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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
⇤.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
).
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 �+.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
+.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
Relating R ,R+,K ,B
Lemma (B. 2014)
Both
KR = q
�2
RK ,
BR
+ = q
�2
R
+B .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
).
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
The linear transformation
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
.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
The linear transformation
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
The linear transformation
Lemma (B. 2014)
Both
K = q
2 K ,
B = q
2 B .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
�.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
�,
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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).
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (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
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (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
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 , ⇤ acts on
Mv as
q
d�2i+1 + q
�d+2i�1
times the identity.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 , ⇤ 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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 , ⇤ 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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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
.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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.
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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 .
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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).
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
The End
Thank you for your attention!
Sarah Bockting-Conrad Tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq (sl2
)
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