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Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(sl2)

Sarah Bockting-Conrad

University of Wisconsin-Madison

June 5, 2014

Introduction

This talk is about tridiagonal pairs and tridiagonal systems. We will focus on a special class of tridiagonal systems known as the q-Racah class. We introduce some linear transformations which act on the on the underlying vector space in an attractive manner. We give two actions of the quantum group Uq(sl2) on the underlying vector space. We describe these actions in detail, and show how they are related.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Definition of a tridiagonal pair

Let K denote a field. Let V denote a vector space over K of finite positive dimension.

Definition

By a tridiagonal pair (or TD pair) on V we mean an ordered pair of linear transformations A : V ! V and A∗ : V ! V satisfying:

• 1. Each of A, A∗ is diagonalizable.
• 2. There exists an ordering {Vi}d

i=0 of the eigenspaces of A such that

A∗Vi ✓ Vi−1 + Vi + Vi+1 (0  i  d), where V−1 = 0 and Vd+1 = 0.

• 3. There exists an ordering {V ∗

i }δ i=0 of the eigenspaces of A∗ such that

AV ∗

i ✓ V ∗ i−1 + V ∗ i + V ∗ i+1

(0  i  δ), where V ∗

−1 = 0 and V ∗ δ+1 = 0.

• 4. There does not exist a subspace W of V such that AW ✓ W ,

A∗W ✓ W , W 6= 0, W 6= V .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Example: Q-polynomial distance-regular graph

Let Γ = Γ(X, E) denote a Q-polynomial distance-regular graph. Let A denote the adjacency matrix of Γ. Fix x 2 X. Let A∗ = A∗(x) denote the dual adjacency matrix of Γ with respect to x. Let W denote an irreducible (A, A∗)-module of C|X|. Then A, A∗ form a TD pair on W .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Connections

There are connections between TD pairs and

I Q-polynomial distance-regular graphs I representation theory I orthogonal polynomials I partially ordered sets I statistical mechanical models I other areas of physics

For further examples, see “Some algebra related to P- and Q-polynomial association schemes” by Ito, Tanabe, and Terwilliger or the survey “An algebraic approach to the Askey scheme of orthogonal polynomials” by Terwilliger.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Standard ordering

Definition

Given a TD pair A, A∗, an ordering {Vi}d

i=0 of the eigenspaces of A is

called standard whenever A∗Vi ✓ Vi−1 + Vi + Vi+1 (0  i  d), where V−1 = 0 and Vd+1 = 0. (A similar discussion applies to A∗.)

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Standard ordering

Definition

Given a TD pair A, A∗, an ordering {Vi}d

i=0 of the eigenspaces of A is

called standard whenever A∗Vi ✓ Vi−1 + Vi + Vi+1 (0  i  d), where V−1 = 0 and Vd+1 = 0. (A similar discussion applies to A∗.)

Lemma (Ito, Terwilliger)

If {Vi}d

i=0 is a standard ordering of the eigenspaces of A, then {Vd−i}d i=0

is standard and no other ordering is standard.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Tridiagonal system

Definition

By a tridiagonal system (or TD system) on V , we mean a sequence Φ = (A; {Vi}d

i=0; A∗; {V ∗ i }d i=0)

that satisfies (1)–(3) below.

• 1. A, A∗ is a tridiagonal pair on V .
• 2. {Vi}d

i=0 is a standard ordering of the eigenspaces of A.

• 3. {V ∗

i }d i=0 is a standard ordering of the eigenspaces of A∗.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Assumptions

Until further notice, we fix a TD system Φ = (A; {Vi}d

i=0; A∗; {V ∗ i }d i=0).

Let Φ⇓ = (A; {Vd−i}d

i=0; A∗; {V ∗ i }d i=0).

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Notation

Throughout this talk, we will focus on Φ and its associated objects. Keep in mind that a similar discussion applies to Φ⇓ and its associated objects. For any object f associated with Φ, we let f ⇓ denote the corresponding

• bject associated with Φ⇓.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Some ratios

For 0  i  d, we let θi (resp. θ∗

i ) denote the eigenvalue of A (resp. A∗)

corresponding to the eigenspace Vi (resp. V ∗

i ).

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Some ratios

For 0  i  d, we let θi (resp. θ∗

i ) denote the eigenvalue of A (resp. A∗)

corresponding to the eigenspace Vi (resp. V ∗

i ).

Lemma (Ito, Tanabe, Terwilliger)

The ratios θi−2 θi+1 θi−1 θi , θ∗

i−2 θ∗ i+1

θ∗

i−1 θ∗ i

are equal and independent of i for 2  i  d 1. This gives two recurrence relations, whose solutions can be written in closed form. The most general case is known as the q-Racah case and is described below.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

q-Racah case

Definition

We say that the TD system Φ has q-Racah type whenever there exist nonzero scalars q, a, b 2 K such that q4 6= 1 and θi = aqd−2i + a−1q2i−d, θ∗

i = bqd−2i + b−1q2i−d

for 0  i  d.

Assumption

Throughout this talk, we assume that Φ has q-Racah type. For simplicity, we also assume that K is algebraically closed.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

First split decomposition of V

Definition

For 0  i  d, define Ui = (V ∗

0 + V ∗ 1 + · · · + V ∗ i ) \ (Vi + Vi+1 + · · · + Vd).

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

First split decomposition of V

Definition

For 0  i  d, define Ui = (V ∗

0 + V ∗ 1 + · · · + V ∗ i ) \ (Vi + Vi+1 + · · · + Vd).

Theorem (Ito, Tanabe, Terwilliger)

V = U0 + U1 + · · · + Ud (direct sum) We refer to {Ui}d

i=0 as the first split decomposition of V .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Second split decomposition of V

Definition

For 0  i  d, define U⇓

i = (V ∗ 0 + V ∗ 1 + · · · + V ∗ i ) \ (V0 + V1 + · · · + Vd−i).

Theorem (Ito, Tanabe, Terwilliger)

V = U⇓

0 + U⇓ 1 + · · · + U⇓ d

(direct sum) We refer to {U⇓

i }d i=0 as the second split decomposition of V .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The maps K, B

Definition

Let K : V ! V denote the linear transformation such that for 0  i  d, Ui is an eigenspace of K with eigenvalue qd−2i. That is, (K qd−2iI)Ui = 0 for 0  i  d.

Definition

Let B : V ! V denote the linear transformation such that for 0  i  d, U⇓

i is an eigenspace of B with eigenvalue qd−2i. That is,

(B qd−2iI)U⇓

i = 0

for 0  i  d. Note: B = K ⇓.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Split decompositions of V

Lemma (Ito, Tanabe, Terwilliger)

Let 0  i  d. A, A∗ act on the first split decomposition in the following way: (A θiI)Ui ✓ Ui+1, (A∗ θ∗

i I)Ui ✓ Ui−1.

A, A∗ act on the second split decomposition in the following way: (A θd−iI)U⇓

i ✓ U⇓ i+1,

(A∗ θ∗

i I)U⇓ i ✓ U⇓ i−1.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The raising maps R, R+

Definition

Let R = A aK a−1K −1. By construction,

I R acts on Ui as A θiI for 0  i  d, I RUi ✓ Ui+1

(0  i < d), RUd = 0.

Definition

Let R⇓ = A a−1B aB−1. By construction,

I R⇓ acts on U⇓ i as A θd−iI for 0  i  d, I R⇓U⇓ i ✓ U⇓ i+1

(0  i < d), R⇓U⇓

d = 0.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Relating R, R+, K, B

Lemma (B. 2014)

Both KR = q−2RK, BR⇓ = q−2R⇓B.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The linear transformation ψ

In 2005, Nomura showed that V =

d/2

X

i=0 d−i

X

j=i

τij(A)Ki, where Ki = Ui \ U⇓

i

and τij(x) = (x θi)(x θi+1) · · · (x θj−1).

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The linear transformation ψ

V =

d/2

X

i=0 d−i

X

j=i

τij(A)Ki,

Definition

Let ψ : V ! V denote the linear transformation such that ψτij(A)v = γijτi,j−1(A)v for 0  i  d/2, i  j  d i, and v 2 Ki. Here γij = [j i]q [d i j + 1]q [d]q and [n]q = qn q−n q q−1 .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The linear transformation ψ

Lemma (B. 2012)

For 0  i  d, both ψUi ✓ Ui−1, ψU⇓

i ✓ U⇓ i−1.

Moreover, ψd+1 = 0. In light of the above result, we refer to ψ as the double lowering

• perator.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The linear transformation ψ

Lemma (B. 2014)

Both Kψ = q2ψK, Bψ = q2ψB.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Relating ψ, R, R+, K, B

Lemma (B. 2014)

Both ψR Rψ =

• q q−1

K K −1 , ψR⇓ R⇓ψ =

• q q−1

B B−1 .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Relation summary

Our relations

Kψ = q2ψK, KR = q−2RK, ψR Rψ =

• q q−1

K K −1 ,

Φ⇓-analogues

Bψ = q2ψB, BR⇓ = q−2R⇓B, ψR⇓ R⇓ψ =

• q q−1

B B−1 ,

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The quantum universal enveloping algebra Uq(sl2)

Definition

The quantum universal enveloping algebra Uq(sl2) is defined to be the unital associative K-algebra with generators e, f , k, k−1 and relations kk−1 = 1 = k−1k, ke = q2ek, kf = q−2fk, ef fe = k k−1 q q−1 . The generators e, f , k, k−1 are known as the Chevalley generators for Uq(sl2).

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Two Uq(sl2)-module structures on V

Theorem (B. 2014)

There exists a Uq(sl2)-module structure on V for which the Chevalley generators act as follows: element of Uq(sl2) e f k k−1 action on V (q q−1)−1ψ (q q−1)−1R K K −1

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Two Uq(sl2)-module structures on V

Theorem (B. 2014)

There exists a Uq(sl2)-module structure on V for which the Chevalley generators act as follows: element of Uq(sl2) e f k k−1 action on V (q q−1)−1ψ (q q−1)−1R K K −1

Theorem (B. 2014)

There exists a Uq(sl2)-module structure on V for which the Chevalley generators act as follows: element of Uq(sl2) e f k k−1 action on V (q q−1)−1ψ (q q−1)−1R⇓ B B−1

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Irreducible Uq(sl2)-submodules of V

Let denote M the subalgebra of End(V ) generated by A. For 0  i  d/2 and v 2 Ki = Ui \ U⇓

i , let Mv denote the M-submodule

• f V generated by v.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Irreducible Uq(sl2)-submodules of V

Let denote M the subalgebra of End(V ) generated by A. For 0  i  d/2 and v 2 Ki = Ui \ U⇓

i , let Mv denote the M-submodule

• f V generated by v.

Lemma (B. 2014)

Let 0  i  d/2 and v 2 Ki. For either of the Uq(sl2)-actions on V from the previous slide, Mv is an irreducible Uq(sl2)-submodule of V with dimension d 2i + 1.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Irreducible Uq(sl2)-submodules of V

Let denote M the subalgebra of End(V ) generated by A. For 0  i  d/2 and v 2 Ki = Ui \ U⇓

i , let Mv denote the M-submodule

• f V generated by v.

Lemma (B. 2014)

Let 0  i  d/2 and v 2 Ki. For either of the Uq(sl2)-actions on V from the previous slide, Mv is an irreducible Uq(sl2)-submodule of V with dimension d 2i + 1.

Lemma (B. 2014)

For either of the Uq(sl2)-actions on V from the previous slide, V can be written as a direct sum of its irreducible Uq(sl2)-submodules.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The Casimir element of Uq(sl2)

Definition

Define the normalized Casimir element Λ of Uq(sl2) by Λ = (q q−1)2ef + q−1k + qk−1, = (q q−1)2fe + qk + q−1k−1.

I Λ is in the center of Uq(sl2). I Λ generates the center of Uq(sl2) whenever q is not a root of unity. I Λ acts as a scalar multiple of the identity on irreducible

Uq(sl2)-modules.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The action of the Casimir element on V

Lemma (B. 2014)

With respect to the first Uq(sl2)-module structure on V , the action of the Λ on V is equal to both ψR + q−1K + qK −1, Rψ + qK + q−1K −1.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The action of the Casimir element on V

Lemma (B. 2014)

With respect to the first Uq(sl2)-module structure on V , the action of the Λ on V is equal to both ψR + q−1K + qK −1, Rψ + qK + q−1K −1.

Lemma (B. 2014)

With respect to the second Uq(sl2)-module structure on V , the action of the Λ on V is equal to both ψR⇓ + q−1B + qB−1, R⇓ψ + qB + q−1B−1.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The action of the Casimir element on V

Lemma (B. 2014)

Let 0  i  d/2 and v 2 Ki. With respect to either of the Uq(sl2)-module structures on V , Λ acts on Mv as qd−2i+1 + q−d+2i−1 times the identity.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The action of the Casimir element on V

Corollary (B. 2014)

Let 0  i  d/2. With respect to either of the Uq(sl2)-module structures on V , Λ acts on MKi as qd−2i+1 + q−d+2i−1 times the identity.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The action of the Casimir element on V

Corollary (B. 2014)

Let 0  i  d/2. With respect to either of the Uq(sl2)-module structures on V , Λ acts on MKi as qd−2i+1 + q−d+2i−1 times the identity.

Lemma (B. 2012)

The following sum is direct. V =

r

X

i=0

MKi, where r = bd/2c.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Comparing the actions of Λ on V

Lemma (B. 2014)

The following coincide: (i) the action of Λ on V for the first Uq(sl2)-module structure on V , (ii) the action of Λ on V for the second Uq(sl2)-module structure on V .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Comparing the actions of Λ on V

Lemma (B. 2014)

The following coincide: (i) the action of Λ on V for the first Uq(sl2)-module structure on V , (ii) the action of Λ on V for the second Uq(sl2)-module structure on V .

Corollary (B. 2014)

The following four expressions are equal: ψR + q−1K + qK −1, Rψ + qK + q−1K −1, ψR⇓ + q−1B + qB−1, R⇓ψ + qB + q−1B−1.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Relating K, B, and their inverses

Theorem (B. 2014)

Each of the following holds: KB−1 = I a−1qψ I aqψ , BK −1 = I aqψ I a−1qψ , K −1B = I a−1q−1ψ I aq−1ψ , B−1K = I aq−1ψ I a−1q−1ψ . Note that since ψ is nilpotent, each of the above expressions is equal to a polynomial in ψ.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

How R, K ±1 and R+, B±1 are related

Lemma (B. 2014)

The following equations hold: B = a2K + (1 a2)K

d

X

i=0

a−iq−iψi, B−1 = a−2K −1 + (1 a−2)K −1

d

X

i=0

aiqiψi, R⇓ = R + (a a−1)

d

X

i=0

(a−iq−iK aiqiK −1)ψi.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

How R, K ±1 and R+, B±1 are related

Lemma (B. 2014)

The following equations hold: K = a−2B + (1 a−2)B

d

X

i=0

aiq−iψi, K −1 = a2B−1 + (1 a2)B−1

d

X

i=0

a−iqiψi, R = R⇓ + (a a−1)

d

X

i=0

(a−iqiB−1 aiq−iB)ψi.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Four equations for ψ

Theorem (B. 2014)

Each of the following expressions is equal to ψ: I BK −1 q(aI a−1BK −1), I KB−1 q(a−1I aKB−1), q(I K −1B) aI a−1K −1B , q(I B−1K) a−1I aB−1K .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Relating K, B, and their inverses

Theorem (B. 2014)

We have aK 2 + aq−1 a−1q q q−1 KB + a−1q−1 aq q q−1 BK + a−1B2 = 0.

Theorem (B. 2014)

We have aB−2 + aq−1 a−1q q q−1 K −1B−1 + a−1q−1 aq q q−1 B−1K −1 + a−1K −2 = 0.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The subalgebra U_

q of Uq(sl2) There exists a second presentation of Uq(sl2) known as the equitable

• presentation. The generators are x, y ±1, z and are subject to the

relations yy −1 = y −1y = 1, qxy q−1yx q q−1 = 1, qyz q−1zy q q−1 = 1, qzx q−1xz q q−1 = 1. Let U∨

q denote the subalgebra of Uq(sl2) generated by x, y −1, z.

The above relations involving K ±1, B±1 helped lead to the discovery of two new presentations for U∨

q .

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The subalgebra U_

q of Uq(sl2)

Theorem (B.,Terwilliger 2014)

Let U denote the K-algebra with generators K, B, A and relations qKA q−1AK q q−1 = aK 2 + a−1I, qBA q−1AB q q−1 = a−1B2 + aI, aK 2 + aq−1 a−1q q q−1 KB + a−1q−1 aq q q−1 BK + a−1B2 = 0. Then U is isomorphic to U∨

q .

The second presentation of U∨

q is obtained by inverting q, a, K, B.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

Summary

Given a TD system Φ on V of q-Racah type, we defined several linear transformations which act on V in an attractive way. We used these linear transformations to obtain two Uq(sl2)-module structures for V and discussed how these actions are related. Using information about Uq(sl2), we derived several new equations relating our linear transformations. These equations helped lead to the discovery of new information about Uq(sl2).

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(

The End

Thank you for your attention!

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(