Convolution The Drinfeld double (I) The RTT equation

Lecture 11

Daniel Bump

May 29, 2019

Vλ

Vµ

W

Vλ

Vµ

W

R(λ−µ)

L(λ)

L(µ) Vλ

Vµ

W

Vλ

Vµ

W

R(λ−µ)

L(λ)

L(µ)

Convolution The Drinfeld double (I) The RTT equation

Convolution

Let H be a finite-dimensional Hopf algebra. Let End(H) be thevector space of all linear transformations of H. Then End(H)has two completely different ring structures. First, it is a ring inwhich the multiplication is the composition of endomorphisms.This ring is isomorphic to Matd(K) where d = dim(H) and K isthe ground field.

The second, unrelated ring structure is called convolution. If fand g are endomorphisms of H, define f ? g ∈ End(H) to be thecomposition:

H H ⊗ H H ⊗ H H .∆ f⊗g µ

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

Convolution The Drinfeld double (I) The RTT equation

The counit and antipode

The map ηε : H → H serves as a unit in the ring End(H). Sincewe are identifying K with its image under the unit map η, we willdenote this map as just ε. To see that it is a unit, note that

(f ? ε)(x) = f (x(1))ε(x(2)) = f (x(1)ε(x(2))) = f (x)

so f ? ε = f and similarly ε ? f = f .

Now the identity map IH ∈ End(H) has a convolution inverse,and that is the antipode. Indeed

(I ? S)(x) = x(1)S(x(2)) = ε(x)

so I ? S = ε and similarly S ? I = ε, and we have noted that ε isthe unit in the convolution ring.

Convolution The Drinfeld double (I) The RTT equation

An isomorphism

Recall that End(H) ∼= H ⊗H∗. In this isomorphism a pure tensorx⊗ λ corresponds to the rank one endomorphism Φx⊗λ definedby

Φx⊗λ(h) = 〈λh〉x.

This isomorphism is a ring isomorphism. Let us check that itrespects multiplication.

(Φx⊗λ ? Φy⊗µ)(h) = 〈λ, h(1)〉x〈µ, h(2)〉y = 〈λµ, h〉xy,

soΦx⊗λ ? Φy⊗µ = Φxy⊗λµ.

Thus denoting Φ : H ⊗ H∗ → End(H) the linear map such thatΦ(x⊗ λ) = Φx⊗λ, we see Φ is an algebra isomorphism.

Convolution The Drinfeld double (I) The RTT equation

Fourier expansions

Choose a basis ei of H, and let ei be the dual basis of H∗. Thenif f ∈ End(H), we have

f = Φ

(∑i

f (ei)⊗ ei).

This is a kind of Fourier expansion. To check it, apply bothsides to a basis vector ej. We have∑

j

Φf (ei)⊗ei(ej) =∑

i

〈ei, ej〉f (ei) = f (ej).

We will sometimes use the summation convention and write

f = Φ(f (ei)⊗ ei

).

Convolution The Drinfeld double (I) The RTT equation

The canonical element of H∗ ⊗ H

As a special case take f to be the identity map. Then we seethat Φ(T) = IH where

T = ei ⊗ ei.

This will be called the canonical element of H ⊗ H∗.We have also seen that the identity map in End(H) isconvolution invertible, and its inverse is the antipode. Takingf = S we have

S = Φ(S(ei)⊗ ei

).

Thus we have proved

T−1 =∑

i S(ei)⊗ ei .

Convolution The Drinfeld double (I) The RTT equation

A counit identity

Another result that we can prove using convolution theory is∑i

ε(ei)⊗ ei = 1H⊗H∗ .

Indeed, under Φ the left-hand side becomes the counitε : H → H (or ηε) which we have noticed is the unit in theconvolution ring. Since Φ is a ring isomorphism, this must bethe identity element of H ⊗ H∗.

Interchanging the roles of H and H∗, we have also∑i

ei ⊗ ε(ei) = 1H⊗H∗ .

Convolution The Drinfeld double (I) The RTT equation

Introduction

In his 1986 ICM talk Drinfeld defined the quantum double thus:

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.

We will prove this in two lectures. Today, we will construct thedual Hopf algebra D(A)∗. The strategy will be to take A∗ ⊗ Aopand use one of its two canonical elements to twist. (It is theother canonical element that provides the R-matrix.)

Convolution The Drinfeld double (I) The RTT equation

Review: Dual Hopf algebra

Let H be a finite-dimensional Hopf algebra. We will assumethat the antipode of H is invertible. It can be shown that this isautomatic for finite-dimensional Hopf algebras, but if onewishes to generalize the construction of the quantum doublethe antipode needs to be invertible.Let H∗ be the dual Hopf algebra. With λ, µ ∈ H∗ and y, y ∈ H

〈Dλ, x⊗ y〉 = 〈λ, µ(x⊗ y)〉 = 〈λ, xy〉.

or in Sweedler notation

〈λ, xy〉 = 〈λ(1), x〉〈λ(2), y〉.

Dually,〈λµ, x〉 = 〈λ, x(1)〉〈µ, x(2)〉.

Furthermore〈S(λ), x〉 = 〈λ, S(x)〉.

Convolution The Drinfeld double (I) The RTT equation

Review: The opposite Hopf algebra

Let (H, µ, η,∆, ε) denote the Hopf algebra H with multiplicationµ, unit η, comultiplication ∆ and unit ε.

Now we may reverse the multiplication, and define µop = µ ◦ τ .So µop(x⊗ y) = µ(y⊗ x) = yx. We do not need to reverse thecomultiplication. It is easy to check that (H, µop, η,∆, ε) is abialgebra. For example consider the Hopf axiom:

H ⊗ H H ⊗ H ⊗ H ⊗ H H ⊗ H ⊗ H ⊗ H

H H ⊗ H

∆⊗∆

µop

1H⊗τ⊗1H

µop⊗µop

∆

This diagram commutes since, both ways to x⊗ y

(yx)(1) ⊗ (yx)(2) = y(1)x(1) ⊗ y(1)x(2).

Convolution The Drinfeld double (I) The RTT equation

Review: The opposite Hopf algebra (continued)

Alternatively, we may reverse the comultiplication and denote∆op = τ ◦∆. Then Let (H, µ, η,∆op, ε) is also a bialgebra. Thereare thus four bialgebras altogether:

H = (H, µ, η,∆, ε), Hop = (H, µop, η,∆, ε),Hcop = (H, µ, η,∆op, ε), Hop cop = (H, µop, η,∆op, ε).

These are Hopf algebras if the antipode is invertible. Theantipode for Hop and Hcop is S−1.

Also if the antipode is invertible, it is an isomorphism from H toHop cop, and similarly from Hcop to Hop.

The algebra (H∗)cop is important in the quasitriangularity storyand we will denote it H◦.

Convolution The Drinfeld double (I) The RTT equation

About the antipode

Assuming that H is a Hopf algebra, we’ve exhibited bialgebrasHop, Hcop and Hop cop. For Hop and Hcop to be Hopf algebras, it isnecessary for them to have antipodes. If the antipode S of H isinvertible, then S−1 is an antipode for both Hop and Hcop.

Let us check this for Hop, leaving Hcop to the reader.

H H ⊗ H H ⊗ H

K H

∆

ε

1H⊗S−1

µτ

η

H H ⊗ H H ⊗ H

K H

∆

ε

S−1⊗1H

µτ

η

The first diagram commutes since µτ(1⊗ S−1)∆(x) =S−1(x(2)) x(1) = S−1(S(x(1))x(2)) = S−1(ε(x)) = ε(x). The secondis similar.

Convolution The Drinfeld double (I) The RTT equation

The canonical element of Aop ⊗ A∗

Let A be a finite-dimensional Hopf algebra. Let ei be a basis ofA, and ei the dual basis of A∗. We consider (omitting asummation)

T = ei ⊗ ei

which we interpret as an element of Aop ⊗ A∗. Note that thisdoes not depend on the choice of basis ei. We will prove

(∆⊗ 1)(T) = T13T23 (1⊗∆)(T) = T13T12

Explicitly

T13T23 = ei ⊗ ej ⊗ eiej, T13T12 = eiej ⊗ ei ⊗ ej.

For the second identity we have used the fact that T ∈ Aop ⊗ A∗,because we have to reverse the multiplication:

T13T12 = (ej ⊗ 1⊗ ej)(ei ⊗ ei ⊗ 1) = eiej ⊗ ei ⊗ ej.

Convolution The Drinfeld double (I) The RTT equation

Proof

With T = ei ⊗ ei, consider (∆⊗ 1)T ∈ Aop ⊗ Aop ⊗ A∗. Expand(∆⊗ 1)T = ei ⊗ ej ⊗ λij,

where λij ∈ A∗ are to be determined. Let ckij be the coefficientsdetermined by

∆ek = ckij ei ⊗ ej.We have λij = ckije

k because

ckijei ⊗ ej ⊗ ek = (∆⊗ 1)(ek ⊗ ek) = (ei ⊗ ej ⊗ λij).On the other hand

ckij = 〈ei ⊗ ej,∆(ek)〉 = 〈eiej, ek〉so

(∆⊗ 1)T = ckij(ei ⊗ ej ⊗ ek) = 〈eiej, ek〉(ei ⊗ ej ⊗ ek) = T13T23.The other identity is proved similarly.

Convolution The Drinfeld double (I) The RTT equation

Reminder: the convolution theory and T−1

We proved earlier that if H is a finite-dimensional Hopf algebraand

T = ei ⊗ ei ∈ H ⊗ H∗

(implied summation) then T−1 = S(ei)⊗ ei. We would like toapply this result with H = Aop. We note that H∗ = (Aop)∗ is thesame as A∗ as an algebra: it has a different comultiplicationthan A∗ but since A and Aop have the same comultiplication, A∗

and H∗ have the same multiplication, and this result applies inAop ⊗ A∗. But we have to remember that the antipode of Aop isS−1, and so in Aop ⊗ A∗

T−1 = S−1(ei)⊗ ei.

Convolution The Drinfeld double (I) The RTT equation

Review: Drinfeld twisting

Proposition (Proved in Lecture 10)Let H be a Hopf algebra and let F be an invertible element ofH ⊗ H. Assume that

F12(∆⊗ 1)(F) = F23(1⊗∆)(F)

and that (1⊗ ε)(F) = (ε⊗ 1)(F) = 1. Define

∆F(x) = F∆(x)F−1.

Then we may replace the comultiplication in H by ∆F to obtainanother Hopf algebra with the same algebra structure.

If H = A∗ ⊗ Aop where A is a finite-dimensional Hopf algebra wewill exhibit an invertible F that allows us to twist H = A◦ ⊗ A.This will give us the dual of the Drinfeld quantum double D(A).

Convolution The Drinfeld double (I) The RTT equation

The twist

We will now work inn H = A∗ ⊗ Aop. We have just proved someidentities in Aop ⊗ A∗, but we will apply those inH ⊗H = A∗ ⊗ Aop ⊗ A∗ ⊗ Aop, which, we note, contains a copy ofAop ⊗ A∗.

Our goal is to exhibit an element F of H ⊗ H that satsifies thehypotheses of the Drinfeld twisting proposition, particularly

F12(∆⊗ 1)(F) = F23(1⊗∆)(F).

We define:F = (1A∗ ⊗ S−1ei)⊗ (ei ⊗ 1A)

By convolution theory

F−1 = (1A∗ ⊗ ei)⊗ (ei ⊗ 1A).

Convolution The Drinfeld double (I) The RTT equation

Proof

We want to show:

F12(∆⊗ 1)(F) = F23(1⊗∆)(F)

We have

(∆⊗ 1)(ei ⊗ ei) = T13T23, (1⊗∆)T = T13T12

and since F−1 = (1A∗ ⊗ ei)⊗ (ei ⊗ 1A),

(∆⊗ 1)F−1 = (F−1)13(F−1)23, (1⊗∆)F−1 = (F−1)13(F−1)12

Therefore

(∆⊗ 1)F = F23F13, (1⊗∆)F = F12F13.

Convolution The Drinfeld double (I) The RTT equation

Proof (concluded)

Now F23 and F12 commute since

F23 = (1A∗ ⊗ 1A)⊗ (1A∗ ⊗ S−1ei)⊗ (ei ⊗ 1A),

F12 = (1A∗ ⊗ S−1ei)⊗ (ei ⊗ 1A)⊗ (1A∗ ⊗ 1A).

So

F12(∆⊗ 1)(F) = F12F23F13 = F23F12F13 = F23(1⊗∆)(F).

We also need (ε⊗ 1)F = (1⊗ ε)F = 1, but this may be deducedfrom the identity

ε(ei)⊗ ei = ei ⊗ ε(ei) = 1

in Aop ⊗ A∗, which follows from convolution theory.

Convolution The Drinfeld double (I) The RTT equation

Summary

The ring that we have constructed is not D(A), but its dualD(A)∗. As a ring, it is A∗ ⊗ Aop. The comultiplication has beenmodified by twisting, that is:

∆F(x) = F∆(x)F−1

whereF = (1A∗ ⊗ S−1ei)⊗ (ei ⊗ 1A).

The property of quasitriangularity is not self-dual. So D(A) willturn out to be quasitriangular, and its category of modules isbraided. The ring we have constructed, D(A)∗ = (A∗ ⊗ Aop)F(where the notation connotes twisting by F is notquasitriangular but dual quasitriangular. This implies that itscategory of comodules is braided.

Convolution The Drinfeld double (I) The RTT equation

Historical origins

The origin of quantum groups came out of the Quantum InverseScattering Method, a technique for studying integrable systemsdeveloped in St. Petersburg by Faddeev and his students,including Kulish, Sklyanin, Reshetikhin, Takhtajan, Korepin,Izergin and Semenov-Tian-Shansky. They found an algebraicstructure underlying applications of the Yang-Baxter equation.For an informative account see the following paper of Faddeev.

Faddeev: History and Perspectives of Quantum Groups

A key feature of this story is the RTT equation, a kind ofparametrized Yang-Baxter equation. Faddeev calls it theFundamental commutation relation and introduces it by theexample of the XXX Heisenberg spin chain Hamiltonian.

https://link.springer.com/article/10.1007/s00032-006-0062-4

Convolution The Drinfeld double (I) The RTT equation

The RTT equation

The equation in question can be written

R(λ− µ)L1(λ)L2(µ) = L2(µ)L1(λ)R(λ− µ).

Here R there are vector spaces V and W such that R acts onV ⊗ V and L acts on V ⊗W. Both sides act on V ⊗ V ⊗W. L1(λ)is the operator L(λ) applied to the first and third component,and L2(µ) is the operator L(µ) acting on the second and thirdcomponent.

In our usual notation we might write this identity

R12(λ− µ)L13(λ)L23(µ) = L23(µ)L13(λ)R12(λ− µ).

Convolution The Drinfeld double (I) The RTT equation

The parametrized Yang-Baxter equation

In addition to the RTT equation

R12(λ− µ)L13(λ)L23(µ) = L23(µ)L13(λ)R12(λ− µ). (1)

We will have another Yang-Baxter equation:

R12(λ− µ)R13(λ− ν)R23(µ− ν) = R23(µ− ν)R13(λ− ν)R12(λ− µ) (2)

We have written the parameter group additively, though there isone unusual case that we are aware of where it is nonabelian.Usually the parameter group is C, C× or an elliptic curve. Theelliptic curve cases arise in the eight-vertex model, or the XYZHamiltonian (Baxter).

Convolution The Drinfeld double (I) The RTT equation

The RTT relation

We already saw a version of the RTT relation in theparametrized Yang-Baxter equation that was used in Lecture 6on solvable lattice models. We recall that there are two vectorspaces V and W. It is sometimes convenient to label each copyof V by a parameter.

Vλ

Vµ

W

Vλ

Vµ

W

R(λ−µ)

L(λ)

L(µ) Vλ

Vµ

W

Vλ

Vµ

W

R(λ−µ)

L(λ)

L(µ)

Convolution The Drinfeld double (I) The RTT equation

The case where W = K

We have written the relation (1) in the form

R12(λ− µ)L13(λ)L23(µ) = L23(µ)L13(λ)R12(λ− µ).where Faddeev just writes

R12(λ− µ)L1(λ)L2(µ) = L2(µ)L1(λ)R12(λ− µ).There are 3 vector spaces involved: Vλ, Vµ and W. So omittingthe 3 subscript means treating W as less imporant. If W = K wedefinitely omit it and the RTT relation looks like:

Vλ

Vµ Vλ

VµL(λ)

L(µ)

R(λ−µ)

=Vλ

Vµ Vλ

VµL(λ)

L(µ)

R(λ−µ)

Convolution The Drinfeld double (I) The RTT equation

The parametrized Yang-Baxter equation

The other parametrized Yang-Baxter equation (2), which doesnot involve L(λ) can be diagrammed this way.

R(µ−ν)

R(λ−ν)

R(λ−µ)

Vλ

Vµ

Vν Vλ

Vµ

Vν

R(µ−ν)

R(λ−ν)

R(λ−µ)

Vλ

Vµ

Vν Vλ

Vµ

Vν

Convolution The Drinfeld double (I) The RTT equation

Quantum group interpretation

In the quantum group interpetation, the Vλ and W should beobjects in a braided category. Potentially this is the category offinite-dimensional modules of a quantum group. In Lecture 6,the quantum group was Uq(ŝl2), where the “hat” denotesaffinization. In this case, there is one two dimensional moduleVλ for each λ ∈ C×. Also in this case, the module W is chosenfrom this family, but in other cases, it might not be.

In a limiting case, the modules Vλ may all be the same. Thenthe RTT relation and the Yang-Baxter equation look like this:

R12R13R23 = R23R13R12, R12T1T2 = T2T1R12,

orR12T13T23 = T23T13R12.

Convolution The Drinfeld double (I) The RTT equation

The FRT construction

Faddeev, Reshetikhin and Takhtajan solved the followingproblem. Given a solution of the Yang-Baxter equation

R12R13R23 = R23R13R12,

produce a quasitriangular Hopf algebra with R-matrix R.Actually they constructed the dual Hopf algebra as follows.They considered the RTT relation in the form:

R12T13T23 = T23T13R12

to be an identity in involving two copies of a matrix T. They tookthe entries in T to be noncommuting indeterminates, subject tothe relations implied by this identity. They showed that the ringgenerated by these indeterminates may be given the structureof a dual quasitriangular Hopf algebra. This is an importantconstruction of quasitriangular Hopf algebras.

Convolution The Drinfeld double (I) The RTT equation

Hopf algebra interpretation

We proved earlier today, for an arbitrary Hopf algebra A, thefollowing identities. Let T = ei ⊗ ei be the canonical element ofA⊗ A∗. Then

(∆⊗ 1)(T) = T13T23, (1⊗∆)(T) = T13T12

That is,

T13T23 = ei ⊗ ej ⊗ eiej, T13T12 = eiej ⊗ ei ⊗ ej.

In the special case where A is quasitriangular, will prove

R12T13T23 = T23T13R12.

Thus the canonical element satisfies the same identity as theLax operator that appears in the RTT equation.

Convolution The Drinfeld double (I) The RTT equation

Proof

Apply τ to the first two components in

(∆⊗ 1)(T) = T13T23 = ei ⊗ ej ⊗ eiej.

We get:(τ∆⊗ 1)(T) = T23T13 = ej ⊗ ei ⊗ eiej.

Remember τ∆(x) = R∆(x)R−1 or

(τ∆⊗ 1)(x⊗ y) = R12∆(x⊗ y)R−112 .

SoT23T13 = R12T13T23R−112

provingR12T13T23 = T23T13R12.

ConvolutionThe Drinfeld double (I)The RTT equation