〉〈1�λ, eiν(2)ej⊗x(2)〉

= 〈a, ν(1)〉ε(ei)ε(x(1))ε(ej)ε(ei)ε(ν(2))ε(ej)〈λ, x(2)〉.


Proof

Since ε(ei)⊗ ei = 1A⊗A∗ we have ε(ei)ε(ei) = ε(1A⊗A∗) = 1 andusing

ν(1)ε(ν(2)) = ν, ε(x(1))x(2) = x,

we see

〈(a � 1)(1 � λ), ν ⊗ x〉 = 〈a, ν〉〈λ, x〉 = 〈a � λ, ν ⊗ x〉

proving (a � 1)(1 � λ) = a � λ.


Proof (continued)

Next consider

〈(1�λ)(a�1), ν⊗x〉 = 〈1�λ, ν(1)⊗S−1(ei)x(1)ej〉〈a�1, eiν(2)ej⊗x(2)〉.

This equals

ε(ν(1))〈λ(1), S−1(ei)〉〈λ(2), x(1)〉〈λ(3), ej〉〈ei, a(1)〉〈ν(2), a(2)〉〈εj, a(3)〉ε(x(2))

= 〈S−1(λ(1)), ei〉〈λ(2), x〉〈λ(3), ej〉〈a(1), ei〉〈a(2), ν〉〈a(3), ε

j〉

Now〈λ(3), ej〉〈εj, a(3)〉 =

〈λ(3), a(3)〉〈S−1(λ(1)), ei〉〈a(1), ei〉 = 〈S−1(λ(1)), a(1)〉


Proof (concluded)

so〈(1 � λ)(a � 1), ν ⊗ x〉 =

〈S−1(λ(1)), a(1)〉〈λ(2), x〉〈λ(3), a(3)〉〈a(2), ν〉 =

〈λ(1), S−1a(1)〉〈λ(3), a(3)〉〈a(2) � λ(2), ν ⊗ x〉.

This proves

(1 � λ)(a � 1) = 〈λ(1), S−1a(1)〉(a(2) � λ(2))〈λ(3), a(3)〉.


The R-matrix

Theorem (Drinfeld)The double D(A) is quasitriangular with R-matrix

R = (ei � 1A◦)⊗ (1A � ei).

By convolution theory, R is invertible and

R−1 = (S(ei) � 1A◦)⊗ (1A � ei).

To verify quasitriangularity we need

τ∆(x) = R∆(x)R−1, x ∈ D(A),

(∆⊗ 1)R = R13R23, (1⊗∆)R13R12.


The braiding axioms

The identities

(∆⊗ 1)R = R13R23, (1⊗∆)R13R12 (1)

follow from the corresponding identities

(∆⊗1)T = T13T23 = ei⊗ej⊗eiej, (1⊗∆T) = T12T13 = eiej⊗ei⊗ej

for the canonical element. These identities are for A⊗ A∗,whereas D(A) is isomorphic as a coalgebra to A⊗ A◦ withA◦ = (A∗)cop. So for the second identity we need

(1⊗ τ∆T) = T13T12 = ejei ⊗ ej ⊗ ei,

and applying this to R = (ei � 1A◦)⊗ (1A � ei), (1) follows.


The coboundary identity

To finish the quasitriangularity proof, we need

R(∆(u)R−1 = τ∆(u), u ∈ D(A). (2)

It is sufficient to check this for u in a set of generators, so wemay assume that u = a � 1A◦ or u = 1A � λ.

The computation is simplified by using a quasitriangularitycriterion due to Radford, Minimal quasitriangular Hopf algebras,J. Algebra 157 (1993), 285-315. I am also following the(possibly hard to find) book Quantum Groups and KnotInvariants by Kassel, Rosso and Turaev.

Radford actually gave 4 equivalent criteria for the coboundarycondition.


Radford’s criteria

Proposition (Radford)Each of the following conditions is equivalent to

R(∆(u)R−1 = τ∆(u), u ∈ D(A). (3)

(i) R(1)u⊗ R(2) = u(2)R(1) ⊗ u(1)R(2)S(u(3)),

(ii) uR(1) ⊗ R(2) = R(1)u(2) ⊗ S(u(1))R(2)u(3),

(iii) R(1) ⊗ R(2)u = u(3)R(1)S−1(u(1))⊗ u(2)R(2),

(iv) R(1) ⊗ uR(2) = u(3)R(1)S−1(u(1))⊗ u(2)R(2),

We will use (i) and (iii) but only prove that (iii) implies (3). Thus:

R∆(u) = R(1⊗ u(2))(u(1) ⊗ 1) = u(4)R(1)S−1(u(2))u(1) ⊗ u(3)R

(2)

= τ∆(u)R.


The coboundary identity for D(A)

It is enough to prove the coboundary condition for a set ofgenerators of D(A). The strategy is to use Radford’s criterion(iii) to handle u of the form a � 1 and criterion (ii) for u = 1 � λ.We will only carry out the first calculation.

We must show

R(1) ⊗ R(2)u = u(3)R(1)S−1(u(1))⊗ u(2)R

(2)

with u = a � 1, R(1) = ei � 1, R(2) = 1 � ei.


Proof

We want to verify

R(1⊗ u) = u(3)R(1)S−1(u(1))⊗ u(2)R

(2)

with R(1) = ei � 1, R(2) = 1 � ei and u = a � 1. The left-hand sideis

(ei � 1)(1 � 1)⊗ (1 � ei)(a � 1)

= 〈ei(1), S

−1a(1)〉〈ei(3), a(3)〉((ei � 1)⊗ (a(2) � ei

(2))).

The right-hand side equals

(a(3)eiS−1(a(1)) � 1)⊗ (a(2) � 1)(1 � ei)

= (a(3)eiS−1(a(1)) � 1)⊗ (a(2) � ei).


Proof (continued)

Let ∆◦ be the comultiplication in A◦. It is the opposite of thecomultiplication ∆ in A∗. Therefore in A

a(3)eiS−1a(1) = 〈ej, a(3)eiS−1a(1)〉ej = 〈ej(3), a(3)〉〈ej

(2), ei〉〈ej(1), S

−1a(1)〉ej.

Using this the right-hand side equals

〈ej(3), a(3)〉〈ej

(2), ei〉〈ej(1), S

−1a(1)〉((ej � 1)⊗ (a(2) � ei)),

or since 〈ej(2), ei〉ei = ej

(2)

〈ej(3), a(3)〉〈ej

(1), S−1a(1)〉((ej � 1)⊗ (a(2) � ej

(2))).

Switching i and j this equals the left-hand side, and the identityis proved.

For the otherside, use Radford’s criterion (ii). We leave thiscase to the reader.


The plan

Now let g = u⊕ h⊕ u− be the triangular decomposition of asemisimple Lie algebra. Drinfeld constructed the quantizedenveloping algebra as follows.

First of all, we have two Lie algebras

b = h⊕ u, b− = h⊕ u−

that are very similar to each other. The quantized envelopingalgebra Uq(u) can be constructed directly, though it is easier towork with a “free” algebra U+ that omits the Serre relations andhas Uq(u) as a quotient, and similarly U−.


The plan (continued)

It is possible to construct a pairing between U+ and U− thatthat is almost a dual pairing, and hence construct the doubleU+ � U−. Unfortunately this contains two copies of h, but asuitable quotient is Uq(g), and it will be possible to see all therelations, including the Serre relations this way.

Strictly speaking Uq(g) is not quasitriangular, since theuniversal R-matrix does not live in Uq(g) but in a completion.The reason that our theorem does not imply quasitriangularityis U+ and U− are infinite dimensional, so ei ⊗ ei is an infinitesum. So quasitriangularity can be obtained (in a suitablesense) but there are further technical issues.


Modular categories: overview

Modular tensor categories are ribbon categories of a particularsort. They are introduced in Chapter II of Turaev’s book, wherehe writes:

As we know, ribbon categories give rise to invariants of links inEuclidean 3-space. Unfortunately, they are too general to yieldsimilar invariants of links in 3-manifolds. This leads to theconcept of modular category which is the key algebraic conceptof this monograph.

Modular categories are closely related to topological quantumfield theories. They can be constructed using quantum groupsat roots of unity or conformal field theory. Interestingly thesecategories come with an action of SL(2,Z) which is the origin ofthe term “modular.” Today we will give the definition.


Abelian categories

Around 1955, Buchsbaum and (independently) Grothendieckaxiomatized categories in which homological algebra works.Their axiomatizations were similar but slightly different. Thenotion of an abelian category eventually stabilized to that usedin Mac Lane’s books Homology and Categories for the WorkingMathematician.

The archetypal abelian category is the category of modulesover a ring, a comfortable category in which the Snake Lemmais proved by diagram chasing. But other naturally occurringabelian categories such as the category of sheaves of abeliangroups over a topological space do not present themselves asmodule categories. Still the Mitchell-Freyd embedding theoremshows that every abelian category can be embedded in amodule category, so proofs using diagram chasing are valid.


Abelian categories (continued)

An additive category is one in which the Hom sets form abeliangroups, the composition law being bilinear; there is alsoassumed to be a 0 element that is both initial and terminal, andfinite products that are also coproducts.

A morphism f : A→ B is a monomorphism (generalizing thenotion of an injective map) if for morphisms g, g′ : C→ A theidentity f ◦ g = f ◦ g′ implies g = g′. The dual property isepimorphism.

If f : A→ B is a morphism, the kernel of f is a morphismi : K → A such that f ◦ i = 0 and for all objects C and g : C→ Aif fg = 0 then g factors uniquely through K. A kernel is amonomorphism. The dual notion is that of a cokernel.


Abelian categories (continued)

In an abelian category, it is assumed that every morphism haveboth a kernel and a cokernel.

It is also assumed that every monomorphism is the kernel of itscokernel, and that every epimorphism is the cokernel of itskernel.

Finally we assume that we may factor any morphism f : A→ Bas f = ψ ◦ φ where for some object C the morphism φ : A→ Cis an epimorphism and ψ : C→ B is a monomorphism.


Tensor categories

The term tensor category is not used consistently by differentauthors. But we will use this term to mean an additive categorywith an additional bilinear bifunctor ⊗ that makes it into amonoidal category. Usually we want the category to be abelian.

The bilinearity assumption means that we have natural andadditive isomorphisms

A⊗ (B⊕ C) ∼= (A⊗ B)⊕ (A⊗ C).

Let K be the unit object in the tensor category C. Then End(K)is a ring; its additive structure comes from the fact that it is anobject in an additive category, and the multiplication comesfrom the monoidal isomorphism K ∼= K ⊗ K. We will call End(K)the ground ring.


Simple objects

We define a simple object in an abelian tensor category to beobject A that is not zero but which has no subobjects. Thus ifi : C→ A is morphism then either i = 0 or i is an epimorphism.

For example the unit object K is simple if End(K) is a field. SeeDeligne and Milne, Tannakian Categories, Proposition 1.17.

Deligne and Milne, Tannakian Categories

If V is any object then since V ∼= K ⊗ V it becomes a vectorspace over k = End(K). If K is an algebraically closed field andV is a simple object that is finite-dimensional over k, thenk = End(V) (Schur’s Lemma).

https://www.jmilne.org/math/xnotes/tc2018.pdf


Semisimple categories

We will define a semisimple category to be an abelian tensorcategory with unit object K in which k = End(K) is a field, allobjects are finite-dimensional vector spaces over K, and if{Vi|i ∈ I} are representatives of the isomorphism classes ofsimple objects, then every object is a finite direct sum of Vi. Insuch a decomposition,

V ∼=⊕

niVi

the multiplicities ni are uniquely determined.


Fusion categories

A Modular tensor category is a semisimple ribbon category Cwith ground field algebraically closed. Frequently thesecategories are discussed under the term fusion categories.Such a category can be constructed from representations of aquantum group at a at root of unity, and alternatively as thefusion category of fields in a Wess-Zumino-Witten (WZW)conformal field theory.

Bakalov and Kirllov: Lectures on tensor categories and modular functions

Fuchs: fusion rules in conformal field theory

In addition to those references see Di Francesco, Mathieu andSénéchal, Conformal Field Theory, chapters 15 and 16.

https://www.math.stonybrook.edu/~kirillov/tensor/tensor.html

https://arxiv.org/pdf/hep-th/9306162.pdf


The quantum dimension

We are assuming that the category C is semisimple. So abelianwith a finite number of nonisomorphic simple objects Vi (i ∈ I)such that every object is uniquely isomorphic to a direct sum ofVi. It is ribbon, so every object has a dual, which is both a rightand a left dual.

Let K denote the unit object in the category If V = Vi then wecan define the quantum dimension to be the morphism K → Kdefined as in previous lectures by the composition

K V ⊗ V∗ KcoevV evV

V V∗ .


The quantum dimension (continued)

From Lecture 9, this definition has implicitly used the ribbonelement, since if we substitute the definition ofevV : V ⊗ V∗ → K, the quantum dimension is actually thequantum trace of 1V : V → V as defined in Lecture 4:

K V ⊗ V∗ V∗ ⊗ V V∗ ⊗ V KcoevV cV,V∗ 1⊗θ−1

V evV

V V∗

θ−1V


The Fusion ring

Let Vi (i ∈ I) be representatives of the isomorphism classes ofsimple objects in the tensor category. We are assuming that thenumber of these is finite. We define nonnegative integers Nk

ij by

Vi ⊗ Vj = NkijVk

(implied summation).

The Grothendieck group of the category is called the Fusionring. It has generators xi corresponding to the simple objects inthe category, with structure constants Nk

ij, so

xixj = Nkijxk.

The ring F with these generators has a conjugation operationc : F that permutes the xi so that c(xi) = xi∗ where Vi∗ = V∗i isthe dual. Also the twist θVi is a scalar θi by Schur’s Lemma.


The S-matrix

Now let Vi and Vj be simple objects. We define s̃i,j to be thescalar that is the morphism K → K defined by the link:

Vi V∗iVj V∗j

Now we impose the assumption that the matrix s̃ = (̃sij) isinvertible. This is called the S-matrix. (“S” for scattering.) Thiscompletes the definition of a modular tensor category.


Modularity

The group SL(2,Z) has two generators

S =

(−1

1

), T =

(1 1

1

),

subject to the relations

S4 = 1, S2T = TS2, (ST)3 = S2.

The modularity consists of an action of SL(2,Z) on thecategory, or at least its Grothendieck group, known as thefusion ring. The matrix s̃4 is not the identity, but it is diagonal, somultiplying it by certain constants gives a matrix s that satisfiess4 = 1. Supplementing it by the matrix t = (δiθij), the relations ofSL(2,Z) are satisfied.


Why modularity?

The fact that there is an action of SL(2,Z) is explained by thefact that the fields in certain conformal field theories can beinterpreted as modular forms. See:

Erik Verlinde, Fusion rules and modular transformations in 2Dconformal field theory. Nuclear Phys. B 300 (1988), no. 3,360-376.

The fusion rings can be constructed alternatively from suchconformal field theories, or from the representation theory ofquantum groups at roots of unity.

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