Also available at http://amc-journal.eu
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 6 (2013) 409–417
Augmented down-up algebras and
uniform posets
Paul Terwilliger ∗, Chalermpong Worawannotai
Department of Mathematics, University of Wisconsin
480 Lincoln Drive, Madison, WI 53706-1388 USA
Received 9 July 2013, accepted 31 August 2013, published online 9 September 2013
Abstract
Motivated by the structure of the uniform posets we introduce the notion of an aug-
mented down-up (or ADU) algebra. We discuss how ADU algebras are related to the
down-up algebras defined by Benkart and Roby. For each ADU algebra we give two pre-
sentations by generators and relations. We also display a Z-grading and a linear basis. In
addition we show that the center is isomorphic to a polynomial algebra in two variables. We
display seven families of uniform posets and show that each gives an ADU algebra module
in a natural way. The main inspiration for the ADU algebra concept comes from the second
author’s thesis concerning a type of uniform poset constructed using a dual polar graph.
Keywords: Uniform poset, dual polar space, dual polar graph, down-up algebra.
Math. Subj. Class.: 06A07, 05E10, 17B37
1 Introduction
In [10] the first author introduced the notion of a uniform poset, and constructed eleven
families of examples from the classical geometries. Among the examples are the polar
spaces Polarb(N, ) and the attenuated spaces Ab(N,M), as well as the posets Altb(N),
Herq(N), and Quadb(N) associated with the alternating, Hermitean, and quadratic forms.
Another example is Hemmeter’s poset Hemb(N). In [12, Proposition 26.4] the second
author constructed a new family of uniform posets using the dual polar graphs. We denote
these posets by Polartopb (N, ) and describe them in Section 5 below.
In [2] Benkart and Roby introduced the down-up algebras, and obtained modules for these
algebras using Altb(N), Herq(N), Quadb(N), and Hemb(N). A down-up algebra mod-
ule is obtained from Polartopb (N, ) in a similar way. However, it appears that the down-
up algebra concept is not sufficiently robust to handle Polarb(N, ) or Ab(N,M). The
∗Corresponding author
E-mail addresses: terwilli@math.wisc.edu (Paul Terwilliger), worawann@math.wisc.edu (Chalermpong
Worawannotai)
Copyright c© 2013 DMFA Slovenije
410 Ars Math. Contemp. 6 (2013) 409–417
same can be said for the generalized down-up algebras [5]. In the present paper we intro-
duce a family of algebras called augmented down-up algebras, or ADU algebras for short.
These algebras seem well suited to handle uniform posets. Indeed, we show that each of
the uniform posets Polarb(N, ), Ab(N,M), Altb(N), Herq(N), Quadb(N), Hemb(N),
Polartopb (N, ) gives an ADU algebra module in a natural way.
The ADU algebras are related to the down-up algebras as follows. Given scalars α, β, γ
the corresponding down-up algebra A(α, β, γ) is defined by generators e, f and relations
e2f = αefe+ βfe2 + γe,
ef2 = αfef + βf2e+ γf.
See [2, p. 308]. To turn this into an ADU algebra we make three adjustments as follows.
Let q denote a nonzero scalar that is not a root of unity. We first require
α = q−2s + q−2t, β = −q−2s−2t
where s, t are distinct integers. Secondly, we add two generators k±1 such that ke = q2ek
and kf = q−2fk. Finally we reinterpret γ as a Laurent polynomial in k for which the
coefficients of ks, kt are zero.
From the above description the ADU algebras are reminiscent of the quantum univeral
enveloping algebra Uq(sl2). To illuminate the difference between these algebras, consider
their center. By [6, p. 27] the center of Uq(sl2) is isomorphic to a polynomial algebra in
one variable. As we will see, the center of an ADU algebra is isomorphic to a polynomial
algebra in two variables.
The results of the present paper are summarized as follows. We define two algebras by
generators and relations, and show that they are isomorphic. We call the common resulting
algebra an ADU algebra. For each ADU algebra we display a Z-grading and a linear basis.
We also show that the center is isomorphic to a polynomial algebra in two variables. We
obtain ADU algebra modules from each of the above seven examples of uniform posets.
We have a remark about the place of down-up algebras and ADU algebras in ring theory.
A down-up algebra can be viewed as an ambiskew polynomial ring [7, Section 3], which in
turn can be viewed as a generalized Weyl algebra [1], [7, Prop. 2.1]. By a comment in [8,
p. 48] that cites a preprint version of the present paper, an ADU algebra can also be viewed
in this way. Hoping to keep our paper accessible to nonexperts in ring theory, we will avoid
this point of view and use only linear algebra.
Recall the natural numbers N = {0, 1, 2, . . .} and integers Z = {0,±1,±2, . . .}.
2 Augmented down-up algebras
Our conventions for the paper are as follows. An algebra is meant to be associative and
have a 1. A subalgebra has the same 1 as the parent algebra. Let F denote a field. Let
λ denote an indeterminate. Let F[λ, λ−1] denote the F-algebra of Laurent polynomials in
λ that have all coefficients in F. Pick ψ ∈ F[λ, λ−1] and write ψ =
∑
i∈Z αiλ
i. By the
support of ψ we mean the set {i ∈ Z|αi 6= 0}. This set is finite.
Fix distinct s, t ∈ Z. Define
F[λ, λ−1]s,t = Span{λi|i ∈ Z, i 6= s, i 6= t}.
P. Terwilliger and C. Worawannotai: Augmented down-up algebras and uniform posets 411
Note that
F[λ, λ−1] = F[λ, λ−1]s,t + Fλs + Fλt (direct sum).
For ψ ∈ F[λ, λ−1] the following are equivalent: (i) ψ ∈ F[λ, λ−1]s,t; (ii) the integers s, t
are not in the support of ψ.
Fix a nonzero q ∈ F that is not a root of unity.
Definition 2.1. For ϕ ∈ F[λ, λ−1]s,t the F-algebra A = Aq(s, t, ϕ) has generators e, f,
k±1 and relations
kk−1 = 1, k−1k = 1,
ke = q2ek, kf = q−2fk,
e2f − (q−2s + q−2t)efe+ q−2s−2tfe2 = eϕ(k), (2.1)
ef2 − (q−2s + q−2t)fef + q−2s−2tf2e = ϕ(k)f. (2.2)
Remark 2.2. Referring to Definition 2.1, consider the special case in which ϕ ∈ F. Then
the relations (2.1), (2.2) become the defining relations for the down-up algebra A(q−2s +
q−2t,−q−2s−2t, ϕ).
Definition 2.3. For φ ∈ F[λ, λ−1]s,t the F-algebra B = Bq(s, t, φ) has generators Cs, Ct,
E, F , K±1 and relations
Cs, Ct are central,
KK−1 = 1, K−1K = 1,
KE = q2EK, KF = q−2FK,
FE = Csq
sKs + Ctq
tKt + φ(qK), (2.3)
EF = Csq
−sKs + Ctq
−tKt + φ(q−1K). (2.4)
Next we describe how the algebras in Definition 2.1 and Definition 2.3 are related.
Definition 2.4. We define an F-linear map F[λ, λ−1] → F[λ, λ−1], ψ 7→ ψs,t as follows.
For ψ ∈ F[λ, λ−1],
ψs,t(λ) = ψ(q
−1λ)− (q−2s + q−2t)ψ(qλ) + q−2s−2tψ(q3λ).
Recall the basis {λi}i∈Z for F[λ, λ−1].
Lemma 2.5. Consider the map ψ 7→ ψs,t from Definition 2.4. For i ∈ Z the vector λi is an
eigenvector for the map. The corresponding eigenvalue is q3i(q−2i − q−2s)(q−2i − q−2t).
This eigenvalue is zero if and only if i ∈ {s, t}.
Proof. Use Definition 2.4.
The following two lemmas are routine consequences of Lemma 2.5.
Lemma 2.6. For the map ψ 7→ ψs,t from Definition 2.4 the image is F[λ, λ−1]s,t and the
kernel is Fλs + Fλt.
412 Ars Math. Contemp. 6 (2013) 409–417
Lemma 2.7. For the map ψ 7→ ψs,t from Definition 2.4 the restriction to F[λ, λ−1]s,t is
invertible.
Let ϕ, φ ∈ F[λ, λ−1]s,t such that ϕ = φs,t. We are going to show that the algebras
Aq(s, t, ϕ) and Bq(s, t, φ) are isomorphic.
Lemma 2.8. For φ ∈ F[λ, λ−1]s,t the following hold in Bq(s, t, φ):
Cs =
q−tFE − qtEF + qtφ(q−1K)− q−tφ(qK)
qs−t − qt−s
K−s, (2.5)
Ct =
q−sFE − qsEF + qsφ(q−1K)− q−sφ(qK)
qt−s − qs−t
K−t. (2.6)
Moreover the algebra Bq(s, t, φ) is generated by E,F,K±1.
Proof. We first verify (2.5). In the expression on the right in (2.5), eliminate FE and EF
using (2.3) and (2.4). After a routine simplification (2.5) is verified. The equation (2.6) is
similarly verified. The last assertion follows from (2.5), (2.6).
Lemma 2.9. For φ ∈ F[λ, λ−1]s,t the following hold in Bq(s, t, φ):
E2F − (q−2s + q−2t)EFE + q−2s−2tFE2 = Eϕ(K), (2.7)
EF 2 − (q−2s + q−2t)FEF + q−2s−2tF 2E = ϕ(K)F. (2.8)
In the above lines ϕ = φs,t.
Proof. We first verify (2.7). In the expression on the left in (2.7), view E2F = E(EF ),
EFE = E(FE), FE2 = (FE)E and eliminate each parenthetical expression using (2.3)
and (2.4). Simplify the result using KE = q2EK along with ϕ = φs,t and Definition 2.4.
The equation (2.7) is now verified. The equation (2.8) is similarly verified.
The following definition is motivated by Lemma 2.8.
Definition 2.10. For ϕ ∈ F[λ, λ−1]s,t let cs, ct denote the following elements in Aq(s, t,
ϕ):
cs =
q−tfe− qtef + qtφ(q−1k)− q−tφ(qk)
qs−t − qt−s
k−s, (2.9)
ct =
q−sfe− qsef + qsφ(q−1k)− q−sφ(qk)
qt−s − qs−t
k−t. (2.10)
In the above lines φ denotes the unique element in F[λ, λ−1]s,t such that ϕ = φs,t.
Lemma 2.11. With the notation and assumptions of Definition 2.10, the elements cs, ct are
central in Aq(s, t, ϕ). Moreover
fe = csq
sks + ctq
tkt + φ(qk), (2.11)
ef = csq
−sks + ctq
−tkt + φ(q−1k). (2.12)
P. Terwilliger and C. Worawannotai: Augmented down-up algebras and uniform posets 413
Proof. We first show that cs is central in Aq(s, t, ϕ). To do this we show cse = ecs,
csf = fcs, csk = kcs. To verify these equations, eliminate each occurrence of cs using
(2.9), and simplify the result using the relations in Definition 2.1. We have shown that cs is
central in Aq(s, t, ϕ). One similarly shows that ct is central in Aq(s, t, ϕ). We now verify
(2.11). In the expression on the right in (2.11), eliminate cs, ct using (2.9), (2.10). After a
routine simplification (2.11) is verified. The equation (2.12) is similarly verified.
Theorem 2.12. Given ϕ, φ ∈ F[λ, λ−1]s,t such that ϕ = φs,t. Then there exists an F-
algebra isomorphism Aq(s, t, ϕ)→ Bq(s, t, φ) that sends
e 7→ E, f 7→ F, k±1 7→ K±1.
The inverse isomorphism sends
Cs 7→ cs, Ct 7→ ct, E 7→ e, F 7→ f, K±1 7→ k±1
where cs, ct are from Definition 2.10.
Proof. Combine Lemmas 2.8, 2.9, 2.11.
Definition 2.13. By an augmented down-up algebra we mean an algebra Aq(s, t, ϕ) from
Definition 2.1 or an algebra Bq(s, t, φ) from Definition 2.3.
Consider the algebra B = Bq(s, t, φ) from Definition 2.3. In Section 3 we are going to
show that the elements Cs, Ct generate the center Z(B), and that Z(B) is isomorphic to
a polynomial algebra in two variables. Because of this and following [6, p. 27], it seems
appropriate to call Cs, Ct the Casimir elements for Bq(s, t, φ).
3 A Z-grading and linear basis for Bq(s, t, φ)
Recall the algebra B = Bq(s, t, φ) from Definition 2.3. In this section we display a Z-
grading for B. We also display a basis for the F-vector space B.
LetA denote an F-algebra. By a Z-grading ofA we mean a sequence {An}n∈Z consisting
of subspaces of A such that
A =
∑
n∈Z
An (direct sum),
and AmAn ⊆ Am+n for all m,n ∈ Z. Let {An}n∈Z denote a Z-grading of A. For n ∈ Z
we call An the n-homogeneous component of A. We refer to n as the degree of An. An
element of A is said to be homogeneous of degree n whenever it is contained in An.
Theorem 3.1. The algebra B has a Z-grading {Bn}n∈Z with the following properties:
(i) The F-vector space B0 has a basis
KhCisC
j
t h ∈ Z, i, j ∈ N. (3.1)
(ii) For n ≥ 1, the F-vector space Bn has a basis
FnKhCisC
j
t h ∈ Z, i, j ∈ N. (3.2)
414 Ars Math. Contemp. 6 (2013) 409–417
(iii) For n ≥ 1, the F-vector space B−n has a basis
EnKhCisC
j
t h ∈ Z, i, j ∈ N. (3.3)
Moreover the union of (3.1)–(3.3) is a basis for the F-vector space B.
Proof. Routinely applying the Bergman diamond lemma [3, Theorem 1.2] one finds that
the union of (3.1)–(3.3) is a basis for the F-vector space B. Let B0 denote the subspace
of B spanned by (3.1). For n ≥ 1 let Bn and B−n denote the subspaces of B spanned by
(3.2) and (3.3), respectively. We show that {Bn}n∈Z is a Z-grading of B. By construction
the sum B =
∑
n∈Z Bn is direct. By construction and since Cs, Ct are central we have
CsBn ⊆ Bn and CtBn ⊆ Bn for n ∈ Z. Using KE = q2EK and KF = q−2FK we find
K±1Bn ⊆ Bn for n ∈ Z. Using (2.3) and (2.4) we find EBn ⊆ Bn−1 and FBn ⊆ Bn+1
for n ∈ Z. By these comments and the construction we see that BmBn ⊆ Bm+n for all
m,n ∈ Z. Therefore {Bn}∈Z is a Z-grading of B. The result follows.
We emphasize a few points from Theorem 3.1.
Corollary 3.2. With respect to the above Z-grading of B, the generatorsCs, Ct, E, F,K±1
are homogeneous with the following degrees:
v Cs Ct E F K
±1
degree of v 0 0 −1 1 0
Corollary 3.3. The homogeneous component B0 is the subalgebra of B generated by
Cs, Ct,K
±1. The algebra B0 is commutative.
Let {λi}2i=0 denote mutually commuting indeterminates.
Corollary 3.4. There exists an F-algebra isomorphism B0 → F[λ±10 , λ1, λ2] that sends
K±1 7→ λ±10 , Cs 7→ λ1, Ct 7→ λ2.
The Z-grading {Bn}n∈Z has the following interpretation.
Lemma 3.5. Consider the F-linear map B → B, ξ 7→ K−1ξK. For n ∈ Z the n-
homogeneous component Bn is an eigenspace of this map. The corresponding eigenvalue
is q2n.
Proof. Use the basis for Bn given in Theorem 3.1, along with the relations KE = q2EK
and KF = q−2FK.
Corollary 3.6. The homogeneous component B0 consists of the elements in B that commute
with K.
Proof. Immediate from Lemma 3.5.
P. Terwilliger and C. Worawannotai: Augmented down-up algebras and uniform posets 415
4 The center of Bq(s, t, φ)
Recall the algebra B = Bq(s, t, φ) from Definition 2.3. In this section we describe the
center Z(B).
Theorem 4.1. The following is a basis for the F-vector space Z(B):
CisC
j
t i, j ∈ N. (4.1)
Proof. By Theorem 3.1 the elements (4.1) are linearly independent over F, so they form
a basis for a subspace of B which we denote by Z ′. We show Z ′ = Z(B). The elements
Cs, Ct are central in B so Z ′ ⊆ Z(B). To obtain the reverse inclusion, pick ξ ∈ Z(B).
The element ξ commutes with K, so ξ ∈ B0 by Corollary 3.6. Recall the basis (3.1) for
B0. Writing ξ in this basis, we find ξ =
∑
h∈ZK
hξh where ξh ∈ Z ′ for h ∈ Z. Using
KE = q2EK and ξE = Eξ we obtain 0 = E
∑
h∈ZK
hξh(q
2h − 1). Combining this
with Theorem 3.1 we find ξh = 0 for all nonzero h ∈ Z. Therefore ξ = ξ0 ∈ Z ′. We have
shown Z ′ = Z(B) and the result follows.
Corollary 4.2. There exists an F-algebra isomorphism Z(B)→ F[λ1, λ2] that sends
Cs 7→ λ1, Ct 7→ λ2.
5 Uniform posets
Recall the algebras Aq(s, t, ϕ) from Definition 2.1. In this section we discuss how these
algebras are related to the uniform posets [10].
Throughout this section we assume that F is the complex number field C. Let P denote a
finite ranked poset with fibers {Pi}Ni=0 [10, p. 194]. Let CP denote the vector space over
C with basis P . Let End(CP ) denote the C-algebra consisting of all C-linear maps from
CP to CP . We now define three elements in End(CP ) called the lowering, raising, and
q-rank operators. For x ∈ P , the lowering operator sends x to the sum of the elements in
P that are covered by x. The raising operator sends x to the sum of the elements in P that
cover x. The q-rank operator sends x to qN−2ix where x ∈ Pi.
In [10] we introduced a class of finite ranked posets said to be uniform. We refer the reader
to that article for a detailed description of these posets. See also [2, p. 306] and [9], [11].
In [10, Section 3] we gave eleven examples of uniform posets. We are going to show
that six of these examples give an Aq(s, t, ϕ)-module. These six examples are listed in
the first six rows of the table below. The remaining row of the table contains an example
Polartopb (N, ) which is defined as follows. Start with the poset Polarb(N, ) which we
denote by P . Using P we define an undirected graph Γ as follows. The vertex set of Γ
consists of the top fiber PN of P . Vertices y, z ∈ PN are adjacent in Γ whenever they are
distinct and cover a common element of P . The graph Γ is often called a dual polar graph
[4, p. 274], [12, Section 16]. Fix a vertex x ∈ PN . Using x we define a partial order ≤
on PN as follows. For y, z ∈ PN let y ≤ z whenever ∂(x, y) + ∂(y, z) = ∂(x, z), where
∂ denotes path-length distance in Γ. We have turned PN into a poset. We call this poset
Polartopb (N, ). Using [12, Proposition 26.4] one checks that Polar
top
b (N, ) is uniform.
Theorem 5.1. In each row of the table below we give an example of a uniform poset P .
For each example we display integers s < t and a Laurent polynomial ϕ ∈ F[λ, λ−1]s,t.
416 Ars Math. Contemp. 6 (2013) 409–417
In each case the vector space CP becomes an Aq(s, t, ϕ)-module such that the generator
e (resp. f ) (resp. k) acts on CP as the lowering (resp. raising) (resp. q-rank) operator for
P . For convenience, for each example we display the element φ ∈ F[λ, λ−1]s,t such that
ϕ = φs,t.
example s t ϕ φ
Polarb(N, ) 0 1 −(q + q−1)(q2N+1+2λ2 + qN−3λ−1) − q
2N+2λ2+qN−1λ−1
(q−q−1)2
Ab(N,M) −1 0 −(q + q−1)qN+2M+1λ − q
N+2M−1
(q−q−1)2 λ
Altb(N) −2 −1 −(q + q−1)q2N+1 − q
2N−2
(q−q−1)2
Herq(N) −2 −1 −(q + q−1)q2N+2 − q
2N−1
(q−q−1)2
Quadb(N) −2 −1 −(q + q−1)q2N+3 −
q2N
(q−q−1)2
Hemb(N) −2 −1 −(q + q−1)q2N+1 − q
2N−2
(q−q−1)2
Polartopb (N, ) −2 −1 −(q + q−1)q2N+3+2 −
q2N+2
(q−q−1)2
In the above table b = q2.
Proof. For each example except the last, our assertions follow routinely from [10, The-
orem 3.2]. For the last example Polartopb (N, ) our assertions follow from [12, Theo-
rem 1.10]. Note that the parameter denoted  in [12, Theorem 1.10] is one more than the
parameter denoted  in [10, p. 201].
6 Acknowledgement
The main inspiration for the ADU algebra concept comes from the second author’s thesis
[12] concerning the uniform poset Polartopb (N, ). To be more precise, it was his discovery
of two central elements that he called C1, C2 [12, Section 28] that suggested to us how to
define an ADU algebra.
References
[1] V. Bavula and D. A. Jordan, Isomorphism problems and groups of automorphisms for general-
ized Weyl algebras, Trans. Amer. Math. Soc. 353 (2001), 769–794.
[2] G. Benkart and T. Roby, Down-up algebras, J. Algebra 209 (1998), 305–344.
[3] G. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178–218.
[4] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag,
Berlin, 1989.
[5] T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279
(2004), 402–421.
P. Terwilliger and C. Worawannotai: Augmented down-up algebras and uniform posets 417
[6] J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6, Amer. Math.
Soc., Providence RI, 1996.
[7] D. A. Jordan, Down-up algebras and ambiskew polynomial rings, J. Algebra 228 (2000), 311–
346.
[8] D. A. Jordan and I. Wells, Simple ambiskew polynomial rings, J. Algebra 382 (2013), 46–70.
[9] S. Miklavic and P. Terwilliger, Bipartite Q-polynomial distance-regular graphs and uniform
posets, J. Algebraic Combin. 38 (2013), 225–242.
[10] P. Terwilliger, The incidence algebra of a uniform poset, in: D. Ray-Chaudhuri (ed.), Coding
theory and design theory, Part I, IMA Vol. Math. Appl., 20, Springer, New York, 1990, 193–
212.
[11] P. Terwilliger, Quantum matroids, in: E. Bannai and A. Munemasa (eds.), Progress in algebraic
combinatorics (Fukuoka, 1993), Adv. Stud. Pure Math., 24, Math. Soc. Japan, Tokyo, 1996,
323–441.
[12] C. Worawannotai, Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of
dual q-Krawtchouk type, Linear Algebra Appl. 438 (2013), 443–497.