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Page 1: Lecture 6 - Stanford Universitysporadic.stanford.edu/quantum/lecture6.pdf · The standard module of Uq(sl2) and its R-matrixThe 6-vertex model Lecture 6 Daniel Bump May 28, 2019 v

The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Lecture 6

Daniel Bump

May 28, 2019

v v v v v

v ′′ v ′′v ′ v ′ v ′ v ′ v ′

σ1 σ2 σ3 σ4 σ5

τ1 τ2 τ3 τ4 τ5

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Uq(sl2)

We remind the reader of Uq(sl2). It has generators E , F and K ;K is “group-line” and has a multiplicative inverse K−1. Therelations are

KEK−1 = q2E , KFK−1 = q−2F EF − FE =K − K−1

q − q−1 .

The counit and comultiplication are

∆(K ) = K ⊗ K ,

∆(E) = E ⊗ K + 1⊗ E , ∆(F ) = F ⊗ 1 + K−1 ⊗ F ,

ε(K ) = 1, ε(E) = ε(F ) = 0.

The antipode satisfies S(K ) = K−1, S(E) = −E , S(F ) = −F .

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Representations of SL2

With a caveat, the following representation theories are thesame:

Finite-dimensional representations of SL(2,R);Finite-dimensional representations of SU(2);Finite-dimensional analytic representations of SL(2,C);Finite-dimensional complex representations of their Liealgebras sl2(R), su2, sl2(C);The enveloping algebras of these Lie algebras;The quantized enveloping algebra Uq(sl2), where q is not aroot of unity.

The caveat is that these categories are the same simpleobjects but Uq(sl2) has an interesting comultiplication andR-matrix so they are not the same as braided categories.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Representations of SL2 (continued)

Each of the above categories is semisimple, that is, everyobject is a direct sum of irreducibles.

There is one unique irreducible for every dimension. Thetwo-dimensional standard representation V generates thecategory: every irreducible is a submodule of

⊗k V for some k .

More precisely, the k -dimensional irreducible module is thesymmetric power Symk−1(V ).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Modules of sl2

The standard module of sl2 has two basis vectors x and y ; withrespect to this basis

E =

(0 10 0

), F =

(0 01 0

), H =

(1 00 −1

),

where H = [E ,F ]. For the 4-dimensional irreducible:

E =

0 1 0 00 0 2 00 0 0 30 0 0 0

, F =

0 0 0 03 0 0 00 2 0 00 0 1 0

, H =

3 0 0 00 1 0 00 0 −1 00 0 0 3

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Modules of Uq(sl2)

The standard module of Uq(sl2) again has basis vectors x andy ; with respect to this basis

E =

(0 10 0

), F =

(0 01 0

), =

(q 00 q−1

).

For the 4-dimensional irreducible:

E =

( 0 1 0 00 0 [2] 00 0 0 [3]0 0 0 0

), F =

( 0 0 0 0[3] 0 0 00 [2] 0 00 0 1 0

), K =

(q3 0 0 00 q 0 00 0 q−1 00 0 0 q−3

)

where the “quantum integer” [n] = (qn − q−n)/(q − q−1).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The standard module of Uq(sl2)

So the quantized standard module has two basis vectors x , y

E =

(0 10 0

), F =

(0 01 0

), K =

(q 00 q−1

),

where recall EF − FE = (K − K−1)/(q − q−1).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The R-matrix

We are not ready to prove that H = Uq(sl2) is (in a suitablesense) quasitriangular. The universal R-matrix lives not in Hbut in a suitable completion, and its existence is a subtle matter.

Nevertheless it is not the universal R-matrix itself, but themorphisms it induces. We will study the R-matrix for thestandard module V of Uq(sl2).

We will determine the endomorphisms of V ⊗ V , then pick outthe ones that satisfy the Yang-Baxter equation.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Endomorphism of V ⊗ V

Let x , y be the standard basis of V . Recall

K (x) = qx , K (y) = q−1y , E(x) = 0, E(y) = x ,

F (x) = y , F (y) = 0,

∆(K ) = K⊗K , ∆(E) = E⊗K +1⊗E , ∆(F ) = F⊗1+K−1⊗F .

We consider an endomorphism τR of V ⊗ V with respect to thebasis x ⊗ x , x ⊗ y , y ⊗ x and y ⊗ y . The eigenspaces of K :

eigenvalue basisq2 x ⊗ x1 x ⊗ y , y ⊗ x

q−2 y ⊗ y

These must be invariant under τR.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

E and F

K (x) = qx , K (y) = q−1y , E(x) = 0, E(y) = x ,

F (x) = y , F (y) = 0

∆(K ) = K⊗K , ∆(E) = E⊗K +1⊗E , ∆(F ) = F⊗1+K−1⊗F .

E(x ⊗ x) = 0E(x ⊗ y) = x ⊗ xE(y ⊗ x) = q(x ⊗ x)E(y ⊗ y) = q−1(x ⊗ y) + y ⊗ x

F (x ⊗ x) = q−1x ⊗ y + y ⊗ xF (x ⊗ y) = y ⊗ yF (y ⊗ x) = q(y ⊗ y)F (y ⊗ y) = 0

E =

0 1 q 00 0 0 q−1

0 0 0 10 0 0 0

, F =

0 0 0 0

q−1 0 0 01 0 0 00 1 q 0

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The R-matrix

Since τR preserves the eigenspaces it has the form( ∗∗ ∗∗ ∗

).

Since it commutes with

E =

0 1 q 00 0 0 q−1

0 0 0 10 0 0 0

, F =

0 0 0 0

q−1 0 0 01 0 0 00 1 q 0

it is easy computation to see that it is a a scalar multiple of

11− qb b

b 1− q−1b1

.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The Yang-Baxter equation

We must now have the Yang-Baxter equation

(τR)12(τR)23(τR)12 = (τR)23(τR)12(τR)23,

or equivalentlyR12R13R23 = R23R13R12.

With

τR =

1

1− qb bb 1− q−1b

1

,

this implies b = 0,q or q−1. We discard the solution b = 0.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The R-matrices

We arrive at two R-matrices for Uq(sl2). We multiply the b = qsolution by q−1:

R =

q−1

1q−1 − q 1

q−1

.

Similarly b = q−1 gives the alternative R-matrix:

R =

q

1 q − q−1

1q

.

Both satisfyR12R13R23 = R23R13R12

and produce H-module endomorphism of V ⊗ V .

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Dual R-matrices

Why are there two R-matrices? Let us consider the case of aQTHA H with universal R-matrix R = R(1) ⊗ R(2). Then it iseasy to see from the axioms that R = R−1

21 is also an R-matrix.(This is proved in Majid Chapter 5.)

Here R21 = R(2)⊗R(1). Now consider the case of sl2. By abuseof notation we will use the same letter R to denote both theuniversal R-matrix and the endomorphism of V that it induces.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Dual R-matrices, continued

So what endomorphism of V ⊗ V does R21 induce? Note thatR21 is R conjugated by the flip endomorphism τ : V → V , whichrespect to the basis x ⊗ x , x ⊗ y , y ⊗ x and y ⊗ y is the matrix

τ =

1

0 11 0

1

.

Consistent with this, for

R =

q−1

1q−1 − q 1

q−1

, R =

q1 q − q−1

1q

,we find that R = τR−1τ .

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

A parametrized Yang-Baxter equation

For later reference, we make note of the following parametrizedYang-Baxter equation.Define

R(z) = xR − yR.

Then

R12(x , y)R13(x , z)R23(y , z) = R23(y , z)R13(x , z)R12(x , y).

If g is a simple Lie algebra there is always a parametrizedYang-Baxter equation associated with Uq(g) or more preciselyits affinization Uq(g). Is not always linear in R and R−1

21 asabove. It can be quadratic: see Jimbo, Introduction to theYang-Baxter equation, Internat. J. Modern Phys. A 4 (1989),equation (5.2).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Statistical mechanics

In statistical mechanics one is concerned with large ensemblesof states. Highly energetic states are less probable.

If T is the temperature, the probability of a state s isproportional to the Boltzmann weight e−kβ(s)/T . Here β is theenergy of the state. The actual probability is

1Z (S)

e−kβ(s)/T ,

whereZ (S) =

∑s

e−kβ(s)/T

is the partition function.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Solvable lattice models

Certain systems may be solved exactly. The first such examplewas the 2-dimensional Ising model (Onsager 1944). Thesix-vertex model was solved by Lieb and Sutherland in the1960’s, then solved another way by Baxter whose methodextended to the eight-vertex model. Baxters results also hadapplications to Heisenberg spin chains in quantum mechanics.

The six-vertex model was proposed by Linus Pauling (1936) inconnection with two-dimensional ice, so these models are alsocalled ice-type models.

The six-vertex model was a key example leading to quantumgroups. Its generalizations have applications far beyond itsorigins in statistical mechanics.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Ice type models

The models that we will be concerned with take place on planargraphs. In using the term graph to describe these arrays we aredeviating from usual terminology, where edges have always twovertices, for we will allow open edges with only a singleendpoint.

Thus we have a set of vertices which are points in the plane,together with edges that are arcs which either join two vertices,or which are attached to only a single vertex. The edges whichare only attached to a single vertex are called boundary edges.Edges attached to two vertices are called interior edges. Everyvertex is adjacent to four edges. Edges can only cross at avertex.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Boltzmann weights

A spin is an element of the set {+,−}. In a state of the systemS every edge of the graph is assigned a spin. The spins of theboundary edges are fixed. Those of the interior edges arevariable.

Every vertex v is assigned a set of Boltzmann weights. Theseassign a value β(v) depending on the configuration of spins onthe adjacent edges. Only six configurations are allowed.

a1(v) a2(v) b1(v) b2(v) c1(v) c2(v)

+

+

+

+

v −−−

−v +

−+

−v −

+

−+

v −+

+

−v +

−−

+

v

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The partition function

A state of the system is an assignment of spins to the edges,with the spins of the boundary edges fixed and part of the datadescribing the system.

The Boltzmann weight of the state:

β(s) =∏

v

β(v).

The partition function

Z (S) =∑s

β(s).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Toroidal boundary conditions

We consider a planar grid. We use use toroidal boundaryconditions so that the left and right boundary edges areidentified. We allow different Boltzmann weights from row torow, even if we are only interested in the case all vi are thesame.

− + − + +

+ + − − +

+ − + − +

− + − + +

+ − − − + +

− − + − − −

− + − + − −

4 3 2 1 0

1

2

3 z3 v3 v3 v3 v3

v2 v2 v2 v2 v2

v1 v1 v1 v1 v1

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Row transfer matrix

We consider the sequence of spins σ = (σi) in a row of verticaledges to be a vector. If σ, τ are two such vectors, the product ofthe Boltzmann weights of the vertices in this row may beregarded as an entry in a matrix Tv (σ, τ). Here all vertices havethe same Boltzmann weight v . But the vertices in different rowsmay have different weights.

If v1, · · · , vk are the Boltzmann weights in the rows then thepartition function

Z (S) = tr(Tv1 · · ·Tvr

).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Commuting transfer matrices

Baxter’s insight was that it is possible to diagonalize the rowtransfer matrix if it can be embedded in a large family ofcommuting row transfer matrices. If this can be done, the modelis called solvable or integrable.

We restrict ourselves to weights that are field free meaning

a1(v) = a2(v), b1(v) = b2(v), c1(v) = c2(v).

We denote these as just a(v), b(v) and c(v). Let

∆(v) =a(v)2 + b(v)2 − c(v)2

2a(v)b(v).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Commuting transfer matrices

The parameter ∆ determines the gross characteristics of thesystem such as the largest eigenvalue of the row transfermatrix. Assuming the Boltzmann weights to be positive andreal, ∆ has physical significance. If ∆ > 1 or ∆ < −1, thesystem is “frozen” and there are correlations between distantspins. If |∆| < 1 the system is in a disordered state. Thus∆ = ±1 are phase transitions.

To study the eigenvalues of Tv , we embedd Tv in a large familyof commuting matrices.

Theorem (Baxter)Suppose v and v ′ are two vertices such that ∆(v) = ∆(v ′)Then Tv and Tv ′ commute.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The Yang-Baxter equation

The key to the last result is a Parametrized Yang-Baxterequation which in retrospect is related to the quantum groupUq(sl2) of the affine Lie algebra sl2. Here q is chosen so that

∆ = 12(q + q−1).

We will describe the relevant Yang-Baxter equations as Baxterunderstood them. We must introduce an auxiliary vertex v ′′ thatwe will draw rotated as follows.

a(v′′) a(v′′) b(v′′) b(v′′) c(v′′) c(v′′)

+

+ +

+

v ′′

− −

−v ′′

+

− +

−v ′′

+ −

+

v ′′

+ +

−v ′′

+

− −

+

v ′′

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The Yang-Baxter equation

Theorem (Baxter)If ∆(v) = ∆(v ′) then there exists a vertex v ′′ such that followingtwo partition functions are equal for every choice of boundaryspins a,b, c,d ,e, f ∈ {+,−}.

a

b

c

d

e

f

v ′′

v

v ′ a

b

c

d

e

f

v ′′

v

v ′

Thus the boundary spins are to be fixed and we sum over thespins of the internal unnumbered edges. The value ∆(v ′′) alsoequals ∆(v) = ∆(v ′).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

A parametrized Yang-Baxter equation

Let us fix q ∈ C×. If x ∈ C× define v(x) to be the vertex withBoltzmann weights

a = q − x2/q, b = 1− x2, c = x(q − 1/q).

Then if v = v(x) and v ′ = v(y), Baxter’s identity may bechecked by direct computation with v ′′ = v(x/y).

This special case implies Baxter’s theorem since given any vwith ∆(v) = 1

2(q + q−1), we may find a λ and x such that

a = λ(q − x2/q), b = λ(1− x2), c = λ x(q − 1/q).

Multiplying the weights by a constant just multiplies the partitionfunctions on both sides by the same constant, and so thisparametrized YBE implies Baxter’s theorem.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The inverse R-matrix

Let v = v(x), v ′ = v(y) and v ′′ = v(x/y). Let v ′′ = c−1v(y/x)where with z = y/x

c = (qz)−2(z − q)(z + q)(qz − 1)(qz + 1).

We mean the Boltzmann weights of v(y/x) are multiplied bythis constant to obtain v ′′. We compute that the partitionfunction of:

a

b c

d

v ′′ v ′′

is δ(a,d) δ(b, c) (Kronecker δ). The same is true if v , v ′′ areswitched.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The inverse R-matrix

In order to show that v ′′ and v ′′ cancel, we will draw them asover-crossings and under-crossing.

a

b c

d

v ′′ v ′′ =

a

b c

d

v ′′ v ′′

This resembles a Reidemeister move, so we give theYang-Baxter equation the same treatment:

a

b

c

d

e

f

v ′′v

v ′

=

a

b

c

d

e

f

v ′′

v

v ′

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Commuting transfer matrices

We may now explain how Baxter proved the commutativity ofthe row transfer matrices in the case of toroidal boundaryconditions. We will fix v and v ′ with ∆(v) = ∆(v ′) and provethat the row transfer matrices Tv and Tv ′ commute.

v v v v v

v ′ v ′ v ′ v ′ v ′

σ1 σ2 σ3 σ4 σ5

τ1 τ2 τ3 τ4 τ5

Here we compute 〈τ,Tv ′Tvσ〉.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Setting up the train argument

We find v ′′ and v ′′ as above and insert them between the rows.This does not change the partition function.

v v v v v

v ′′ v ′′

v ′ v ′ v ′ v ′ v ′

σ1 σ2 σ3 σ4 σ5

τ1 τ2 τ3 τ4 τ5

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The train argument

Repeatedly using the Yang-Baxter equation, the braid moves tothe right.

v ′ v ′ v ′ v ′ v ′

v ′′ v ′′v v v v v

σ1 σ2 σ3 σ4 σ5

τ1 τ2 τ3 τ4 τ5

After repeatedly applying the Yang-Baxter equation, the braidv ′′ moves across the two rows. Note that v and v ′ are nowswitched. Due to the toroidal boundary conditions, the braidsv ′′ and v ′′ are again in juxtaposition and may be discarded.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Edges as modules

The parameter ∆ is fixed. Moreover we choose and fix q suchthat ∆ = 1

2(q + q−1).

We postulate a category some of whose objects are vectorspaces Vz indexed by a nonzero complex number z. Everyedge will be assigned a Vz such that opposite edges throughthe same vertex v will be assigned the same Vz .

Vz Vz

Vw

Vw

or

Vz

Vz

Vw

Vw

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Interpretation of Boltzmann weights

The module Vz will be a two-dimensional vector space withbasis v+ = v+(z), v− = v−(z) indexed by the possible spins ±.At a vertex

Vz Vz

Vw

Vw or

Vz

Vz

Vw

Vw

we take the Boltzmann weights with

a = λ(q − x2/q), b = λ(1− x2), c = λ x(q − 1/q),

where x = z/w .

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Example

We can assign Vz ’s consistently, for example in this case

a

b

c

d

e

f

v ′′v

v ′

we want v , v ′ and v ′′ to have Boltzmann weights

a = λ(q − z2/q), b = λ(1− z2), c = λ z(q − 1/q),

with z taking values x , y , x/y respectively. So we label theedges thus:

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Example (continued)

a

b

c

d

e

f

Vx

Vy

V1

All the Boltzmann weights are then correct, and similarly for theother side of the Yang-Baxter equation.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Interpretation of Boltzmann weights (continued)

Now we will interpret the Boltzmann weights as describing anendomorphism R(z,w) ∈ End(Vz ⊗ Vw ). With a,b, c as abovethis is the transformation

ab cc b

a

with respect to the basis

v+(z)⊗ v+(w), v+(z)⊗ v−(w),

v−(z)⊗ v−(w) v−(z)⊗ v−(w).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The R-matrix

In other words, if α, β, γ, δ are spins, and we defineRγ,δα,β = Rγ,δ

α,β(z,w) to be the Boltzmann weight of the vertex at

α

β

γ

δ

v

or α

β γ

δ

v

thenR(z,w) : vα ⊗ vβ 7→

∑γ,δ

Rγ,δα,β(vγ ⊗ vβ).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The left side of the Yang-Baxter equation

Now let us determine the matrix S such that the partitionfunction of:

α

β

γ

δ

µ

ν

Vx

Vy

Vz

is Sδ,µ,να,β,γ . Regard this as an endomorphism of Vy ⊗ Vx ⊗ Vz :

S(vα ⊗ vβ ⊗ vγ) =∑δ,µ,ν

Sδ,µ,να,β,γ(vδ ⊗ vµ ⊗ vν).

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The left side of the Yang-Baxter equation

We have to sum over the possible spins of the three interioredges, labeled φ, ψ, ξ in the following:

α

β

γ

δ

µ

ν

φ

ψ

ξ

Vx

Vy

Vz

Vx

Vy

Vz

ThusSδ,µ,να,β,γ =

∑φ,ψ,ξ

Rφ,ξα,β(x , y)Rδ,ξ

φ,γ(x , z)Rµ,νψ,ξ (y , z).

In matrix notation,

S = R(x , y)12R(x , z)13R(y , z)23.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

The parametrized Yang-Baxter equation reappears

The other side of the Yang-Baxter equation is treated similarly,and so it is equivalent to the matrix identity:

R(x , y)12R(x , z)13R(y , z)23 = R(y , z)23R(x , z)13R(x , y)12.

This is equivalent to the parametrized Yang-Baxter equation weencountered earlier in this lecture.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Affine sl2

To get this R-matrix from the theory of quantum groups, weneed a quantum group with one two-dimensional module Vz foreach z ∈ C×. This is Uq(sl2) for the affine Lie algebra sl2. It canbe constructed as a central extension of the “loop” Lie algebrasl2 ⊗ C[t , t−1], the Lie algebra defined by

[X ⊗ tk ,Y ⊗ t`] = [X ,Y ]⊗ tk+`.

The central extension is needed for infinite-dimensionalrepresentations. If V is a finite-dimensional sl2-module andz ∈ C× then there is a representation Vz of sl2 ⊗ C[t , t−1] inwhich

(X ⊗ tk ) · v = zkX · v .

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Relation with Uq(sl2)

A variant replaces this R-matrix

R(x , y) =

q − z2/q

1− z2 z(q − 1/q)z(q − 1/q) 1− z2

q − z2/q

(where z = x/y ) by

R(x , y) =

q − z2/q

1− z2 z2(q − 1/q)(q − 1/q) 1− z2

q − z2/q

.

It is not obviously equivalent but still true that the parametrizedYang-Baxter holds:

R(x , y)12R(x , z)13R(y , z)23 = R(y , z)23R(x , z)13R(x , y)12.

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The standard module of Uq(sl2) and its R-matrix The 6-vertex model

Relation with Uq(sl2) (continued)

This has the advantage that we may specialize z → 0 andobtain

R =

q

1q − 1/q 1

q

.

This is the R-matrix for Uq(sl2) in its standard 2-dimensionalrepresentation.

(Comparing the R-matrix from earlier in this lecture, q hasbecome q−1, so more correctly this is the R-matrix forUq−1(sl2).)