tridiagonal pairs of q-racah type and the quantum …...the quantum universal enveloping algebra u...

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 theq-Racah class.

We introduce some linear transformations which act on the on theunderlying vector space in an attractive manner.

We give two actions of the quantum group Uq(sl2) on the underlyingvector space. We describe these actions in detail, and show how they arerelated.

Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2

)

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

DefinitionBy 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}di=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 }�i=0

of the eigenspaces of A⇤such that

AV ⇤i ✓ V ⇤

i�1

+ V ⇤i + V ⇤

i+1

(0 i �),

where V ⇤�1

= 0 and V ⇤�+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 .


)

Example: Q-polynomial distance-regular graph

Let � = �(X ,E ) denote a Q-polynomial distance-regular graph.

Let A denote the adjacency matrix of �.

Fix x 2 X . Let A⇤ = A

⇤(x) denote the dual adjacency matrix of � withrespect to x .

Let W denote an irreducible (A,A⇤)-module of C|X |.

Then A,A⇤ form a TD pair on W .


)

Connections

There are connections between TD pairs and

IQ-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.


)

Standard ordering

DefinitionGiven a TD pair A,A⇤, an ordering {Vi}di=0

of the eigenspaces of A iscalled 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}di=0

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

is standard and no other ordering is standard.


)

Tridiagonal system

DefinitionBy a tridiagonal system (or TD system) on V , we mean a sequence

� = (A; {Vi}di=0

;A⇤; {V ⇤i }di=0

)

that satisfies (1)–(3) below.

1. A,A⇤ is a tridiagonal pair on V .

2. {Vi}di=0

is a standard ordering of the eigenspaces of A.

3. {V ⇤i }di=0

is a standard ordering of the eigenspaces of A

⇤.


)

Assumptions

Until further notice, we fix a TD system

� = (A; {Vi}di=0

;A⇤; {V ⇤i }di=0

).

Let�+ = (A; {Vd�i}di=0

;A⇤; {V ⇤i }di=0

).


)

Notation

Throughout this talk, we will focus on � and its associated objects. Keepin mind that a similar discussion applies to �+ and its associated objects.

For any object f associated with �, we let f + denote the correspondingobject associated with �+.


)

Some ratios

For 0 i d , we let ✓i (resp. ✓⇤i ) denote the eigenvalue of A (resp. A⇤)corresponding to the eigenspace Vi (resp. V

⇤i ).

Lemma (Ito, Tanabe, Terwilliger)

The ratios

✓i�2

� ✓i+1

✓i�1

� ✓i,

✓⇤i�2

� ✓⇤i+1

✓⇤i�1

� ✓⇤i

are equal and independent of i for 2 i d � 1.

This gives two recurrence relations, whose solutions can be written inclosed form. The most general case is known as the q-Racah case and isdescribed below.


)

q-Racah case

DefinitionWe say that the TD system � has q-Racah type whenever there existnonzero scalars q, a, b 2 K such that q4 6= 1 and

✓i = aq

d�2i + a

�1

q

2i�d ,

✓⇤i = bq

d�2i + b

�1

q

2i�d

for 0 i d .

Assumption

Throughout this talk, we assume that � has q-Racah type. Forsimplicity, we also assume that K is algebraically closed.


)

First split decomposition of V

DefinitionFor 0 i d , define

Ui = (V ⇤0

+ V

⇤1

+ · · ·+ V

⇤i ) \ (Vi + Vi+1

+ · · ·+ Vd).

Theorem (Ito, Tanabe, Terwilliger)

V = U

0

+ U

1

+ · · ·+ Ud (direct sum)

We refer to {Ui}di=0

as the first split decomposition of V .


)

Second split decomposition of V

DefinitionFor 0 i d , define

U

+i = (V ⇤

0

+ V

⇤1

+ · · ·+ V

⇤i ) \ (V

0

+ V

1

+ · · ·+ Vd�i ).

Theorem (Ito, Tanabe, Terwilliger)

V = U

+0

+ U

+1

+ · · ·+ U

+d (direct sum)

We refer to {U+i }di=0

as the second split decomposition of V .


)

The maps K ,B

DefinitionLet K : V ! V denote the linear transformation such that for 0 i d ,Ui is an eigenspace of K with eigenvalue q

d�2i . That is,

(K � q

d�2iI )Ui = 0

for 0 i d .

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

+i is an eigenspace of B with eigenvalue q

d�2i . That is,

(B � q

d�2iI )U+

i = 0

for 0 i d .

Note: B = K

+.


)

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� ✓i I )Ui ✓ Ui+1

, (A⇤ � ✓⇤i I )Ui ✓ Ui�1

.

A,A⇤act on the second split decomposition in the following way:

(A� ✓d�i I )U+i ✓ U

+i+1

, (A⇤ � ✓⇤i I )U+i ✓ U

+i�1

.


)

The raising maps R ,R+

DefinitionLet

R = A� aK � a

�1

K

�1.

By construction,

IR acts on Ui as A� ✓i I for 0 i d ,

IRUi ✓ Ui+1

(0 i < d), RUd = 0.

DefinitionLet

R

+ = A� a

�1

B � aB

�1.

By construction,

IR

+ acts on U

+i as A� ✓d�i I for 0 i d ,

IR

+U

+i ✓ U

+i+1

(0 i < d), R

+U

+d = 0.


)

Relating R ,R+,K ,B

Lemma (B. 2014)

Both

KR = q

�2

RK ,

BR

+ = q

�2

R

+B .


)

The linear transformation

In 2005, Nomura showed that

V =

d/2X

i=0

d�iX

j=i

⌧ij(A)Ki ,

whereKi = Ui \ U

+i

and⌧ij(x) = (x � ✓i )(x � ✓i+1

) · · · (x � ✓j�1

).


)


V =

d/2X

i=0

d�iX

j=i

⌧ij(A)Ki ,

DefinitionLet : V ! V denote the linear transformation such that

⌧ij(A)v = �ij⌧i,j�1

(A)v

for 0 i d/2, i j d � i , and v 2 Ki . Here

�ij =[j � i ]q [d � i � j + 1]q

[d ]q

and

[n]q =q

n � q

�n

q � q

�1

.


)


Lemma (B. 2012)

For 0 i d, both

Ui ✓ Ui�1

,

U+i ✓ U

+i�1

.

Moreover, d+1 = 0.

In light of the above result, we refer to as the double lowering

operator.


)


Lemma (B. 2014)

Both

K = q

2 K ,

B = q

2 B .


)

Relating ,R ,R+,K ,B

Lemma (B. 2014)

Both

R � R =�q � q

�1

� �K � K

�1

�,

R+ � R

+ =�q � q

�1

� �B � B

�1

�.


)

Relation summaryOur relations

K = q

2 K ,

KR = q

�2

RK ,

R � R =�q � q

�1

� �K � K

�1

�,

�+-analogues

B = q

2 B ,

BR

+ = q

�2

R

+B ,

R+ � R

+ =�q � q

�1

� �B � B

�1

�,


)

The quantum universal enveloping algebra Uq

(sl2)

DefinitionThe quantum universal enveloping algebra Uq(sl2) is defined to bethe unital associative K-algebra with generators

e, f , k , k

�1

and relations

kk

�1 = 1 = k

�1

k ,

ke = q

2

ek ,

kf = q

�2

fk ,

ef � fe =k � k

�1

q � q

�1

.

The generators e, f , k , k�1 are known as the Chevalley generators forUq(sl2).


)

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)�1 (q � q

�1)�1

R 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)�1 (q � q

�1)�1

R

+B B

�1


)

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

of 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.


)

The Casimir element of Uq

(sl2)

DefinitionDefine the normalized Casimir element ⇤ of Uq(sl2) by

⇤ = (q � q

�1)2ef + q

�1

k + qk

�1,

= (q � q

�1)2fe + qk + q

�1

k

�1.

I ⇤ is in the center of Uq(sl2).

I ⇤ generates the center of Uq(sl2) whenever q is not a root of unity.

I ⇤ acts as a scalar multiple of the identity on irreducibleUq(sl2)-modules.


)

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 ⇤ on V is equal to both

R + q

�1

K + qK

�1,

R + qK + q

�1

K

�1.

Lemma (B. 2014)

With respect to the second Uq(sl2)-module structure on V , the action of

the ⇤ on V is equal to both

R+ + q

�1

B + qB

�1,

R

+ + qB + q

�1

B

�1.


)


Lemma (B. 2014)

Let 0 i d/2 and v 2 Ki .

With respect to either of the Uq(sl2)-module structures on V , ⇤ acts on

Mv as

q

d�2i+1 + q

�d+2i�1

times the identity.


)


Corollary (B. 2014)

Let 0 i d/2.

With respect to either of the Uq(sl2)-module structures on V , ⇤ acts on

MKi as

q

d�2i+1 + q

�d+2i�1

times the identity.

Lemma (B. 2012)

The following sum is direct.

V =rX

i=0

MKi ,

where r = bd/2c.


)

Comparing the actions of ⇤ on V

Lemma (B. 2014)

The following coincide:

(i) the action of ⇤ on V for the first Uq(sl2)-module structure on V ,

(ii) the action of ⇤ on V for the second Uq(sl2)-module structure on V .

Corollary (B. 2014)

The following four expressions are equal:

R + q

�1

K + qK

�1,

R + qK + q

�1

K

�1,

R+ + q

�1

B + qB

�1,

R

+ + qB + q

�1

B

�1.


)

Relating K , B , and their inverses

Theorem (B. 2014)

Each of the following holds:

KB

�1 =I � a

�1

q

I � aq ,

BK

�1 =I � aq

I � a

�1

q ,

K

�1

B =I � a

�1

q

�1

I � aq

�1 ,

B

�1

K =I � aq

�1

I � a

�1

q

�1 .

Note that since is nilpotent, each of the above expressions is equal to apolynomial in .


)

How R ,K±1 and R+,B±1 are related

Lemma (B. 2014)

The following equations hold:

B = a

2

K + (1� a

2)KdX

i=0

a

�iq

�i i ,

B

�1 = a

�2

K

�1 + (1� a

�2)K�1

dX

i=0

a

iq

i i ,

R

+ = R + (a� a

�1)dX

i=0

(a�iq

�iK � a

iq

iK

�1) i .


)

How R ,K±1 and R+,B±1 are related

Lemma (B. 2014)

The following equations hold:

K = a

�2

B + (1� a

�2)BdX

i=0

a

iq

�i i ,

K

�1 = a

2

B

�1 + (1� a

2)B�1

dX

i=0

a

�iq

i i ,

R = R

+ + (a� a

�1)dX

i=0

(a�iq

iB

�1 � a

iq

�iB) i .


)

Four equations for

Theorem (B. 2014)

Each of the following expressions is equal to :

I � BK

�1

q(aI � a

�1

BK

�1),

I � KB

�1

q(a�1

I � aKB

�1),

q(I � K

�1

B)

aI � a

�1

K

�1

B

,q(I � B

�1

K )

a

�1

I � aB

�1

K

.


)

Relating K , B , and their inverses

Theorem (B. 2014)

We have

aK

2 +aq

�1 � a

�1

q

q � q

�1

KB +a

�1

q

�1 � aq

q � q

�1

BK + a

�1

B

2 = 0.

Theorem (B. 2014)

We have

aB

�2 +aq

�1 � a

�1

q

q � q

�1

K

�1

B

�1 +a

�1

q

�1 � aq

q � q

�1

B

�1

K

�1 + a

�1

K

�2 = 0.


)

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 therelations yy�1 = y

�1

y = 1,

qxy � q

�1

yx

q � q

�1

= 1,qyz � q

�1

zy

q � q

�1

= 1,qzx � q

�1

xz

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 oftwo new presentations for U_

q .


)

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

�1

AK

q � q

�1

= aK

2 + a

�1

I ,qBA� q

�1

AB

q � q

�1

= a

�1

B

2 + aI ,

aK

2 +aq

�1 � a

�1

q

q � q

�1

KB +a

�1

q

�1 � aq

q � q

�1

BK + a

�1

B

2 = 0.

Then U is isomorphic to U

_q .

The second presentation of U_q is obtained by inverting q, a,K ,B .


)

Summary

Given a TD system � on V of q-Racah type, we defined several lineartransformations which act on V in an attractive way.

We used these linear transformations to obtain two Uq(sl2)-modulestructures for V and discussed how these actions are related.

Using information about Uq(sl2), we derived several new equationsrelating our linear transformations.

These equations helped lead to the discovery of new information aboutUq(sl2).


)

The End

Thank you for your attention!


)

tridiagonal pairs of q-racah type and the quantum …...the quantum universal enveloping algebra u...

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