Lecture 13 - sporadic.stanford.edusporadic.stanford.edu/quantum/lecture13.pdfLecture 13 Daniel Bump May 29, 2019 V i V j. Review: Uq(sln) The Drinfeld Double (III)Modular Tensor Categories

Review: Uq(sln) The Drinfeld Double (III) Modular Tensor Categories (II)

Lecture 13

Daniel Bump

May 29, 2019

V∗i Vj


Review: The quantized enveloping algebra

Today we will show how Uq(sln) can be obtained from theDrinfeld double construction.

In Lecture 10 we gave generators and relations for this algebra,but we did not give motivation for them or prove that the algebraas described was correct. But we want to have this in mind solet us recall this presentation from Lecture 10.

Uq(sln) has generators Ki (invertible) together with Ei and Fi.The Ki together with the Ei generate a Hopf subalgebra Uq(b)(corresponding to the positive Borel subgroup of SL(n)) and theK − I with the Fi generate the Hopf subalgebra Uq(b−)corresponding to the negative Borel.


Review: the Cartan matrix

Let

aij = 〈α∨i , αj〉 =

2 if i = j;−1 if j = i± 1;0 if |j− i| > 2,

These entries make up the Cartan matrix. So for SL(4):

A =

2 −1 0 0−1 2 −1 00 −1 2 −10 0 −1 2


The algebra Uq(sln) has generators Ki (invertible), Ei and Fi

subject to the relations

KiEjK−1i = q〈αi,αj〉Ej, KiFjK−1

i = q−〈αi,αj〉Fj,

EiFj − FjEi = δijKi − K−1

iq− q−1 ,

and the quantum Serre relations,

EiEj = EjEi if |i− j| > 1,

E2i Ej − (q + q−1)EiEjEi + EjE2

i = 0.

The comultiplication is given on generators by:

∆(Ki) = Ki ⊗ Ki,

∆(Ei) = Ei ⊗ Ki + 1⊗ Ei, ∆(Fi) = Fi ⊗ 1 + K−1i ⊗ Fi.


Hopf pairings

Let A and B be Hopf algebras, together with a pairing A×B→ K(the ground field) that satisfies the following conditions.

〈a, bb′〉 = 〈a(1), b〉〈a(2), b′〉

and〈aa′, b〉 = 〈a, b(2)〉〈a′, b(1)〉.

Note the reversal in the second identity. We also require

〈a, 1〉 = ε(a), 〈1, b〉 = ε(b), 〈S(a), b〉 = 〈a, S−1b〉.

This is called a Hopf pairing. For example, the dual pairingbetween A and Aop = (A∗)cop satisfies these conditions.


Generalized double construction

Given two Hopf algebras A,B with a Hopf pairing, we maydefine the double A� B by the formulas of Lecture 12. Thecomultiplication is the same as A⊗ B, and the multiplication isgiven by:

(a� 1)(1� λ) = a� λ,

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

This is a Hopf algebra. (We omit giving a formula for theantipode.)

We should be cautious about asserting quasitriangularity, but inour application we will end up with a QTHA.


Application to quantized enveloping algebras

Following Kassel, Rosso and Turaev, we construct Uq(sln).Roughly the idea is construct the double of Uq(b) where b is theBorel subalgebra. This has generators Ki (invertible) and Ei

(i 6 n− 1) subject to relations

KiKj = KjKi, KiEjK−1i = qaijEj (1)

where the Cartan matrix entries are aij and the quantum Serrerelations. However we omit the Serre relation and denote byU+ the free Lie algebra with the relations (1). Similarly U− isgenerated by K̃i and Fi subject to

K̃iK̃j = K̃jK̃i, K̃iFjK̃−1i = q−aijFj.


The Hopf pairing (continued)

The comultiplication in U+ is defined by

∆(Ki) = Ki ⊗ Ki, ∆(Ei) = Ei ⊗ Ki + 1⊗ Ei,

It is easy to see that Ki ⊗ Ki and Ei ⊗ Ki + 1⊗ Ei satisfy thesame relations as Ki and Ei, so such an algebrahomomorphism ∆ : U+ → U+ ⊗ U+. Also S(Ki) = K−1

i ,S(Ei) = −Ei is an antipode, so U+ is a Hopf algebra.

To recapitulate U+ is the quotient of a free Hopf algebra ongenerators Ki, K−1

i Ei modulo the relations

KiKj = KjKi, KiK−1i = K−1

i Ki = 1,

KiEj = qaijEjKi, K−1i Ej = q−aijEjK−1

i ,


The Hopf pairing

PropositionThere is a Hopf pairing between U+ and U− such that

〈Ki, K̃j〉 = q−aij , 〈Ei,Fj〉 =δij

q− q−1 ,

〈Ei, K̃j〉 = 〈Ki,Fj〉 = 0 .

We only sketch the proof. Consider a pair of Hopf algebras Aand B that are free algebras on generators {ai} free algebrasand {bj}. Assume that ∆(ai) is a linear combination of aj ⊗ ak

and similarly for ∆(bk). Then a Hopf pairing on A× B may bedefined arbitrarily on generators 〈ai, bj〉, and extended all ofA× B using the comultiplications.


The Hopf pairing (continued)

Thus we define the Hopf pairing on the free Lie algebras havingU+ and U− as quotients, and there are only a few relations tobe checked for compatibility.

For example we need to check that KiEj − qaijEjKi has trivialpairing with the generators K̃j and Fj.

Now we may construct the double U+ � U−.

Let I+ be the kernel of the pairing, that is,

I+ = {t ∈ U+ | 〈t,U−〉 = 0}.

It is a two-sided ideal of U+.


The quantum Serre relations

It may be checked the quantum Serre relations:

E2i Ej − (q + q−1)EiEjEi + EjE2

i , j = i± 1,

EiEj − EjEj (|i− j| > 1)

are in the kernel I+ of the Hopf pairing. (It is enough to checkthat they pair trivially with the generators K̃i and Fi. Thereforewe may divide by the ideal that they generate This almost givesUq(sln) except that we have two copies of the Cartan subgroup,Ki and K̃i. However we may further divide by the idealgenerated by Ki − K̃i to obtain Uq(sln), with all generators andrelations implemented.


Reminder (Lecture 9) Dual Objects

Let V be an object in a rigid monoidal category. We recall theassumptions we made of the dual, which we call the left dualV∗. It comes with morphisms evV : V∗ ⊗ V → I andcoevV : I → V ⊗ V∗ subject to

(1V⊗evV)◦(coevV ⊗1V) = 1V , (evV ⊗1V∗)◦(1V∗⊗coevV) = 1V∗ .

V

evV

coevV

V

V∗=

V

V

V∗

evV

coevV

V∗

V=

V∗

V∗


Reminder (Lecture 9): The right dual

Dually, we can ask for a right dual ∗V with morphismsevV : V ⊗ ∗V → I and coevV : I → ∗V ⊗ V subject to

(evV ⊗1V) ◦ (1V ⊗ coevV) = 1V ,

(1∗V ⊗ evV) ◦ (coevV ⊗1∗V) = 1V∗ .

V

evV

coevV

V

∗V=

V

V

∗V

evV

coevV

∗V

V=

∗V

∗V


Reminder (Lecture 9): Left and right duals

The notions of left and right dual are not really different. V∗ is aright dual of V if and only if V is a left dual of V∗. If every objectin the category has both a right and a left dual, then bydefinition

∗(V∗) = V, (∗V)∗ = V,

since the only difference between the defining properties

V∗

evV

coevV

V∗

V =

V∗

V∗

V

evV

coevV

V

∗V =

V

V

is the labelling of V, V∗ and ∗V.


Reminder (Lecture 9): Duals in ribbon categories

PropositionIn a ribbon category, every left dual is also a right dual.

We describe the morphisms evV : V ⊗ V∗ → I andcoevV : I → V∗ ⊗ V.

evV = evV ◦(1V∗ ⊗ θ−1V ) ◦ cV,V∗

V V∗

evV

=

V V∗

θ−1V

evV


Duals in ribbon categories (continued)

coevV = cV,V∗ ◦ (1V∗ ⊗ θ−1V ) ◦ coevV

V∗ V

coevV

=

V∗ V

θ−1V

coevV


References for Modular Tensor Categories

Last time I gave some references. In addition to Turaev’s book:

Bakalov and Kirllov: Lectures on tensor categories and modular functions

Fuchs: fusion rules in conformal field theory

Di Francesco, Mathieu and Sénéchal, Conformal FieldTheory, chapters 15 and 16.

This list of references is infinitely expandable.

Erik Verlinde, Fusion rules and modular transformations in2D conformal field theory. Nuclear Phys. B 300 (1988), no.3, 360-376.Moore and Seiberg, Lectures on RCFT (on-linesomewhere)

Fuchs, Runkel and Schwegert, Construction of RCFT Correlators I: Parition Functions

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

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

https://arxiv.org/pdf/hep-th/0204148.pdf


Review: Modular tensor categories

We recall the definition of a modular tensor category. This is asemisimple ribbon category with a finite number of simpleobjects Vi (i ∈ I). We assume that the ground ring K = End(I) isa field, where I is the unit object. This implies that I is itself asimple object. Moreover, we assume that End(Vi) = End(K) forall simple objects Vi. Finally, we assume that the S-matrix (̃sij) isinvertible, where s̃ij is the scalar (i.e. endomorphism of I)defined by the Hopf link:

Vi V∗iVj V∗j


The Fusion Ring

The monoidal structure gives the Grothendieck group of thecategory a multiplication that makes it into a ring. This ring is afree abelian group on the isomorphism classes of simplemodules. Thus if i ∈ I let [i] = [Vi] be the class of arepresentative simple module Vi. We can decompose Vi ⊗ Vj

into simple modules Vk with structure constants Nkij. Thus

[i][j] = Nkij[k], Vi ⊗ Vj =

⊕k

NkijVk.

We sum the repeated subscript k. The Nkij are nonnegative

integers. Taking the quantum dimension, which is multiplicativeby Lecture 4:

didj = Nkijdk, di = dim(Vi).


Remark about notation

It is understood that we use coevVi and evVi in this definition.Thus:

Vi V∗iVj V∗j

RemarkIn a MTC if Vi are the objects (i ∈ I) then V∗i is Vi∗ for somei∗ ∈ I. So the duality (conjugation) is implemented as apermutation of the index set.


Reminder of Lecture 3

In Lecture 3 we proved that

(evV ⊗1W)(1V∗ ⊗ cW,V) = (1W ⊗ evV)(c−1W,V∗ ⊗ 1V)

W

V∗ W V

=W

V∗ W V

We will call this the coevaluation crossing identity. Theevaluation crossing identity is similar:

W

V W V∗=

W

V W V∗


Symmetry of the S-matrix

We now observe that s̃ij = s̃ji. To see this use the crossingidentities to move the link around:

Vi V∗iVj V∗j =Vi V∗i

Vj V∗j= Vi V∗iVj V∗j


Alternative trace principle

Let f : V → V be a morphism in a ribbon category. We cancompute the trace in two different ways:

fV

V∗ = fV

V∗

To see this, replace evV by evV(1⊗ θ−1V )cV,V∗ in the first figure,

and in the second, replace coevV by cV,V∗(θV ⊗ 1) coevV .

fV

V∗

θ−1V

θ−1V

VV∗

f


Consequences of the alternative trace principle

If f : Vi → Vi is any morphism, since Vi is simple, End(Vi) = K.Thus f is just a scalar. In particular, θVi is a scalar which we willdenote θi.

We will denote the quantum dimension of Vi as di. Applying thealternative trace principle to 1V : V → V gives di = di∗, where i∗is the index such that Vi∗ = V∗i .

Remembering that θV∗i

= θ∗Vi, the alternative trace principle

implies that θi = θi∗.

θi

ViV∗i = θV∗

i

V∗iVi


Alternative definition of s̃ij

We proves̃ij = tr

(c−1

Vi∗,Vjc−1

Vj,V∗i

).

Indeed use the principle of the alternative trace to flip one circle:

Vi V∗iVj V∗j

V∗i Vj


Reminder: the ribbon axiom

c−1U,W ◦ c−1

W,U = (θ−1U ⊗ θ

−1W ) ◦ θU⊗W = θU⊗W ◦ (θ−1

U ⊗ θ−1W )

U W

U W

=

θU⊗W

θ−1U θ−1

W

U W

U W


Another formula for the S-matrix

We will prove:

PropositionWe have

s̃ij = θ−1i θ−1

j tr(θVi∗⊗Vj)

= θ−1i θ−1

j Nki∗,jθkdk.

Remember:

Vi V∗iVj V∗jV∗i Vj


Proof

Use the ribbon axiom:

θV∗i ⊗V

θV−1i∗

θV−1j

V∗i Vj

Thuss̃ij = θ−1

i θ−1j tr(θVi∗⊗Vj).


The first relation

We just proveds̃ij = θ−1

i θ−1j tr(θVi∗⊗Vj).

NowVi∗ ⊗ Vj = ⊕kNk

i∗,jVk

and tr(θVi∗⊗Vj) can be computed by summing the traces onthese summands. Therefore

s̃ij = θ−1i θ−1

j Nki∗,jθkdk.


Modular Meanings

Bakalov and Kirillov say:

The appearance of the modular group in tensor categories mayseem mysterious; however there is a simple geometricalexplanation, based on the fact that to each modular tensorcategory one can associate a 2 + 1 dimensional TQFT. Thisshows that in fact we have an action of the mapping classgroup of any oriented 2-dimensional surface on the appropriateobjects in MTC. This is the key idea in [Turaev’s book].

Thus SL(2,Z) is the mapping class group of the torus.


More Modular Meanings

In Chapter IV Section 5, Turaev defines a more general action.Concerning this, Turaev (p.190) states:

It is this relationship which suggested the terms modularfunctor and modular category.

Yet this is not the whole story since in certain “rational”conformal field theories the fields are actually modular forms!


The group SL(2,Z)

The group SL(2,Z) mentioned in the last quote is the mappingclass group of the torus, i.e. the group of group ofhomeomorphisms modulo isotropy. The larger group SL(2,R)acts on the upper half plane H = {z = x + iy ∈ C|y > 0} via(

a bc d

): z 7→ az + b

cz + d.

The subgroup Γ = SL(2,Z) acts discontinuously.

A modular form is a function f that satisfies

f (z) = (cz + d)−kf(

az + bcz + d

)for(

a bc d

)in SL(2,Z) or a subgroup of finite index.


Two important elements

Note that −I ∈ SL(2,Z) acts trivially on H, so the action is notfaithful. Let

S =

(−1

1

), T =

(1 1

1

)These generate SL(2,Z). Since S2 = −I, S has order 4 as anelement of the group but order 2 in its action on H.

T : z→ z + 1 is the translation by 1. If a holomorphic function fis invariant under SL(2,Z) it is invariant under T and so it has aFourier expansion:

f (z) =∑

anqn, q = e2πiz.

Although T has infinite order, ST has order 3 in its action on Hor 6 as an element of SL(2,Z).

S2 = −I, (ST)3 = −I.


Digression on SL(2,Z) (continued)

Here is the well-known fundamental domain for SL(2,Z):

e2πi/3 i

We have marked the fixed points i and e2πi/3 of S and ST.

S =

(−1

1

), T =

(1 1

1

)

S : z→ −1z, T : z→ z + 1.


Looking ahead

Let c = (δi,i∗) be the conjugation map, s = c̃s = s̃c.

We will see later that given a modular tensor category, there isa (projective) representation of SL(2,Z) in which the role ofS ∈ SL(2,Z) is played by s.

The role of the translation T is played by the matrix t = (δijθi) oftwists.

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Lecture 13 - sporadic.stanford.edusporadic.stanford.edu/quantum/lecture13.pdfLecture 13 Daniel Bump May 29, 2019 V i V j. Review: Uq(sln) The Drinfeld Double (III)Modular Tensor Categories