ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 14 (2018) 55-65
https://doi.org/10.26493/1855-3974.1283.456
(Also available at http://amc-journal.eu)
Tridiagonal pairs of q-Racah type, the Bockting operator and L-operators for Uq(L(sl2))
Paul Terwilliger
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI53706-1388 USA
Received 11 January 2017, accepted 18 March 2017, published online 5 May 2017
We describe the Bockting operator ^ for a tridiagonal pair of q-Racah type, in terms of a certain L-operator for the quantum loop algebra Uq(L(sl2)).
Keywords: Bockting operator, tridiagonal pair, Leonard pair. Math. Subj. Class.: 17B37, 15A21
1 Introduction
In the theory of quantum groups there exists the concept of an L-operator; this was introduced in [20] to obtain solutions for the Yang-Baxter equation. In linear algebra there exists the concept of a tridiagonal pair; this was introduced in [13] to describe the irreducible modules for the subconstituent algebra of a Q-polynomial distance-regular graph. Recently some authors have connected the two concepts. In [1], [4] Pascal Baseilhac and Kozo Koizumi use L-operators for the quantum loop algebra Uq(L(sl2)) to construct a family of finite-dimensional modules for the q-Onsager algebra Oq; see [2, 3, 5, ?] for related work. A finite-dimensional irreducible Oq-module is essentially the same thing as a tridiagonal pair of q-Racah type [?, Section 12], [23, Section 3]. In [22, Section 9], Kei Miki uses similar L-operators to describe how Uq(L(sl2)) is related to the q-tetrahedron algebra Klq. A finite-dimensional irreducible Klq-module is essentially the same thing as a tridiagonal pair of q-geometric type [16, Theorem 2.7], [14, Theorems 10.3, 10.4]. Following Baseilhac, Koizumi, and Miki, in the present paper we use L-operators for Uq(L(sl2)) to describe the Bockting operator ^ associated with a tridiagonal pair of q-Racah type. Before going into detail, we recall some notation and basic concepts. Throughout this paper
E-mail address: terwilli@math.wisc.edu (Paul Terwilliger)
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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Ars Math. Contemp. 14 (2018) 117-128
F denotes a field. Let V denote a vector space over F with finite positive dimension. For an F-linear map A : V ^ V and a subspace W C V, we say that W is an eigenspace of A whenever W = 0 and there exists 6 G F such that W = {v G V|Av = 6v}; in this case 6 is called the eigenvalue of A associated with W. We say that A is diagonalizable whenever V is spanned by the eigenspaces of A.
Definition 1.1. (See [13, Definition 1.1].) Let V denote a vector space over F with finite positive dimension. By a tridiagonal pair (or TD pair) on V we mean an ordered pair of F-linear maps A : V ^ V and A* : V ^ V that satisfy the following four conditions:
(i)	Each of A, A* is diagonalizable.
(ii)	There exists an ordering {Vi}d=0 of the eigenspaces of A such that
A*Vi C + Vi + V+1	(0 < i < d),	(1.1)
where V_1 = 0 and Vd+1 = 0.
(iii)	There exists an ordering {Vj*}f=0 of the eigenspaces of A* such that
AV* C Vi_i + V; + Vi+i	(0 < i < S),	(1.2)
where V_1 =0 and V/+1 = 0.
(iv)	There does not exist a subspace W C V such that AW C W, A* W C W, W = 0, W = V.
We refer the reader to [12,13,17] for background on TD pairs, and here mention only a few essential points. Let A, A* denote a TD pair on V, as in Definition 1.1. By [13, Lemma 4.5] the integers d and S from (1.1) and (1.2) are equal; we call this common value the diameter of A, A*. An ordering of the eigenspaces for A (resp. A*) is called standard whenever it satisfies (1.1) (resp. (1.2)). Let {Vi}d=0 denote a standard ordering of the eigenspaces of A. By [13, Lemma 2.4] the ordering {Vd_i}d=0 is standard and no further ordering is standard. A similar result holds for the eigenspaces of A*. Until the end of this section fix a standard ordering {Vi}d=0 (resp. {Vi*}d=0) of the eigenspaces for A (resp. A*). For 0 < i < d let 6i (resp. 6*) denote the eigenvalue of A (resp. A*) for the eigenspace Vi (resp. Vi*). By construction {6i}d=0 are mutually distinct and contained in F. Moreover {6* }d=0 are mutually distinct and contained in F. By [13, Theorem 11.1] the expressions
6i_2 - 6i+1 6*_2 - 6i*+1
/3/3	/3*	/3*
6i_1 - 6i 6j_1 - 6j
are equal and independent of i for 2 < i < d - 1. For this constraint the solutions can be given in closed form [13, Theorem 11.2]. The "most general" solution is called q-Racah, and will be described shortly.
We now recall the split decomposition [13, Section 4]. For 0 < i < d define
Ui = (V0* + V1* + • • • + V/) n (V0 + V1 + • • • + Vd_i).
For notational convenience define U_1 = 0 and Ud+1 = 0. By [13, Theorem 4.6] the sum V = Ed=0 Ui is direct. By [13, Theorem 4.6] both
U0 + U1 + • • • + Ui = V0* + V1* + • • • + Vi*, Ui + Ui+1 + • • • + Ud = V0 + V1 + • • • + Vd_i
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for 0 < i < d. Let I: V ^ V denote the identity map. By [13, Theorem 4.6] both
(A - ed-il)Ui C Ui+1,	(A* - 0*T)Ui C Ui-i	(1.3)
for 0 < i < d.
We now describe the q-Racah case. Pick a nonzero q e F such that q4 = 1. We say that A, A* has q-Racah type whenever there exist nonzero a, b e F such that both
0i = aq2i-d + a-1qd-2i,	0* = bq2i-d + b-1qd-2i	(1.4)
for 0 < i < d. For the rest of this section assume that A, A* has q-Racah type. For 1 < i < d we have q2i = 1; otherwise 0i = 0o. Define an F-linear map K : V ^ V such that for 0 < i < d, Ui is an eigenspace of K with eigenvalue qd-2i. Thus
(K - qd-2i7)Ui = 0	(0 < i < d).	(1.5)
Note that K is invertible. For 0 < i < d the following holds on Ui:
aK + a-1K-1 = 0d-i7.	(1.6)
Define an F-linear map R : V ^ V such that for 0 < i < d, R acts on Ui as A — 0d-i1. By (1.6),
A = aK + a-1K-1 + R.	(1.7)
By the equation on the left in (1.3),
RUi C Ui+1	(0 < i < d).	(1.8)
We now recall the Bockting operator By [8, Lemma 5.7] there exists a unique F-linear map ^ : V ^ V such that both
VU C Ui-1	(0 < i < d),	(1.9)
^R - R^ = (q - q-1)(K - K-1).	(1.10)
The known properties of ^ are described in [7, 8, ?]. Suppose we are given A, A*, R, K in matrix form, and wish to obtain ^ in matrix form. This can be done using (1.8), (1.9), (1.10) and induction on i. The calculation can be tedious, so one desires a more explicit description of In the present paper we give an explicit description of in terms of a certain L-operator for Uq(L(sl2)). According to this description, ^ is equal to -a times the ratio of two components for the L-operator. Theorem 5.4 is our main result.
The paper is organized as follows. In Section 2 we review the algebra Uq (L(sl2)) in its Chevalley presentation. In Section 3 we recall the equitable presentation for Uq(L(sl2)). In Section 4 we discuss some L-operators for Uq(L(sl2)). In Section 5 we use these L-operators to describe
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Ars Math. Contemp. 14 (2018) 117-128
2 The quantum loop algebra Uq (L (sl2))
Recall the integers Z = {0, ±1, ±2,...} and natural numbers N = {0,1, 2,...}. We will be discussing algebras. An algebra is meant to be associative and have a 1. Recall the field F. Until the end of Section 4, fix a nonzero q G F such that q2 = 1. Define
q" — q "
q =
[n]q =
n G Z.
All tensor products are meant to be over F.
Definition 2.1. (See [10, Section 3.3].) Let (L(sl2)) denote the F-algebra with generators Ej, Fj, K^1 (i G {0,1}) and relations
KiKi KcKi KiEi

= 1, = 1,
q2EiKi
K - i Ki
1,
KiEj = q-2Ej Ki
EiFj — Fj Ei
Ki — Ki di'j q — q-1
KiKo = 1,
KiFi = q-2FiKi,
KiFj = q2Fj Ki, i
« =
EfEj - [3]qE2EjEi + [3]qEiEjE2 - EjEf = 0,	i = j,
- [3]qEj2EjFi + [3]qFiFjF2 - FjF3 = 0,	i = j.
We call Ei, Fi, K^1 the Chevalley generators for (L(sl2)).
Lemma 2.2. (See [18, p. 35].) We turn (L(sl2)) into a Hopf algebra as follows. The coproduct A satisfies
A(Ki) = Ki << Ki,	A(K-1) = K-1 < K-1,
A(Ei) = Ei <g> 1 + Ki << Ei,	A(Fi) = 1 <g> Fi + Fi << K-1.
The counit e satisfies
e(Ki) = 1, e(K-1) = 1,	e(Ei) = 0, e(Fi) = 0.
The antipode S satisfies
S(Ki ) = Ki
i
«(K-1) = Ki, S(Ei ) = —K-1E
i
«(Fi) = —FiKi.
We now discuss the (L(sl2))-modules.
Lemma 2.3. (See [10, Section 4].) There exists a family of (L(sl2))-modules
V(d,t)	0 = d G N,	0= t G F
with this property: V(d, t) has a basis {vj}d=0 such that
(2.1)
K1vi = qd-2ivi	(0 < i < d),
EiVi = [d — i + 1]qVi-1	(1 < i < d),
FiVi = [i + 1]qVi+1	(0 < i < d — 1),
Kovi = q2i-dvi	(0 < i < d),
EoVi = t[i + 1]qVi+1	(0 < i < d — 1),
E1V0 F1Vd =
0, 0,
FoVi = t	[d — i + 1]q Vi-1
(1 < i < d),
Eovd = 0, Fovo = 0.
The module V(d, t) is irreducible provided that q2i = 1 for 1 < i < d.
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Definition 2.4. Referring to Lemma 2.3, we call V(d,t) an evaluation module for Uq(L(sl2)). We call d the diameter. We call t the evaluation parameter.
Example 2.5. For 0 = t g F the Uq(L(sl2))-module V(1,t) is described as follows. With respect to the basis v0, v\ from Lemma 2.3, the matrices representing the Chevalley generators are
Ei :
Eo :
0 1 0 0
00 t 0
Fi : Fo :
00 10
0 t-1 00
Ki : Ko :
q 0 0 q-1
q-1 0 0q
Lemma 2.6. (See [19, p. 58].) Let U and V denote Uq(L(sl2))-modules. Then U < V becomes a Uq (L(sl2))-module as follows. For u G U and v G V,
Ki(u < v) = Ki(u) < Ki(v), Kr\u < v) = K-i(u) < K— (v), Ei(u < v) = Ei(u) < v + Ki(u) < Ei(v), Fi(u < v) = u < Fi(v) + Fi(u) < K-rl(v).
Definition 2.7. (See [11, p. 110].) Up to isomorphism, there exists a unique Uq(L(sl2))-module of dimension 1 on which each u G Uq(L(sl2)) acts as e(u)I, where e is from Lemma 2.2. This Uq (L(sl2))-module is said to be trivial.
Proposition 2.8. (See [22, Theorem 3.2].) Assume that F is algebraically closed with characteristic zero, and q is not a root of unity. Let V denote a nontrivial finite-dimensional irreducible Uq (L(sl2))-module on which each eigenvalue of Ki is an integral power of q. Then V is isomorphic to a tensor product of evaluation Uq (L(sl2))-modules.
3 The equitable presentation for Uq (L (sl2))
In this section we recall the equitable presentation for Uq(L(sl2)). Let Z4 = Z/4Z denote the cyclic group of order 4. In a moment we will discuss some objects Xij. The subscripts i,j are meant to be in Z4.
Lemma 3.1. (See [15, Theorem 2.1], [22, Proposition 4.2].) The algebra Uq(L(sl2)) has a presentation by generators
Xoi, Xi2, and the following relations:
X23, X30, X13, X31
(3.1)
X13X31 — 1, X31X13 — 1, 9X23X30 — q 1X3oX23 — 1
9X01X12 — q X12X01
9X12X23 — q X23X12
— 1,
q	— q-1
qX30X01	— q 1Xo1X3o
q	— q-1
qX23X31	— q 1X31X23
— 1,
q—q-
— 1,
q—	q-1
qXo1X13 —	q 1X13Xo1
q—	q-1
qX13X30 —	q 1X3oX13
— 1,
q — q-1
9X31X12 — q 1X12X31
q — q-1
X3 ,i+1 Xi+2,i+3 — [3] q Xi,i+1Xi+2,i+3Xi,i+1 + [3]q Xi,i+1Xi+2,i+3 Xi,i+1 — Xi+2,i+3Xi,i+1 — 0.
q—q-
— 1,
1
1
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Ars Math. Contemp. 14 (2018) 117-128
An isomorphism with the presentation in Definition 2.1 sends
Xoi ^ Ko + q(q - q-1)KoFo, X23 ^ Ki + q(q - q-1)KiFi, X13 ^ Ki,	X31 ^ Ko.
X12 ^ Ki - (q - q-i)Ei, X3o ^ Ko - (q - q-i)Eo,
The inverse isomorphism sends
ii
Ei ^ (Xi3 - Xi2)(q - q ) , Eo ^ (X3i - X3o)(q - q-i)-
Fi ^ (X3iX23 - 1)q-i(q - q-i)-i, Fo ^ (X^Xoi - 1)q-i(q - q-i)-i,
Ki ^ X
i3,
Ko ^ X3i.
Note 3.2. For notational convenience, we identify the copy of Uq (L(sl2)) given in Definition 2.1 with the copy given in Lemma 3.1, via the isomorphism given in Lemma 3.1.
Definition 3.3. Referring to Lemma 3.1, we call the generators (3.1) the equitable generators for Uq(L(sl2)).
Lemma 3.4. (See [24, Theorem 3.4].) From the equitable point of view the Hopf algebra Uq (L(sl2)) looks as follows. The coproduct A satisfies
A(Xis) = X13 ® X13,	A(Xsi) = X31 ® X31,
A(Xoi) = (X01 - X31) ® 1 + X31 ® X01, A(Xi2) = (X12 - X13) ® 1 + X13 ® X12, A(X23) = (X23 - X13) ® 1 + X13 ® X23, A(X30) = (X30 - X31) ® 1 + X31 ® X30.
The counit e satisfies
e(Xi3) = 1,
e(Xi2) = 1,
e(X3i) = 1,
e(X23) = 1,
e(Xoi) = 1,
e(X3o) = 1.
The antipode S satisfies
S(X3i) = X13,	S (X13) = X31,
S(XoO = 1 + X13 — X13X01,	S (X12) = 1 + X31 — X31X12,
S(X23) = 1 + X31 — X31X23, S (X30) = 1 + X13 — X13X30. 4 Some L-operators for Uq (L (sl2 ))
In this section we recall some L-operators for Uq(L(sl2)), and describe their basic properties.
We recall some notation. Let A denote the coproduct for a Hopf algebra H. Then the opposite coproduct Aop is the composition
A°P : H
H <g> H

H < H.
Definition 4.1. (See [22, Section 9.1].) Let V denote a Uq(L(st2))-module and 0 = t e F. Consider an F-linear map
L : V < V(1,t) ^ V < V(1,t).
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61
We call this map an L-operator for V with parameter t whenever the following diagram commutes for all u G Uq(L(sl2)):
V	<g> V(1,t)
L
V	<g> V(1,t)
A(u)
A°p(u)
>	V <g> V(1,t)
L
>	V <g> V(1,t)
Definition 4.2. (See [22, Section 9.1].) Let V denote a Uq(L(sh))-module and 0 = t G F. Consider any F-linear map
L : V <g> V(1,t) ^ V <g> V(1,t).
For r,s G {0,1} define an F-linear map Lrs : V ^ V such that for v G V,
L(v ® vo) = Loo(v) ® vo + Lio(v) <g> vi, L(v ® vi) = Loi(v) ® vo + Lii(v) ® vi.
Here vo, vi is the basis for V(1, t) from Lemma 2.3.
(4.1)
(4.2)
(4.3)
Lemma 4.3. Referring to Definition 4.2, the map (4.1) is an L-operator for V with parameter t if and only if the following equations hold on V:
K1L00 = L00K1, K1L10 = q2LioKi,
KiLoi = q LoiKi, K1L11 = LiiKi;
LooEi — qEiLoo = Lio, LioEi — q 1 EiLio = 0,
LoiEi — qEiLoi = Lii — LooKi, L11E1 — q 1E1L11 = —LioKi;
FiLoo — q LooFi = Loi, FiLio — q 1LioFi = L11 — KoLo
FiLoi — qLoiFi = 0,
F1L11 — qLiiFi = —KoLoi;
KoLoo = LooKo, KoLio = q 2LioKo
KoLoi = q LoiKo, KoLii = LiiKo;
LooEo — q EoLoo = —tLoiKo, LioEo — qEoLio = tLoo — tLiiKo,
LoiEo — q EoLoi = 0, LiiEo — qEoLii = tLoi;
FoLoo — qLooFo = —t KiLio,
FoLoi — q 1LoiFo = t 1Loo — t 1 Ki L11,
FoLio — qLioFo = 0,	Fo L11 — q 1Lii Fo = t 1Lio.
Proof. This is routinely checked.
□
Example 4.4. (See [21, Appendix], [22, Proposition 9.2].) Referring to Definition 4.2, assume that V is an evaluation module V(d, such that q2i = 1 for 1 < i < d. Consider the matrices that represent the Lrs with respect to the basis {v}d=0 for V(d, from Lemma 2.3. Then the following are equivalent:
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Ars Math. Contemp. 14 (2018) 117-128
(i)	the map (4.1) is an L-operator for V with parameter t;
(ii)	the matrix entries are given in the table below (all matrix entries not shown are zero):
operator (i,i — 1)-entry (i,i) -entry
Loo Loi Lio Lii
Here £ G F.
0
[i]q qi-i 0 0

(i — 1, i)-entry
£
q-q-1
0 0
-d+1-ß-1tq-
q-q-1
0 0
[d — i +1]q qi-d^-it£
Lemma 4.5. (See [22, Proposition 9.3].) Let U and V denote Uq(L(s[2))-modules, and consider the Uq(L(sl2))-module U ( V from Lemma 2.6. Let 0 = t G F. Suppose we are given L-operators for U and V with parameter t. Then there exists an L-operator for U ( V with parameter t such that for r,s G {0,1},
Lrs(u <g) v) = Lro(u) ® Los(v) + Lri(u) <g> Lis(v)
u G U, v G V. (4.4)
Proof. For r, s G {0,1} define an F-linear map Lrs : U (g> V ^ U (g> V that satisfies (4.4). Using (4.4) and Lemma 2.6 one checks that the Lrs satisfy the equations in Lemma 4.3. The result follows by Lemma 4.3.	□
Corollary 4.6. Adopt the notation and assumptions of Proposition 2.8. Then for 0 = t G F there exists a nonzero L-operator for V with parameter t.
Proof. By Proposition 2.8 along with Example 4.4 and Lemma 4.5.
□
1
q
£
q
£
0
5 TD pairs and L-operators
In Section 1 we discussed a TD pair A, A* on V. We now return to this discussion, adopting the notation and assumptions that were in force at the end of Section 1. Recall the scalars q, a, b from (1.4). Recall the map K from above (1.5).
Proposition 5.1. (See [17, p. 103].) Assume that F is algebraically closed with characteris-ticzero, and q is not a root of unity. Then the vector space V becomesa Uq (L(sl2))-module on which K = X3\, K= X\3 and
A = aXoi + a-1Xi2,	A* = bX23 + b-1X3o.
Proof. This is how [17, p. 103] looks from the equitable point of view.	□
Note 5.2. The Uq(L(s[2))-module structure from Proposition 5.1 is not unique in general.
We now investigate the Uq (L(sl2))-module structure from Proposition 5.1. Recall the map R from above (1.7).
Lemma 5.3. Assume that the vector space V becomes a Uq(L(sl2))-module on which K = X31, K-1 = X13 and
A = aXo1 + a-1X12,	A* = bX23 + b-1X3o.
On this module,
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63
(i)	R looks as follows in the equitable presentation:
R = a(Xoi - X31)+ a-1(Xu - X13).	(5.1)
(ii)	R looks as follows in the Chevalley presentation:
R =(q - q-1)(aqKoFo - a-1Ei).	(5.2)
Proof. (i) In line (1.7) eliminate A, K, K-1 using the assumptions of the present lemma. (ii) Evaluate the right-hand side of (5.1) using the identifications from Lemma 3.1 and Note 3.2.	□
We now present our main result. Recall the Bockting operator i from (1.9), (1.10).
Theorem 5.4. Assume that the vector space V becomes a Uq(L(sl2))-module on which K = X31, K-1 = X13 and
A = aXo1 + a-1X12,	A* = bX23 + b-1X3o.
Consider an L-operator for V with parameter a2. Then on V,
i = -a(Loo)-1 L01	(5.3)
provided that Loo is invertible.
Proof. Let ip denote the expression on the right in (5.3). We show i = ip. To do this, we show that ip satisfies (1.9), (1.10). Concerning (1.9), by Lemma 4.3 the equation Koip = q2'ipKo holds on V. By Lemma 3.1, Note 3.2, and the construction, we obtain K0 = X31 = K on V. By these comments Kip = q2ipK on V. By this and (1.5) we obtain ipU.i C Ui-1 for 0 < i < d. So ip satisfies (1.9). Next we show that ip satisfies (1.10). Since Loo is invertible and KoK1 = 1 it suffices to show that on V,
Loo(iR - Rip) = (q - q-1)Loo(Ko - K1).	(5.4)
By this and (5.2) it suffices to show that on V,
aqLoo(iKoFo - KoFoi) - a-1Loo(iE1 - E1$) + LooK - Ko) = 0. (5.5)
We examine the terms in (5.5). By Lemma 4.3 and the construction, the following hold on V:
LooiKoFo = -aLo1KoFo
= -aq-2KoLo1Fo
= -aq-1Ko(FoLo1 - a-2Loo + a-2K1Ln)
and
LooKoFoi = KoLooFoi
^1o + FoLoo) q-1Ko(a-2K1L1oi - aFoLo!)
= q 1Ko(a 2K1LW + FoLoo)'
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Ars Math. Contemp. 14 (2018) 117-128
and
LooV>Ei = -aLoiEi
= -a(qEiLoi + Lii - LooKi) = -a(qEiLoi + Lii - KiLoo)
and
LooEi'ip = (Lio + qEiLoo)i> = Lio^A - qaEiLoi
and
LooKi = KiLoo,
LooKo = KoLoo.
To verify (5.5), evaluate its left-hand side using the above comments and simplify the result using K0Ki = 1. The computation is routine, and omitted. We have shown that ^ satisfies
Acknowledgment
The author thanks Sarah Bockting-Conrad and Edward Hanson for giving this paper a close reading and offering valuable suggestions. The author also thanks Pascal Baseilhac for many conversations concerning quantum groups, L-operators, and tridiagonal pairs.
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(1.10). The result follows.
□
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P Terwilliger: Tridiagonal pairs of q-Racah type, the Bockting operator and...
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