ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P3.01 / 363–386
https://doi.org/10.26493/1855-3974.2669.58c
(Also available at http://amc-journal.eu)
A compact presentation for the alternating
central extension of the positive part of Uq(ŝl2)
Paul M. Terwilliger *
Department of Mathematics, University of Wisconsin,
480 Lincoln Drive, Madison, WI 53706-1388 USA
Received 6 July 2021, accepted 24 August 2021, published online 9 June 2022
Abstract
This paper concerns the positive part U+q of the quantum group Uq(ŝl2). The algebra
U+q has a presentation involving two generators that satisfy the cubic q-Serre relations.
We recently introduced an algebra U+q called the alternating central extension of U+q . We
presented U+q by generators and relations. The presentation is attractive, but the multitude
of generators and relations makes the presentation unwieldy. In this paper we obtain a
presentation of U+q that involves a small subset of the original set of generators and a very
manageable set of relations. We call this presentation the compact presentation of U+q .
Keywords: q-Onsager algebra, q-Serre relations, q-shuffle algebra, tridiagonal pair.
Math. Subj. Class. (2020): 17B37, 05E14, 81R50
1 Introduction
The algebra Uq(ŝl2) is well known in representation theory [15] and statistical mechanics
[20]. This algebra has a subalgebra U+q called the positive part. The algebra U
+
q has a
presentation involving two generators (said to be standard) and two relations, called the
q-Serre relations. The presentation is given in Definition 2.1 below.
Our interest in U+q is motivated by some applications to linear algebra and combina-
torics; these will be described shortly. Before going into detail, we have a comment about q.
In the applications, either q is not a root of unity, or q is a root of unity with exponent large
enough to not interfere with the rest of the application. To keep things simple, throughout
the paper we will assume that q is not a root of unity.
*The author thanks Pascal Baseilhac for many conversations about U+q and its central extension U+q . The
author thanks Kazumasa Nomura for giving this paper a close reading and offering many valuable comments.
E-mail address: terwilli@math.wisc.edu (Paul M. Terwilliger)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
364 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
Our first application has to do with tridiagonal pairs [17]. A tridiagonal pair is roughly
described as an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional
vector space, that each act on the eigenspaces of the other one in a block-tridiagonal fashion
[17, Definition 1.1]. There is a type of tridiagonal pair said to be q-geometric [18, Defini-
tion 2.6]; for this type of tridiagonal pair the eigenvalues of each map form a q2-geometric
progression. A finite-dimensional irreducible U+q -module on which the standard genera-
tors are not nilpotent, is essentially the same thing as a tridiagonal pair of q-geometric type
[18, Theorem 2.7]; these U+q -modules are described in [18, Section 1]. See [13, 24] for
more background on tridiagonal pairs.
Our next application has to do with distance-regular graphs [1, 14, 32]. Consider a
distance-regular graph Γ that has diameter d ≥ 3 and classical parameters (d, b, α, β)
[14, p. 193] with b = q2 and α = q2 − 1. The condition on α implies that Γ is formally
self-dual in the sense of [14, p. 49]. Let A denote the adjacency matrix of Γ, and let A∗
denote the dual adjacency matrix with respect to any vertex of Γ [19, Section 7]. Then by
[19, Lemma 9.4], there exist complex numbers r, s, r∗, s∗ with r, r∗ nonzero such that
rA + sI , r∗A∗ + s∗I satisfy the q-Serre relations. As mentioned in [19, Example 8.4],
the above parameter restriction is satisfied by the bilinear forms graph [14, p. 280], the
alternating forms graph [14, p. 282], the Hermitean forms graph [14, p. 285], the quadratic
forms graph [14, p. 290], the affine E6 graph [14, p. 340], and the extended ternary Golay
code graph [14, p. 359].
Our next application has to do with uniform posets [23, 27]. Let GF(b) denote a finite
field with b elements, and let N,M denote positive integers. Let H denote a vector space
over GF(b) that has dimension N +M . Let h denote a subspace of H with dimension M .
Let P denote the set of subspaces of H that have zero intersection with h. For x, y ∈ P
define x ≤ y whenever x ⊆ y. The relation ≤ is a partial order on P , and the poset P
is ranked with rank N . The poset P is called an attenuated space poset, and denoted by
Ab(N,M) [21], [27, Example 3.1]. By [27, Theorem 3.2] the poset Ab(N,M) is uniform
in the sense of [27, Definition 2.2]. It is shown in [21, Lemma 3.3] that for Ab(N,M)
the raising matrix R and the lowering matrix L satisfy the q-Serre relations, provided that
b = q2.
Our last application has to do with q-shuffle algebras. Let F denote a field, and let x, y
denote noncommuting indeterminates. Let V denote the free associative F-algebra with
generators x, y. By a letter in V we mean x or y. For an integer n ≥ 0, by a word of length
n in V we mean a product of letters v1v2 · · · vn. The words in V form a basis for the vector
space V . In [25, 26] M. Rosso introduced an algebra structure on V , called the q-shuffle
algebra. For letters u, v their q-shuffle product is u ⋆ v = uv + q⟨u,v⟩vu, where ⟨u, v⟩ = 2
(resp. ⟨u, v⟩ = −2) if u = v (resp.u ̸= v). By [25, Theorem 13], in the q-shuffle algebra
V the elements x, y satisfy the q-Serre relations. Consequently there exists an algebra
homomorphism ♮ from U+q into the q-shuffle algebra V , that sends the standard generators
of U+q to x, y. By [26, Theorem 15] the map ♮ is injective.
Next we recall the alternating elements in U+q [30]. Let v1v2 · · · vn denote a word in
V . This word is called alternating whenever n ≥ 1 and vi−1 ̸= vi for 2 ≤ i ≤ n. Thus the
alternating words have the form · · ·xyxy · · · . The alternating words are displayed below:
x, xyx, xyxyx, xyxyxyx, . . .
y, yxy, yxyxy, yxyxyxy, . . .
yx, yxyx, yxyxyx, yxyxyxyx, . . .
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 365
xy, xyxy, xyxyxy, xyxyxyxy, . . .
By [30, Theorem 8.3] each alternating word is contained in the image of ♮. An element
of U+q is called alternating whenever it is the ♮-preimage of an alternating word. For ex-
ample, the standard generators of U+q are alternating because they are the ♮-preimages of
the alternating words x, y. It is shown in [30, Lemma 5.12] that for each row in the above
display, the corresponding alternating elements mutually commute. A naming scheme for
alternating elements is introduced in [30, Definition 5.2].
Next we recall the alternating central extension of U+q [29]. In [30] we displayed two
types of relations among the alternating elements of U+q ; the first type is [30, Proposi-
tions 5.7, 5.10, 5.11] and the second type is [30, Propositions 6.3, 8.1]. The relations
in [30, Proposition 5.11] are redundant; they follow from the relations in [30, Proposi-
tions 5.7, 5.10] as pointed out in [2, Propositions 3.1, 3.2] and [5, Remark 2.5]; see also
Corollary 6.3 below. The relations in [30, Proposition 6.3] are also redundant; they follow
from the relations in [30, Propositions 5.7, 5.10] as shown in the proof of [30, Proposi-
tion 6.3]. By [30, Lemma 8.4] and the previous comments, the algebra U+q is presented by
its alternating elements and the relations in [30, Propositions 5.7, 5.10, 8.1]. For this presen-
tation it is natural to ask what happens if the relations in [30, Proposition 8.1] are removed.
To answer this question, in [29, Definition 3.1] we defined an algebra U+q by generators and
relations in the following way. The generators, said to be alternating, are in bijection with
the alternating elements of U+q . The relations are the ones in [30, Propositions 5.7, 5.10].
By construction there exists a surjective algebra homomorphism U+q → U+q that sends
each alternating generator of U+q to the corresponding alternating element of U+q . In
[29, Lemma 3.6, Theorem 5.17] we adjusted this homomorphism to get an algebra isomor-
phism U+q → U+q ⊗F[z1, z2, . . .], where {zn}∞n=1 are mutually commuting indeterminates.
By [29, Theorem 10.2] the alternating generators form a PBW basis for U+q . The algebra
U+q is called the alternating central extension of U+q .
We mentioned above that the algebra U+q is presented by its alternating generators and
the relations in [30, Propositions 5.7, 5.10]. This presentation is attractive, but the multitude
of generators and relations makes the presentation unwieldy. In this paper we obtain a
presentation of U+q that involves a small subset of the original set of generators and a very
manageable set of relations. This presentation is given in Definition 3.1 below; we call it
the compact presentation of U+q . At first glance, it is not clear that the algebra presented in
Definition 3.1 is equal to U+q . So we denote by U the algebra presented in Definition 3.1,
and eventually prove that U = U+q . After this result is established, we describe some
features of U+q that are illuminated by the presentation in Definition 3.1.
Our investigation of U+q is inspired by some recent developments in statistical mechan-
ics, concerning the q-Onsager algebra Oq . In [9] Baseilhac and Koizumi introduce a current
algebra Aq for Oq , in order to solve boundary integrable systems with hidden symmetries.
In [12, Definition 3.1] Baseilhac and Shigechi give a presentation of Aq by generators and
relations. This presentation and the discussion in [12, Section 4] suggest that Aq is related
to Oq in roughly the same way that U+q is related to U+q . The relationship between Aq and
Oq was conjectured in [7, Conjectures 1, 2] and [28, Conjectures 4.5, 4.6, 4.8], before be-
ing settled in [31, Theorems 9.14, 10.2, 10.3, 10.4]. The articles [3, 4, 6, 7, 8, 9, 10, 11, 12]
contain background information on Oq and Aq .
Earlier in this section, we indicated how U+q has applications to tridiagonal pairs,
distance-regular graphs, and uniform posets. Possibly U+q appears in these applications,
and this possibility should be investigated in the future.
366 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
This paper is organized as follows. In Section 2 we review some facts about U+q . In Sec-
tion 3, we introduce the algebra U and give an algebra homomorphism U+q → U . In Sec-
tion 4, we introduce the alternating generators for U and establish some formulas involving
these generators. In Sections 5, 6 we use these formulas and generating functions to show
that the alternating generators for U satisfy the relations in [30, Propositions 5.7, 5.10].
Using this result, we prove that U = U+q . Theorem 6.2 and Corollary 6.5 are the main
results of the paper. In Section 7 we describe some features of U+q that are illuminated
by the presentation in Definition 3.1. Appendix A contains a list of relations involving the
generating functions from Section 5.
2 The algebra U+q
We now begin our formal argument. For the rest of the paper, the following notational
conventions are in effect. Recall the natural numbers N = {0, 1, 2, . . .}. Let F denote a
field. Every vector space and tensor product mentioned is over F. Every algebra mentioned
is associative, over F, and has a multiplicative identity. Fix a nonzero q ∈ F that is not a
root of unity. Recall the notation
[n]q =
qn − q−n
q − q−1
n ∈ N.
For elements X,Y in any algebra, define their commutator and q-commutator by
[X,Y ] = XY − Y X, [X,Y ]q = qXY − q−1Y X.
Note that
[X, [X, [X,Y ]q]q−1 ] = X
3Y − [3]qX2Y X + [3]qXYX2 − Y X3.
Definition 2.1 ([22, Corollary 3.2.6]). Define the algebra U+q by generators W0, W1 and
relations
[W0, [W0, [W0,W1]q]q−1 ] = 0, [W1, [W1, [W1,W0]q]q−1 ] = 0. (2.1)
We call U+q the positive part of Uq(ŝl2). The generators W0,W1 are called standard. The
relations (2.1) are called the q-Serre relations.
We will use the following concept.
Definition 2.2 ([16, p. 299]). Let A denote an algebra. A Poincaré-Birkhoff-Witt (or PBW)
basis for A consists of a subset Ω ⊆ A and a linear order < on Ω such that the following is
a basis for the vector space A:
a1a2 · · · an n ∈ N, a1, a2, . . . , an ∈ Ω, a1 ≤ a2 ≤ · · · ≤ an.
We interpret the empty product as the multiplicative identity in A.
In [16, p. 299] Damiani obtains a PBW basis for U+q that involves some elements
{Enδ+α0}∞n=0, {Enδ+α1}∞n=0, {Enδ}∞n=1. (2.2)
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 367
These elements are defined recursively as follows:
Eα0 = W0, Eα1 = W1, Eδ = q
−2W1W0 −W0W1 (2.3)
and for n ≥ 1,
Enδ+α0 =
[Eδ, E(n−1)δ+α0 ]
q + q−1
, Enδ+α1 =
[E(n−1)δ+α1 , Eδ]
q + q−1
, (2.4)
Enδ = q
−2E(n−1)δ+α1W0 −W0E(n−1)δ+α1 . (2.5)
Proposition 2.3 ([16, p. 308]). A PBW basis for U+q is obtained by the elements (2.2) in
the linear order
Eα0 < Eδ+α0 < E2δ+α0 < · · · < Eδ < E2δ < E3δ < · · · < E2δ+α1 < Eδ+α1 < Eα1 .
The elements (2.2) satisfy many relations [16]. We mention a few for later use.
Lemma 2.4 ([16, p. 300]). For i, j ∈ N with i > j the following hold in U+q .
(i) Assume that i− j = 2r + 1 is odd. Then
Eiδ+α0Ejδ+α0 = q
−2Ejδ+α0Eiδ+α0
− (q2 − q−2)
r∑
ℓ=1
q−2ℓE(j+ℓ)δ+α0E(i−ℓ)δ+α0 ,
Ejδ+α1Eiδ+α1 = q
−2Eiδ+α1Ejδ+α1
− (q2 − q−2)
r∑
ℓ=1
q−2ℓE(i−ℓ)δ+α1E(j+ℓ)δ+α1 .
(ii) Assume that i− j = 2r is even. Then
Eiδ+α0Ejδ+α0 = q
−2Ejδ+α0Eiδ+α0 − qj−i+1(q − q−1)E2(r+j)δ+α0
− (q2 − q−2)
r−1∑
ℓ=1
q−2ℓE(j+ℓ)δ+α0E(i−ℓ)δ+α0 ,
Ejδ+α1Eiδ+α1 = q
−2Eiδ+α1Ejδ+α1 − qj−i+1(q − q−1)E2(r+j)δ+α1
− (q2 − q−2)
r−1∑
ℓ=1
q−2ℓE(i−ℓ)δ+α1E(j+ℓ)δ+α1 .
Lemma 2.5. The following (i) – (iii) hold in U+q .
(i) (See [16, p. 307].) For positive i, j ∈ N,
EiδEjδ = EjδEiδ. (2.6)
(ii) (See [16, p. 307].) For i, j ∈ N,
[Eiδ+α0 , Ejδ+α1 ]q = −qE(i+j+1)δ. (2.7)
368 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
(iii) For i ∈ N,
[W0, Eiδ+α0 ]q
q − q−1
=
i∑
ℓ=0
Eℓδ+α0E(i−ℓ)δ+α0 , (2.8)
[Eiδ+α1 ,W1]q
q − q−1
=
i∑
ℓ=0
E(i−ℓ)δ+α1Eℓδ+α1 . (2.9)
Proof. (iii) To verify (2.8) and (2.9), use Lemma 2.4 to write each term in the PBW basis
for U+q from Proposition 2.3. We give the details for (2.8). Referring to (2.8), let ∆ denote
the right-hand side minus the left-hand side. We show that ∆ = 0. This is quickly verified
for i = 0, so assume that i ≥ 1. For i even (resp. i odd) write i = 2r (resp. i = 2r + 1).
Using Lemma 2.4 we obtain ∆ =
∑r
ℓ=0 αℓEℓδ+α0E(i−ℓ)δ+α0 , where for i even,
α0 = 1 + q
−2 − q
q − q−1
+
q−3
q − q−1
,
αℓ = 1 + q
−2 − (q2 − q−2)
ℓ∑
k=1
q−2k − (q + q−1)q−2ℓ−1 (1 ≤ ℓ ≤ r − 1),
αr = 1− (q − q−1)
r∑
k=1
q1−2k − q−i
and for i odd,
α0 = 1 + q
−2 − q
q − q−1
+
q−3
q − q−1
,
αℓ = 1 + q
−2 − (q2 − q−2)
ℓ∑
k=1
q−2k − (q + q−1)q−2ℓ−1 (1 ≤ ℓ ≤ r).
For either case αℓ = 0 for 0 ≤ ℓ ≤ r, so ∆ = 0. We have verified (2.8). For (2.9) the
details are similar, and omitted.
3 An extension of U+q
In this section we introduce the algebra U . In Section 6 we will show that U coincides with
the alternating central extension U+q of U+q .
Definition 3.1. Define the algebra U by generators W0, W1, {G̃k+1}k∈N and relations
(i) [W0, [W0, [W0,W1]q]q−1 ] = 0,
(ii) [W1, [W1, [W1,W0]q]q−1 ] = 0,
(iii) [G̃1,W1] = q
[[W0,W1]q,W1]
q2−q−2 ,
(iv) [W0, G̃1] = q
[W0,[W0,W1]q ]
q2−q−2 ,
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 369
(v) for k ≥ 1,
[G̃k+1,W1] =
[[[G̃k,W0]q,W1]q,W1]
(1− q−2)(q2 − q−2)
,
(vi) for k ≥ 1,
[W0, G̃k+1] =
[W0, [W0, [W1, G̃k]q]q]
(1− q−2)(q2 − q−2)
,
(vii) for k, ℓ ∈ N,
[G̃k+1, G̃ℓ+1] = 0.
For notational convenience define G̃0 = 1.
Note 3.2. Referring to Definition 3.1, the relation (iii) (resp. (iv)) is obtained from (v)
(resp. (vi)) by setting k = 0.
Lemma 3.3. There exists a unique algebra homomorphism ♭ : U+q → U that sends W0 7→
W0 and W1 7→ W1.
Proof. Compare Definitions 2.1, 3.1.
In Corollary 6.7 we will show that ♭ is injective. Let ⟨W0,W1⟩ denote the subalgebra
of U generated by W0,W1. Of course ⟨W0,W1⟩ is the ♭-image of U+q . For the elements
(2.2) of U+q , the same notation will be used for their ♭-images in ⟨W0,W1⟩.
4 Augmenting the generating set for U
Some of the relations in Definition 3.1 are nonlinear. Our next goal is to linearize the
relations by adding more generators.
Definition 4.1. We define some elements in U as follows. For k ∈ N,
W−k =
[G̃k,W0]q
q − q−1
, (4.1)
Wk+1 =
[W1, G̃k]q
q − q−1
, (4.2)
Gk+1 = G̃k+1 +
[W1,W−k]
1− q−2
. (4.3)
For notational convenience define G0 = 1.
Lemma 4.2. For k ∈ N the following hold in U:
G̃kW0 = q
−2W0G̃k + (1− q−2)W−k,
G̃kW1 = q
2W1G̃k + (1− q2)Wk+1.
Proof. These are reformulations of (4.1) and (4.2).
370 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
The following is a generating set for U :
{W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G̃k+1}k∈N. (4.4)
The elements of this set will be called alternating. We seek a presentation of U , that
has the above generating set and all relations linear. We will obtain this presentation in
Theorem 6.2.
Next we obtain some formulas that will help us prove Theorem 6.2. We will show that
for n ∈ N,
Wn+1 =
n∑
k=0
Ekδ+α1G̃n−k(−1)kqk
(q − q−1)2k
, (4.5)
W−n =
n∑
k=0
Ekδ+α0G̃n−k(−1)kq3k
(q − q−1)2k
. (4.6)
We will prove (4.5), (4.6) by induction on n. Note that (4.5), (4.6) hold for n = 0, since
W1 = Eα1 and W0 = Eα0 . We will give the main induction argument after a few lemmas.
For the rest of this section k and ℓ are understood to be in N.
Lemma 4.3. Pick n ∈ N, and assume that (4.5), (4.6) hold for n, n− 1, . . . , 1, 0. Then
[W0,Wn+1] = [W−n,W1]. (4.7)
Proof. The commutator [W0,Wn+1] is equal to
W0Wn+1 −Wn+1W0
=
n∑
k=0
W0Ekδ+α1G̃n−k(−1)kqk
(q − q−1)2k
−
n∑
k=0
Ekδ+α1G̃n−kW0(−1)kqk
(q − q−1)2k
=
n∑
k=0
W0Ekδ+α1G̃n−k(−1)kqk
(q − q−1)2k
−
n∑
k=0
Ekδ+α1
(
q−2W0G̃n−k + (1− q−2)Wk−n
)
(−1)kqk
(q − q−1)2k
=
n∑
k=0
(
W0Ekδ+α1 − q−2Ekδ+α1W0
)
G̃n−k(−1)kqk
(q − q−1)2k
−
n∑
k=0
Ekδ+α1Wk−n(−1)kqk−1
(q − q−1)2k−1
= −
n∑
k=0
E(k+1)δG̃n−k(−1)kqk
(q − q−1)2k
−
n∑
k=0
Ekδ+α1Wk−n(−1)kqk−1
(q − q−1)2k−1
= −
n∑
k=0
E(k+1)δG̃n−k(−1)kqk
(q − q−1)2k
−
n∑
k=0
Ekδ+α1(−1)kqk−1
(q − q−1)2k−1
n−k∑
ℓ=0
Eℓδ+α0G̃n−k−ℓ(−1)ℓq3ℓ
(q − q−1)2ℓ
= −
n∑
p=0
E(p+1)δG̃n−p(−1)pqp
(q − q−1)2p
−
n∑
p=0
( ∑
k+ℓ=p
q2ℓEkδ+α1Eℓδ+α0
)
G̃n−p(−1)pqp−1
(q − q−1)2p−1
.
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 371
The commutator [W−n,W1] is equal to
W−nW1 −W1W−n
=
n∑
k=0
Ekδ+α0G̃n−kW1(−1)kq3k
(q − q−1)2k
−
n∑
k=0
W1Ekδ+α0G̃n−k(−1)kq3k
(q − q−1)2k
=
n∑
k=0
Ekδ+α0
(
q2W1G̃n−k + (1− q2)Wn−k+1
)
(−1)kq3k
(q − q−1)2k
−
n∑
k=0
W1Ekδ+α0G̃n−k(−1)kq3k
(q − q−1)2k
=
n∑
k=0
(
q2Ekδ+α0W1 −W1Ekδ+α0
)
G̃n−k(−1)kq3k
(q − q−1)2k
−
n∑
k=0
Ekδ+α0Wn−k+1(−1)kq3k+1
(q − q−1)2k−1
= −
n∑
k=0
E(k+1)δG̃n−k(−1)kq3k+2
(q − q−1)2k
−
n∑
k=0
Ekδ+α0Wn−k+1(−1)kq3k+1
(q − q−1)2k−1
= −
n∑
k=0
E(k+1)δG̃n−k(−1)kq3k+2
(q − q−1)2k
−
n∑
k=0
Ekδ+α0(−1)kq3k+1
(q − q−1)2k−1
n−k∑
ℓ=0
Eℓδ+α1G̃n−k−ℓ(−1)ℓqℓ
(q − q−1)2ℓ
= −
n∑
p=0
E(p+1)δG̃n−p(−1)pq3p+2
(q − q−1)2p
−
n∑
p=0
( ∑
k+ℓ=p
q2kEkδ+α0Eℓδ+α1
)
G̃n−p(−1)pqp+1
(q − q−1)2p−1
.
By these comments
[W−n,W1]− [W0,Wn+1] =
n∑
p=0
CpG̃n−p(−1)pqp
(q − q−1)2p
,
where for 0 ≤ p ≤ n,
Cp = E(p+1)δ + q
−1(q − q−1)
∑
k+ℓ=p
q2ℓEkδ+α1Eℓδ+α0
− q2p+2E(p+1)δ − q(q − q−1)
∑
k+ℓ=p
q2kEkδ+α0Eℓδ+α1
= (1− q2p+2)E(p+1)δ − (1− q2)
∑
k+ℓ=p
q2ℓ
(
q−2Ekδ+α1Eℓδ+α0 − Eℓδ+α0Ekδ+α1
)
= (1− q2p+2)E(p+1)δ − (1− q2)
∑
k+ℓ=p
q2ℓE(p+1)δ
372 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
=
(
1− q2p+2 − (1− q2)
p∑
ℓ=0
q2ℓ
)
E(p+1)δ
= 0.
The result follows.
Lemma 4.4. Pick n ∈ N, and assume that (4.5), (4.6) hold for n, n− 1, . . . , 1, 0. Then
[G̃n, Eδ] = 0. (4.8)
Proof. Using Lemma 4.3,
0 = (q − q−1)
(
[W−n,W1]− [W0,Wn+1]
)
= [[G̃n,W0]q,W1]− [W0, [W1, G̃n]q]
= [G̃n, [W0,W1]q]
= −q[G̃n, Eδ].
Lemma 4.5. Pick n ∈ N, and assume that (4.5), (4.6) hold for n, n− 1, . . . , 1, 0. Then
[W−n,W0] = 0. (4.9)
Proof. The commutator [W−n,W0] is equal to
W−nW0 −W0W−n
=
n∑
k=0
Ekδ+α0G̃n−kW0(−1)kq3k
(q − q−1)2k
−
n∑
k=0
W0Ekδ+α0G̃n−k(−1)kq3k
(q − q−1)2k
=
n∑
k=0
Ekδ+α0
(
q−2W0G̃n−k + (1− q−2)Wk−n
)
(−1)kq3k
(q − q−1)2k
−
n∑
k=0
W0Ekδ+α0G̃n−k(−1)kq3k
(q − q−1)2k
=
n∑
k=0
[W0, Ekδ+α0 ]qG̃n−k(−1)k−1q3k−1
(q − q−1)2k
+
n∑
k=0
Ekδ+α0Wk−n(−1)kq3k−1
(q − q−1)2k−1
=
n∑
k=0
[W0, Ekδ+α0 ]qG̃n−k(−1)k−1q3k−1
(q − q−1)2k
+
n∑
k=0
Ekδ+α0(−1)kq3k−1
(q − q−1)2k−1
n−k∑
ℓ=0
Eℓδ+α0G̃n−k−ℓ(−1)ℓq3ℓ
(q − q−1)2ℓ
=
n∑
p=0
[W0, Epδ+α0 ]qG̃n−p(−1)p−1q3p−1
(q − q−1)2p
+
n∑
p=0
( ∑
k+ℓ=p
Ekδ+α0Eℓδ+α0
)
G̃n−p(−1)pq3p−1
(q − q−1)2p−1
.
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 373
By these comments
[W−n,W0] =
n∑
p=0
SpG̃n−p(−1)p−1q3p−1
(q − q−1)2p−1
where
Sp =
[W0, Epδ+α0 ]q
q − q−1
−
∑
k+ℓ=p
Ekδ+α0Eℓδ+α0 (0 ≤ p ≤ n).
By (2.8) we have Sp = 0 for 0 ≤ p ≤ n. The result follows.
Lemma 4.6. Pick n ∈ N, and assume that (4.5), (4.6) hold for n, n− 1, . . . , 1, 0. Then
[Wn+1,W1] = 0. (4.10)
Proof. The commutator [Wn+1,W1] is equal to
Wn+1W1 −W1Wn+1
=
n∑
k=0
Ekδ+α1G̃n−kW1(−1)kqk
(q − q−1)2k
−
n∑
k=0
W1Ekδ+α1G̃n−k(−1)kqk
(q − q−1)2k
=
n∑
k=0
Ekδ+α1
(
q2W1G̃n−k + (1− q2)Wn−k+1
)
(−1)kqk
(q − q−1)2k
−
n∑
k=0
W1Ekδ+α1G̃n−k(−1)kqk
(q − q−1)2k
=
n∑
k=0
[Ekδ+α1 ,W1]qG̃n−k(−1)kqk+1
(q − q−1)2k
−
n∑
k=0
Ekδ+α1Wn−k+1(−1)kqk+1
(q − q−1)2k−1
=
n∑
k=0
[Ekδ+α1 ,W1]qG̃n−k(−1)kqk+1
(q − q−1)2k
−
n∑
k=0
Ekδ+α1(−1)kqk+1
(q − q−1)2k−1
n−k∑
ℓ=0
Eℓδ+α1G̃n−k−ℓ(−1)ℓqℓ
(q − q−1)2ℓ
=
n∑
p=0
[Epδ+α1 ,W1]qG̃n−p(−1)pqp+1
(q − q−1)2p
−
n∑
p=0
( ∑
k+ℓ=p
Ekδ+α1Eℓδ+α1
)
G̃n−p(−1)pqp+1
(q − q−1)2p−1
.
By these comments
[Wn+1,W1] =
n∑
p=0
TpG̃n−p(−1)pqp+1
(q − q−1)2p−1
374 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
where
Tp =
[Epδ+α1 ,W1]q
q − q−1
−
∑
k+ℓ=p
Ekδ+α1Eℓδ+α1 (0 ≤ p ≤ n).
By (2.9) we have Tp = 0 for 0 ≤ p ≤ n. The result follows.
Proposition 4.7. The equations (4.5), (4.6) hold in U for n ∈ N.
Proof. The proof is by induction on n. We assume that (4.5), (4.6) hold for
n, n− 1, . . . , 1, 0, and show that (4.5), (4.6) hold for n+ 1. Concerning (4.5),
Wn+2 =
qW1G̃n+1 − q−1G̃n+1W1
q − q−1
by (4.2)
= W1G̃n+1 − q−1
[G̃n+1,W1]
q − q−1
= W1G̃n+1 −
[[[G̃n,W0]q,W1]q,W1]
(q − q−1)2(q2 − q−2)
by Definition 3.1(v)
= W1G̃n+1 −
[[W−n,W1]q,W1]
(q − q−1)(q2 − q−2)
by (4.1)
= W1G̃n+1 −
[[W−n,W1],W1]q
(q − q−1)(q2 − q−2)
= W1G̃n+1 −
[[W0,Wn+1],W1]q
(q − q−1)(q2 − q−2)
by Lemma 4.3
= W1G̃n+1 −
[[W0,W1]q,Wn+1]
(q − q−1)(q2 − q−2)
by Lemma 4.6
= W1G̃n+1 +
q[Eδ,Wn+1]
(q − q−1)(q2 − q−2)
by (2.3)
= W1G̃n+1 + q
n∑
k=0
[Eδ, Ekδ+α1G̃n−k](−1)kqk
(q − q−1)2k+1(q2 − q−2)
by (4.5) and induction
= W1G̃n+1 + q
n∑
k=0
[Eδ, Ekδ+α1 ]G̃n−k(−1)kqk
(q − q−1)2k+1(q2 − q−2)
by Lemma 4.4
= W1G̃n+1 +
n∑
k=0
E(k+1)δ+α1G̃n−k(−1)k+1qk+1
(q − q−1)2k+2
by (2.4)
= Eα1G̃n+1 +
n+1∑
k=1
Ekδ+α1G̃n+1−k(−1)kqk
(q − q−1)2k
=
n+1∑
k=0
Ekδ+α1G̃n+1−k(−1)kqk
(q − q−1)2k
.
We have shown that (4.5) holds for n+ 1. Concerning (4.6),
W−n−1 =
qG̃n+1W0 − q−1W0G̃n+1
q − q−1
by (4.1)
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 375
= W0G̃n+1 − q
[W0, G̃n+1]
q − q−1
= W0G̃n+1 − q2
[W0, [W0, [W1, G̃n]q]q]
(q − q−1)2(q2 − q−2)
by Definition 3.1(vi)
= W0G̃n+1 − q2
[W0, [W0,Wn+1]q]
(q − q−1)(q2 − q−2)
by (4.2)
= W0G̃n+1 − q2
[W0, [W0,Wn+1]]q
(q − q−1)(q2 − q−2)
= W0G̃n+1 − q2
[W0, [W−n,W1]]q
(q − q−1)(q2 − q−2)
by Lemma 4.3
= W0G̃n+1 − q2
[W−n, [W0,W1]q]
(q − q−1)(q2 − q−2)
by Lemma 4.5
= W0G̃n+1 + q
3 [W−n, Eδ]
(q − q−1)(q2 − q−2)
by (2.3)
= W0G̃n+1 + q
3
n∑
k=0
[Ekδ+α0G̃n−k, Eδ](−1)kq3k
(q − q−1)2k+1(q2 − q−2)
by (4.6) and induction
= W0G̃n+1 + q
3
n∑
k=0
[Ekδ+α0 , Eδ]G̃n−k(−1)kq3k
(q − q−1)2k+1(q2 − q−2)
by Lemma 4.4
= W0G̃n+1 +
n∑
k=0
E(k+1)δ+α0G̃n−k(−1)k+1q3k+3
(q − q−1)2k+2
by (2.4)
= Eα0G̃n+1 +
n+1∑
k=1
Ekδ+α0G̃n+1−k(−1)kq3k
(q − q−1)2k
=
n+1∑
k=0
Ekδ+α0G̃n+1−k(−1)kq3k
(q − q−1)2k
.
We have shown that (4.6) holds for n+ 1.
Lemma 4.8. The following relations hold in U . For n ∈ N,
[W0,Wn+1] = [W−n,W1], [G̃n, Eδ] = 0,
[W−n,W0] = 0, [Wn+1,W1] = 0.
Proof. By Lemmas 4.3 – 4.6 and Proposition 4.7.
Lemma 4.9. The following relations hold in U . For k ∈ N,
(i) [Gk+1,W1]q = [W1, G̃k+1]q;
(ii) [W0, Gk+1]q = [G̃k+1,W0]q .
Proof. (i) We have
[Gk+1,W1]q − [W1, G̃k+1]q =
[
G̃k+1 +
[W1,W−k]
1− q−2
,W1
]
q
− [W1, G̃k+1]q
376 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
= (q + q−1)[G̃k+1,W1]−
[[W−k,W1],W1]q
1− q−2
= (q + q−1)[G̃k+1,W1]−
[[W−k,W1]q,W1]
1− q−2
= (q + q−1)[G̃k+1,W1]−
[[[G̃k,W0]q,W1]q,W1]
(1− q−2)(q − q−1)
= 0.
(ii) We have
[W0, Gk+1]q − [G̃k+1,W0]q =
[
W0, G̃k+1 +
[Wk+1,W0]
1− q−2
]
q
− [G̃k+1,W0]q
= (q + q−1)[W0, G̃k+1]−
[W0, [W0,Wk+1]]q
1− q−2
= (q + q−1)[W0, G̃k+1]−
[W0, [W0,Wk+1]q]
1− q−2
= (q + q−1)[W0, G̃k+1]−
[W0, [W0, [W1, G̃k]q]q]
(1− q−2)(q − q−1)
= 0.
5 Generating functions
The alternating generators of U are displayed in (4.4). In the previous section we described
how these generators are related to W0 and W1. Our next goal is to describe how the
alternating generators are related to each other. It is convenient to use generating functions.
Definition 5.1. We define some generating functions in an indeterminate t. Referring to
(4.4),
G(t) =
∑
n∈N
Gnt
n, G̃(t) =
∑
n∈N
G̃nt
n,
W−(t) =
∑
n∈N
W−nt
n, W+(t) =
∑
n∈N
Wn+1t
n.
Lemma 5.2. For the algebra U ,
[W0, G(t)]q
q − q−1
= W−(t),
[G̃(t),W0]q
q − q−1
= W−(t),
[W0,W
−(t)] = 0,
[W0,W
+(t)]
1− q−2
= t−1(G̃(t)−G(t))
and
[G(t),W1]q
q − q−1
= W+(t),
[W1, G̃(t)]q
q − q−1
= W+(t),
[W1,W
+(t)] = 0,
[W1,W
−(t)]
1− q−2
= t−1(G(t)− G̃(t)).
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 377
Proof. Use Definition 4.1 and Lemmas 4.8, 4.9.
For the rest of this section, let s denote an indeterminate that commutes with t.
Lemma 5.3. For the algebra U ,
[W−(s),W−(t)] = 0, [W+(s),W+(t)] = 0,
[W−(s),W+(t)] + [W+(s),W−(t)] = 0,
s[W−(s), G(t)] + t[G(s),W−(t)] = 0,
s[W−(s), G̃(t)] + t[G̃(s),W−(t)] = 0,
s[W+(s), G(t)] + t[G(s),W+(t)] = 0,
s[W+(s), G̃(t)] + t[G̃(s),W+(t)] = 0,
[G(s), G(t)] = 0, [G̃(s), G̃(t)] = 0,
[G̃(s), G(t)] + [G(s), G̃(t)] = 0
and also
[W−(s), G(t)]q = [W
−(t), G(s)]q, [G(s),W
+(t)]q = [G(t),W
+(s)]q,
[G̃(s),W−(t)]q = [G̃(t),W
−(s)]q, [W
+(s), G̃(t)]q = [W
+(t), G̃(s)]q,
t−1[G(s), G̃(t)]− s−1[G(t), G̃(s)] = q[W−(t),W+(s)]q − q[W−(s),W+(t)]q,
t−1[G̃(s), G(t)]− s−1[G̃(t), G(s)] = q[W+(t),W−(s)]q − q[W+(s),W−(t)]q,
[G(s), G̃(t)]q − [G(t), G̃(s)]q = qt[W−(t),W+(s)]− qs[W−(s),W+(t)],
[G̃(s), G(t)]q − [G̃(t), G(s)]q = qt[W+(t),W−(s)]− qs[W+(s),W−(t)].
Proof. We refer to the generating functions A(s, t), B(s, t), . . . , S(s, t) from Appendix A.
The present lemma asserts that for the algebra U these generating functions are all zero.
To verify this assertion, we refer to the canonical relations in Appendix A. We will use
induction with respect to the linear order
I(s, t),M(s, t), N(s, t), A(s, t), B(s, t), Q(s, t), D(s, t), E(s, t), F (s, t),
G(s, t), R(s, t), S(s, t), H(s, t),K(s, t), L(s, t), P (s, t), C(s, t), J(s, t).
For each element in this linear order besides I(s, t), there exists a canonical relation that
expresses the given element in terms of the previous elements in the linear order. So in
U the given element is zero, provided that in U every previous element is zero. Note that
in U we have I(s, t) = 0 by Definition 3.1(vii). By these comments and induction, in
U every element in the linear order is zero. We have shown that in U each of A(s, t),
B(s, t), . . . , S(s, t) is zero.
6 The main results
In this section we present our main results, which are Theorem 6.2 and Corollary 6.5.
Recall the alternating generators (4.4) for U .
378 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
Lemma 6.1. The following relations hold in U . For k, ℓ ∈ N we have
[W0,Wk+1] = [W−k,W1] = (1− q−2)(G̃k+1 −Gk+1), (6.1)
[W0, Gk+1]q = [G̃k+1,W0]q = (q − q−1)W−k−1, (6.2)
[Gk+1,W1]q = [W1, G̃k+1]q = (q − q−1)Wk+2, (6.3)
[W−k,W−ℓ] = 0, [Wk+1,Wℓ+1] = 0, (6.4)
[W−k,Wℓ+1] + [Wk+1,W−ℓ] = 0, (6.5)
[W−k, Gℓ+1] + [Gk+1,W−ℓ] = 0, (6.6)
[W−k, G̃ℓ+1] + [G̃k+1,W−ℓ] = 0, (6.7)
[Wk+1, Gℓ+1] + [Gk+1,Wℓ+1] = 0, (6.8)
[Wk+1, G̃ℓ+1] + [G̃k+1,Wℓ+1] = 0, (6.9)
[Gk+1, Gℓ+1] = 0, [G̃k+1, G̃ℓ+1] = 0, (6.10)
[G̃k+1, Gℓ+1] + [Gk+1, G̃ℓ+1] = 0 (6.11)
and also
[W−k, Gℓ]q = [W−ℓ, Gk]q, [Gk,Wℓ+1]q = [Gℓ,Wk+1]q, (6.12)
[G̃k,W−ℓ]q = [G̃ℓ,W−k]q, [Wℓ+1, G̃k]q = [Wk+1, G̃ℓ]q, (6.13)
[Gk, G̃ℓ+1]− [Gℓ, G̃k+1] = q[W−ℓ,Wk+1]q − q[W−k,Wℓ+1]q, (6.14)
[G̃k, Gℓ+1]− [G̃ℓ, Gk+1] = q[Wℓ+1,W−k]q − q[Wk+1,W−ℓ]q, (6.15)
[Gk+1, G̃ℓ+1]q − [Gℓ+1, G̃k+1]q = q[W−ℓ,Wk+2]− q[W−k,Wℓ+2], (6.16)
[G̃k+1, Gℓ+1]q − [G̃ℓ+1, Gk+1]q = q[Wℓ+1,W−k−1]− q[Wk+1,W−ℓ−1]. (6.17)
Proof. The relations (6.1) – (6.3) are from Definition 4.1 and Lemmas 4.8, 4.9. The rela-
tions (6.4) – (6.17) follow from Definition 5.1 and Lemma 5.3.
Theorem 6.2. The algebra U has a presentation by generators
{W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G̃k+1}k∈N
and the relations in Lemma 6.1.
Proof. It suffices to show that the relations in Definition 3.1 are implied by the relations in
Lemma 6.1. The relation (iii) in Definition 3.1 is obtained from the equation on the left in
(6.3) at k = 0, by eliminating G1 using [W0,W1] = (1−q−2)(G̃1−G1). The relation (iv)
in Definition 3.1 is obtained from the equation on the left in (6.2) at k = 0, by eliminating
G1 using [W0,W1] = (1 − q−2)(G̃1 − G1). For k ≥ 1 the relation (v) in Definition 3.1
is obtained from the equation on the left in (6.3), by eliminating Gk+1 using [W−k,W1] =
(1 − q−2)(G̃k+1 − Gk+1) and evaluating the result using [G̃k,W0]q = (q − q−1)W−k.
For k ≥ 1 the relation (vi) in Definition 3.1 is obtained from the equation on the left in
(6.2), by eliminating Gk+1 using [W0,Wk+1] = (1− q−2)(G̃k+1 −Gk+1) and evaluating
the result using [W1, G̃k]q = (q − q−1)Wk+1. The relation (vii) in Definition 3.1 is from
(6.10). The relation (i) in Definition 3.1 is obtained from [W0,W−1] = 0, by eliminating
W−1 using [G̃1,W0]q = (q − q−1)W−1 and evaluating the result using Definition 3.1(iv).
The relation (ii) in Definition 3.1 is obtained from [W1,W2] = 0, by eliminating W2 using
[W1, G̃1]q = (q − q−1)W2 and evaluating the result using Definition 3.1(iii).
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 379
It is apparent from the proof of Theorem 6.2 that the relations in Lemma 6.1 are redundant
in the following sense.
Corollary 6.3. The relations in Lemma 6.1 are implied by the relations listed in (i) – (iii)
below:
(i) (6.1) – (6.3);
(ii) (6.4) with k = 0 and ℓ = 1;
(iii) the relations on the right in (6.10).
Proof. By Lemma 6.1 the relations (6.1) – (6.17) are implied by the relations in Defini-
tions 3.1, 4.1. The relations listed in (i) – (iii) are used in the proof of Theorem 6.2 to
obtain the relations in Definition 3.1. The relations listed in (i) imply the relations in Defi-
nition 4.1. The result follows.
The relations in Lemma 6.1 first appeared in [30, Propositions 5.7, 5.10, 5.11]. It was
observed in [2, Propositions 3.1, 3.2] and [5, Remark 2.5] that the relations (6.1) – (6.11)
imply the relations (6.12) – (6.17). This observation motivated the following definition.
Definition 6.4 ([29, Definition 3.1]). Define the algebra U+q by generators
{W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G̃k+1}k∈N
and the relations (6.1) – (6.11). The algebra U+q is called the alternating central extension
of U+q .
Corollary 6.5. We have U = U+q .
Proof. By Theorem 6.2, Corollary 6.3, and Definition 6.4.
Definition 6.6. By the compact presentation of U+q we mean the presentation given in
Definition 3.1. By the expanded presentation of U+q we mean the presentation given in
Theorem 6.2.
Corollary 6.7. The map ♭ from Lemma 3.3 is injective.
Proof. By Corollary 6.5 and [29, Proposition 6.4].
7 The subalgebra of U+q generated by {G̃k+1}k∈N
Let G̃ denote the subalgebra of U+q generated by {G̃k+1}k∈N. In this section we describe
G̃ and its relationship to ⟨W0,W1⟩.
The following notation will be useful. Let z1, z2, . . . denote mutually commuting inde-
terminates. Let F[z1, z2, . . .] denote the algebra consisting of the polynomials in z1, z2, . . .
that have all coefficients in F. For notational convenience define z0 = 1.
Lemma 7.1 ([29, Lemma 3.5]). There exists an algebra homomorphism U+q →
F[z1, z2, . . .] that sends
W−n 7→ 0, Wn+1 7→ 0, Gn 7→ zn, G̃n 7→ zn
for n ∈ N.
380 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
Proof. By Theorem 6.2 and the nature of the relations in Lemma 6.1.
Corollary 7.2 ([29, Theorem 10.2]). The generators {G̃k+1}k∈N of G̃ are algebraically
independent.
Proof. By Lemma 7.1 and since {zk+1}k∈N are algebraically independent.
The following result will help us describe how G̃ is related to ⟨W0,W1⟩.
Lemma 7.3. For n ∈ N,
G̃nW1 = W1G̃n +
n∑
k=1
Ekδ+α1G̃n−k(−1)k+1qk+1
(q − q−1)2k−1
, (7.1)
G̃nW0 = W0G̃n +
n∑
k=1
Ekδ+α0G̃n−k(−1)kq3k−1
(q − q−1)2k−1
. (7.2)
Proof. To obtain (7.1), eliminate Wn+1 from (4.5) using (4.2), and solve the resulting
equation for G̃nW1. To obtain (7.2), eliminate W−n from (4.6) using (4.1), and solve the
resulting equation for G̃nW0.
Shortly we will describe how G̃ is related to ⟨W0,W1⟩. This description involves the
center Z of U+q . To prepare for this description, we have some comments about Z . In
[29, Sections 5, 6] we introduced some algebraically independent elements Z1, Z2, . . . that
generate the algebra Z . For notational convenience define Z0 = 1. Using {Zn}n∈N we
obtain a basis for Z that is described as follows. For n ∈ N, a partition of n is a sequence
λ = {λi}∞i=1 of natural numbers such that λi ≥ λi+1 for i ≥ 1 and n =
∑∞
i=1 λi. The set
Λn consists of the partitions of n. Define Λ = ∪n∈NΛn. For λ ∈ Λ define Zλ =
∏∞
i=1 Zλi .
The elements {Zλ}λ∈Λ form a basis for the vector space Z . Next we describe a grading
for Z . For n ∈ N let Zn denote the subspace of Z with basis {Zλ}λ∈Λn . For example
Z0 = F1. The sum Z =
∑
n∈N Zn is direct. Moreover ZrZs ⊆ Zr+s for r, s ∈ N.
By these comments the subspaces {Zn}n∈N form a grading of Z . Note that Zn ∈ Zn for
n ∈ N. Next we describe how Z is related to ⟨W0,W1⟩.
Lemma 7.4 ([29, Proposition 6.5]). The multiplication map
⟨W0,W1⟩ ⊗ Z → U+q
w ⊗ z 7→ wz
is an algebra isomorphism.
For n ∈ N let Un denote the image of ⟨W0,W1⟩ ⊗ Zn under the multiplication map. By
construction the sum U+q =
∑
n∈N Un is direct.
In the next two lemmas we describe how G̃ is related to Z .
Lemma 7.5 ([29, Lemmas 3.6, 5.9]). For n ∈ N,
G̃n ∈
n∑
k=0
⟨W0,W1⟩Zk, G̃n − Zn ∈
n−1∑
k=0
⟨W0,W1⟩Zk.
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 381
For λ ∈ Λ define G̃λ =
∏∞
i=1 G̃λi . By Corollary 7.2 the elements {G̃λ}λ∈Λ form a basis
for the vector space G̃.
Lemma 7.6. For n ∈ N and λ ∈ Λn,
G̃λ ∈
n∑
k=0
Uk, G̃λ − Zλ ∈
n−1∑
k=0
Uk.
Proof. By Lemma 7.5 and our comments above Lemma 7.4 about the grading of Z .
Next we describe how G̃ is related to ⟨W0,W1⟩.
Proposition 7.7. The multiplication map
⟨W0,W1⟩ ⊗ G̃ → U+q
w ⊗ g 7→ wg
is an isomorphism of vector spaces.
Proof. The multiplication map is F-linear. The multiplication map is surjective by
Lemma 7.3 and since U+q is generated by W0, W1, G̃. We now show that the multiplicaton
map is injective. Consider a vector v ∈ ⟨W0,W1⟩ ⊗ G̃ that is sent to zero by the multi-
plication map. We show that v = 0. Write v =
∑
λ∈Λ aλ ⊗ G̃λ, where aλ ∈ ⟨W0,W1⟩
for λ ∈ Λ and aλ = 0 for all but finitely many λ ∈ Λ. To show that v = 0, we must
show that aλ = 0 for all λ ∈ Λ. Suppose that there exists λ ∈ Λ such that aλ ̸= 0. Let
C denote the set of natural numbers m such that Λm contains a partition λ with aλ ̸= 0.
The set C is nonempty and finite. Let n denote the maximal element of C. By construction∑
λ∈Λn aλ ⊗ Zλ is nonzero. By Lemma 7.4,∑
λ∈Λn
aλZλ ̸= 0. (7.3)
By construction
0 =
∑
λ∈Λ
aλG̃λ =
n∑
k=0
∑
λ∈Λk
aλG̃λ =
∑
λ∈Λn
aλG̃λ +
n−1∑
k=0
∑
λ∈Λk
aλG̃λ. (7.4)
Using (7.4),
∑
λ∈Λn
aλZλ =
∑
λ∈Λn
aλ(Zλ − G̃λ)−
n−1∑
k=0
∑
λ∈Λk
aλG̃λ. (7.5)
The left-hand side of (7.5) is contained in Un. By Lemma 7.6 the right-hand side of (7.5)
is contained in
∑n−1
k=0 Uk. The subspaces Un and
∑n−1
k=0 Uk have zero intersection because
the sum
∑n
k=0 Uk is direct. This contradicts (7.3), so aλ = 0 for λ ∈ Λ. Consequently
v = 0, as desired. We have shown that the multiplication map is injective. By the above
comments the multiplication map is an isomorphism of vector spaces.
382 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
ORCID iDs
Paul M. Terwilliger https://orcid.org/0000-0003-0942-5489
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384 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
Appendix A
Recall the algebra U from Definition 3.1. In this appendix we list some relations that hold
in U . We will define an algebra U∨ that is a homomorphic preimage of U . All the results
in this appendix are about U∨.
Define the algebra U∨ by generators
{W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G̃k+1}k∈N
and the following relations. For k ∈ N,
[W0,Wk+1] = [W−k,W1] = (1− q−2)(G̃k+1 −Gk+1), (A.1)
[W0, Gk+1]q = [G̃k+1,W0]q = (q − q−1)W−k−1, (A.2)
[Gk+1,W1]q = [W1, G̃k+1]q = (q − q−1)Wk+2, (A.3)
[W0,W−k] = 0, [W1,Wk+1] = 0. (A.4)
For notational convenience, define G0 = 1 and G̃0 = 1.
For U∨ we define the generating functions W−(t), W+(t), G(t), G̃(t) as in Defini-
tion 5.1. In terms of these generating functions, the relations (A.1) – (A.4) become the
relations in Lemma 5.2. Let s denote an indeterminate that commutes with t. Define
A(s, t) = [W−(s),W−(t)],
B(s, t) = [W+(s),W+(t)],
C(s, t) = [W−(s),W+(t)] + [W+(s),W−(t)],
D(s, t) = s[W−(s), G(t)] + t[G(s),W−(t)],
E(s, t) = s[W−(s), G̃(t)] + t[G̃(s),W−(t)],
F (s, t) = s[W+(s), G(t)] + t[G(s),W+(t)],
G(s, t) = s[W+(s), G̃(t)] + t[G̃(s),W+(t)],
H(s, t) = [G(s), G(t)],
I(s, t) = [G̃(s), G̃(t)],
J(s, t) = [G̃(s), G(t)] + [G(s), G̃(t)]
and also
K(s, t) = [W−(s), G(t)]q − [W−(t), G(s)]q,
L(s, t) = [G(s),W+(t)]q − [G(t),W+(s)]q,
M(s, t) = [G̃(s),W−(t)]q − [G̃(t),W−(s)]q,
N(s, t) = [W+(s), G̃(t)]q − [W+(t), G̃(s)]q,
P (s, t) = t−1[G(s), G̃(t)]− s−1[G(t), G̃(s)]− q[W−(t),W+(s)]q + q[W−(s),W+(t)]q,
Q(s, t) = t−1[G̃(s), G(t)]− s−1[G̃(t), G(s)]− q[W+(t),W−(s)]q + q[W+(s),W−(t)]q,
R(s, t) = [G(s), G̃(t)]q − [G(t), G̃(s)]q − qt[W−(t),W+(s)] + qs[W−(s),W+(t)],
S(s, t) = [G̃(s), G(t)]q − [G̃(t), G(s)]q − qt[W+(t),W−(s)] + qs[W+(s),W−(t)].
P. M. Terwilliger: A compact presentation for the alternating central extension of the positive . . . 385
By linear algebra,
C(s, t) =
(q + q−1)(P (s, t) +Q(s, t))− (s−1 + t−1)(R(s, t) + S(s, t))
(q2 − s−1t)(q2 − st−1)q−1
, (A.5)
J(s, t) =
(q + q−1)(R(s, t) + S(s, t))− (s+ t)(P (s, t) +Q(s, t))
(q2 − s−1t)(q2 − st−1)q−2
. (A.6)
Using Lemma 5.2 we routinely obtain
[W0, A(s, t)] = 0,
[W0, B(s, t)]
1− q−2
=
G(s, t)− F (s, t)
st
,
[W0, C(s, t)]
1− q−2
=
E(s, t)−D(s, t)
st
,
[W0, D(s, t)]q
q − q−1
= (s+ t)A(s, t),
[E(s, t),W0]q
q − q−1
= (s+ t)A(s, t),
[W0, F (s, t)]q
1− q−2
= S(s, t)− (q + q−1)H(s, t),
[G(s, t),W0]q
1− q−2
= S(s, t)− (q + q−1)I(s, t),
[W0, H(s, t)]q2
q − q−1
= K(s, t),
[I(s, t),W0]q2
q − q−1
= M(s, t),
[W0, J(s, t)]
q − q−1
= M(s, t)−K(s, t)
and
[W0,K(s, t)]q
q2 − q−2
= A(s, t),
[W0, L(s, t)]q
q − q−1
= P (s, t)− (s−1 + t−1)H(s, t),
[M(s, t),W0]q
q2 − q−2
= A(s, t),
[N(s, t),W0]q
q − q−1
= Q(s, t)− (s−1 + t−1)I(s, t),
[P (s, t),W0]
q − q−1
= (s−1 + t−1)K(s, t)− (q + q−1)s−1t−1E(s, t),
[W0, Q(s, t)]
q − q−1
= (s−1 + t−1)M(s, t)− (q + q−1)s−1t−1D(s, t),
[W0, R(s, t)]
q − q−1
= (s−1 + t−1)(E(s, t)−D(s, t)),
[W0, S(s, t)]
q2 − q−2
= M(s, t)−K(s, t)
and
[W1, A(s, t)]
1− q−2
=
D(s, t)− E(s, t)
st
, [W1, B(s, t)] = 0,
[W1, C(s, t)]
1− q−2
=
F (s, t)−G(s, t)
st
,
[D(s, t),W1]q
1− q−2
= R(s, t)− (q + q−1)H(s, t),
[W1, E(s, t)]q
1− q−2
= R(s, t)− (q + q−1)I(s, t), [F (s, t),W1]q
q − q−1
= (s+ t)B(s, t),
[W1, G(s, t)]q
q − q−1
= (s+ t)B(s, t),
[H(s, t),W1]q2
q − q−1
= L(s, t),
[W1, I(s, t)]q2
q − q−1
= N(s, t),
[W1, J(s, t)]
q − q−1
= L(s, t)−N(s, t)
386 Ars Math. Contemp. 22 (2022) #P3.01 / 363–386
and
[K(s, t),W1]q
q − q−1
= P (s, t)− (s−1 + t−1)H(s, t), [L(s, t),W1]q
q2 − q−2
= B(s, t),
[W1,M(s, t)]q
q − q−1
= Q(s, t)− (s−1 + t−1)I(s, t), [W1, N(s, t)]q
q2 − q−2
= B(s, t),
[W1, P (s, t)]
q − q−1
= (s−1 + t−1)L(s, t)− (q + q−1)s−1t−1G(s, t),
[Q(s, t),W1]
q − q−1
= (s−1 + t−1)N(s, t)− (q + q−1)s−1t−1F (s, t),
[W1, R(s, t)]
q2 − q−2
= L(s, t)−N(s, t), [W1, S(s, t)]
q − q−1
= (s−1 + t−1)(F (s, t)−G(s, t)).
We just listed 38 relations, including (A.5), (A.6). These 38 relations are called canonical.