Lecture 15 - Stanford Universitysporadic.stanford.edu/quantum/lecture15.pdf · 2019-05-29 · The...

The Racah-Speiser algorithm The Kac-Walton formula

Lecture 15

Daniel Bump

May 29, 2019

−ρ


Motivation

Let g be a semisimple Lie algebra and let k be an integer thatwe will sometimes call the level. There is associated with thesedata a modular tensor category C(g, k) which we will describewithout trying to give complete proofs.

The category can be constructed in at least two ways,

Motivated by CFT, from integrable representations of levelk of the affine Lie algebra g;From the representation theory of Uq(g) where q = e2πi/κ

Next week we will discuss the second construction. Meanwhilewe will describe (without proof) the simple objects, ribbonelement and monoidal operation algorithmically (theKac-Walton formula).


Motivation (continued)

However the Kac-Walton formula is very similar to a method ofdecomposing tensor products of irreducible representations ofa Lie group, so we review that first.

This method was periodically discovered by different people. Inmy Lie groups book it is called the Brauer-Klimyk algorithm butin physics literature it is often called the Racah-Speiseralgorithm. We will begin with that, then show how to modify it tocompute fusion products in the Kac-Walton formula.


Weights

Let G be a complex reductive Lie algebra such as GL(r) orSL(r), and let T be a maximal torus. Thus if G = GL(r) or SL(r)we may take T to be the diagonal subgroup. The group X∗(T) ofrational characters of T is a torus: for GL(r) we may identifyX∗(T) = Zr, with λ = (λ1, · · · , λr) being the character

z =

z1. . .

zr

7→ zλ =

r∏i=1

zλii .

For SL(r) the group X∗(T) is the quotient Zr/Z where Z isembedded diagonally in Zr.

Let π be an irreducible representation of G. Since T is abelian,restricting G to T it decomposes into one-dimensionalrepresentations, that is weights, with certain multiplicities.


Example

Let G = GL(3). Then G has a 15-dimensional irreduciblerepresentation whose weights are as in the following table.

(3, 1, 0)

The doubly circled weights have multiplicity 2.


Roots

The group G acts on its Lie algebra g via the adjointrepresentation. The root system Φ consists of the nonzeroweights α in the adjoint representation. The roots arepartitioned into positive and negative ones. For GL(3) thepositive roots are:

(1,−1, 0), (0, 1,−1), (1, 0,−1).

The Weyl vector ρ which is half the sum of the positive rootsplays a special role. For GL(3) it is (1, 0,−1). The simple rootsare the positive roots that can’t be decomposed into otherpositive roots. For GL(3) the simple roots are

α1 = (1,−1, 0), (0, 1,−1).


The GL(3) root system

α1

α2

Here is the GL(3) root system. Positive roots: •. Negativeroots: ◦. The simple roots are α1 and α2. The shaded region isthe positive Weyl chamber

C = {λ|〈λ, αi〉 > 0 for simple roots αi}


Dominant weights

The positive Weyl chamber:

C = {λ|〈λ, αi〉 > 0 for simple roots αi}

We are embedding the weight lattice in an ambient real vectorspace V and C is a subset of V. Here 〈 , 〉 is a W-invariant innerproduct on Λ, normalized so 〈α, α〉 = 2 if α is a short root. TheWeyl group W is generated by the reflections si in thehyperplanes Hi perpendicular to the αi, which are the walls of C.

α1

α2 H1

H2


The highest weight

We define a partial order on Λ, in which λ < µ if λ− µ is a sumwith nonnegative coefficients of the simple roots.

If π is an irreducible representation, there is a unique weight λof π that is maximal or “highest” with respect to this partialorder. The weight λ is dominant. We denote π = πλ.

Theorem (Weyl)The association λ↔ πλ is a bijection between the dominantweights and the irreducible representations of G.


The highest weight

(3, 1, 0)

The highest weight vector (3, 1, 0) of this 15-dimensionalrepresentation is drawn in red.


The Weyl Character formula

Let z ∈ T. Let χλ(z) = tr πλ(z) be the character of πλ.

Theorem (Weyl)We have

χλ(z) = ∆−1∑w∈W

(−1)`(w)zw(λ+ρ)

where the Weyl denominator is

∆ =∑w∈W

(−1)`(w)zρ = zρ∏α∈Φ+

(1− z−α).

We will omit the proof, but we point out an importantinterpretation.


BGG Category O

Let B = TU be the positive Borel subgroup of G, where T is thegiven maximal torus, and U is the maximal unipotent of U. ItsLie algebra u is generated by the positive roots.

Let λ be a weight of G, not necessarily dominant. We call apossibly infinite-dimensional Π representation of G a highestweight representation with highest weight λ if it is generated bya vector vλ for some weight λ that is fixed by U, and such that

Π(z)vλ = zλvλ

for z ∈ T. There is a nice abelian category O that contains allhighest weight representations, due to Bernstein, Gelfand andGelfand.


Interpretation of the Weyl Character formula

If λ is a weight, not necessarily dominant, there is a universalhighest weight module M(λ) such that every highest weightrepresentation with highest weight λ is a quotient of M(λ). It isconstructed by inducing λ from B to G. Its character can becomputed using the triangular decomposition (Lecture 10):

U(g) = U(u)⊗ U(h)⊗ U(u−).

The character of M(λ) is

zλ∏α∈Φ+

(1− z−α)−1.


Interpretation of the Weyl Character formula (continued)

Now if λ is dominant, write the Weyl character formula

χλ(z) =∑w∈W

(−1)`(w)zw(λ+ρ)−ρ∏α∈Φ+

(1− z−α)−1.

So in the Grothendieck group of category O this expresses πλas an alternating sum of Verma modules.


The Brauer-Klimyk or Racah-Speiser algorithm

Let χλ be the character of πλ. We expand it in terms of weightµ, each with multiplicity K(λ, µ):

χλ(z) =∑µ

K(λ, µ)zµ.

for z ∈ T. Then we can try to decompose χλχν , which is thecharacter of πλ ⊗ πν .

The simplest case is where ν + µ is dominant for all weights µof λ. Then

χλ ∗ χν =∑µ

K(λ, µ)χµ+ν ,

πλ ∗ πν =⊕µ

K(λ, µ)πµ+ν ,


Proof

We substitute the weight expansion for χλ and the Weylcharacter formula for χν :

χλχν = ∆−1∑µ

K(λ, µ) zµ∑

w

(−1)`(w) zw(ν+ρ).

Interchange the order of summation, so that the sum over ν isthe inner sum, and make the variable change ν −→ w(ν). SinceK(λ, µ) = K(λ,wµ), we get

∆−1∑

w

∑ν

K(λ, µ) (−1)`(w) zw(ν+µ+ρ).

Now we may interchange the order of summation again andapply the Weyl character formula to obtain

∑K(λ, µ)χν+µ.


Example

λ+ ν

As long as µ+ ν is dominant for every weight µ of λ, we get thedecomposition of πλ ⊗ πµ as∑

µ

K(λ, µ)πν+µ.


What if µ+ ν is not dominant?

The proof of the Brauer-Klimyk-Steinberg-Racah-Speiserformula goes through but we have to reinterpret

∆−1∑w∈W

(−1)`(w)zµ+ν+ρ

PropositionLet λ be given, not assumed dominant. Write λ = wξ wherew ∈ W and ξ is dominant. Let η = ξ + w−1ρ− ρ. Then

∆−1∑

w

(−1)wzw(λ+ρ) =

{(−1)`(w)χη if η is dominant0 otherwise.

This is easily proved by making a change of variables in theWeyl character formula. It may be seen that η is dominantunless ξ lies on a wall of the positive Weyl chamber.


The algorithm, continued

We may describe the algorithm as follows. There is a modifiedaction of the Weyl group known as the “dot” action in which

w · λ = w(λ+ ρ)− ρ.

The point fixed by this action is −ρ. Let H be the set ofhyperplanes perpendicular to the roots. It includes the walls ofC. If µ+ ν lies on one of these hyperplanes, it contributes zeroto the sum: ∑

µ

K(λ, µ)∆−1∑w∈W

(−1)`(w)zµ+ν+ρ

But if µ+ ν does not lie on one of these hyperplanes, thenη = w · (µ+ ν) is dominant for some w ∈ W and we get a term

±K(λ, µ)χη.


Example

−ρ

The green weights lie on a hyperplane through −ρ so they arediscarded. Three other red weights to the left of the hyperplaneare reflected and subtracted.


After cancellation

−ρ

χ(3,1,0)χ(3,3,0) = χ(4,3,3)+χ(4,4,2)+χ(5,3,2)+χ(5,4,1)+χ(6,3,1)+χ(6,4,0)


The affine Weyl group (I)

The Weyl group W is generated by the reflections in thehyperplanes Hα perpendicular to the roots, or just reflections si

in the hyperplanes Hαi perpendicular to the simple roots. It is aCoxeter group.

The set of hyperplanes Hα is stable under W. The positive Weylchamber C is a connected component of the complement of theHα, and it is a fundamental domain for W.

These facts have analogs for the affine Weyl group which is aninfinite Coxeter group that we will now describe.


The affine Weyl group (II)

Fix a positive integer k, to be called the level.

The root system Φ has a highest root θ. For GL(r) it is(1, 0, · · · , 0,−1). Let α0 = −θ. The roots {α0, α1, · · · , αr} havethe property that 〈αi, αj〉 6 0 if i 6= j.

Consider the family of affine linear maps rα,m on V defined byrα,m(v) = 〈α, v〉+ m. Let Hα,m = {v|rα,m = 0}.

DefinitionThe group generated by the set of reflections in thehyperplanes Hα,m is the affine Weyl group of level k.


The fundamental alcove

The affine Weyl group Waffk is a Coxeter group. Recall this

means that W has generators s0, s1, · · · , sr with relations s2i = 1

and braid relations of the type

sisjsi · · · = sjsisj · · ·

where the number of factors on both sides is the order of sisj.

A fundamental domain for Waffk consists of the fundamental

alcove (of level k):

Fk = 〈αi, v〉 > 0, (i = 1, · · · , r), 〈α0, v〉 > −k.


Example

Here is the SL(3) fundamental alcove of level 3, and theweights that are contained in it.

There exists modular tensor category C(g, k) whose simpleobjects are in bijection with the weights λ inside Fk. Themonoidal operation, called fusion is given by the Kac-Waltonalgorithm.


The Kac-Walton algorithm

The Kac-Walton algorithm is the affine analog of theRacah-Speiser algorithm. We compute the decomposition

[λ] ∗ [µ] =∑ν

Nνλ,µ[ν].

We begin with the character

χλ =∑µ

K(λ, µ)zµ

which is known from the Weyl character formula. If the weightsµ+ ν are all in the fundamental alcove, then the fusion

[λ] ∗ [µ] =∑

K(λ, µ)[µ+ ν].

So in this case the fusion product coincides with the tensorproduct.


The Kac-Walton algorithm

If some µ+ ν are not in Fk imitate the Speiser-Racah methodusing reflections in hyperplanes parallel to the walls of thefundamental alcove, offset by 1.

−ρ

This describes the monoidal composition in C(g, k).


The dual Coxeter number

The dual Coxeter number h∨ appears in many formulas. Thiscan be defined as h∨ = 〈ρ, θ∨〉 where θ∨ is the longest coroot.

Cartan type h∨

Ar r + 1Br 2r − 1Cr r + 1Dr 2r − 2E6 12E7 18E8 30F4 9G2 4


The modular category

This modular tensor category arises two different ways.

Motivated by CFT, from integrable representations of levelk of the affine Lie algebra g;From the representation theory of Uq(g) where q = eπi/`

Here ` = k + h∨ in the simply-laced case, and can be madeexplicit in every case.

In the first construction, the monoidal structure on theserepresentations was found by physicists and made rigorous byKazhdan and Lusztig. In both cases one actually obtains aribbon category that is not semisimple but can besemisimplified by a simple procedure.


The twist and quantum dimension

The twist is given by the formula

θλ = q〈λ,λ+2ρ〉 .

The quantum dimension is

dλ =∏α∈Φ+

[〈λ+ ρ, α〉]q[〈ρ, α〉]q

.

Here as usual [n]q = (qn − q−n)/(q− q−1). This can becompared with the usual Weyl dimension formula, in the Liegroup case:

dim(πλ) =∏α∈Φ+

〈λ+ ρ, α〉〈ρ, α〉

.


The S-matrix

The S-matrix can be computed from the formula (Lecture 13):

sλ,µ =∑

k

Nνλ∗,µθνdνθλθµ

.

There is another formula:

sλµ =

∑w(−1)`(w)q2〈λ+ρ,w(µ+ρ)〉∑

w(−1)`(w)q2〈ρ,w(ρ)〉

We hope to cover further interesting formulas (such as theVerlinde formula) that are valid in a general MTC in our last fewlectures.

Lecture 15 - Stanford Universitysporadic.stanford.edu/quantum/lecture15.pdf · 2019-05-29 · The...

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