DIFFERENTIAL POSETS AND DOWN-UP ALGEBRAS

Georgia Benkart

Abstract. Down-up algebras originated in the study of differential posets. In this paper we discuss their combinatorial origins, representations, and structure. Downup algebras exhibit many of the important features of the universal enveloping algebra U(𝔰𝔩_{2}) of the Lie algebra 𝔰𝔩_{2} including a Poincaré-Birkhoff-Witt type basis and a wellbehaved representation theory. They have many interesting connections with Weyl algebras, quantum groups, and Witten’s deformations of U(𝔰𝔩_{2}).

Differential posets.

In [St2], Stanley introduced a class of partially ordered sets, which he termed differential posets. Many of the remarkable enumerative and combinatorial properties of these posets involve counting saturated chains y_{1} ≺ y_{2} ≺ ⋯ ≺ y_{k} or Hasse walks y_{1}, y_{2}, …, y_{k}, (where either y_{i+1} covers y_{i} or y_{i} covers y_{i+1}). Essential in the computations are two operators, d (down) and u (up), which are defined on the complex vector space CP having basis the elements of the poset P. If y ∈ P, then d(y) is the sum of all elements that y covers and u(y) is the sum of all elements that cover y. For many posets the down and up operators give well-defined linear transformations of CP. Precursors of the operators d and u appeared in [St1] and [P], where they were used to show posets are Sperner or rank unimodal.

The characterizing property of an r-differential poset is that the down and up operators satisfy du − ud = rI for some positive integer r (see [St2, Thm. 2.2]), where I is the identity transformation on CP. Thus, the poset affords a representation of the Weyl algebra, (the associative algebra with generators y, x subject to the relation yx − xy = 1), via the mapping y ↦ d/r, and x ↦ u. Since the Weyl algebra also can be realized as differential operators y ↦ d/dx and x ↦ x (multiplication by x) on C[x], Stanley referred to the posets satisfying du−ud = rI as r-differential or simply differential when r = 1. Fomin [F] studied essentially the same class of posets for r = 1, calling them Y -graphs”. This terminology comes from the fact that Young’s lattice Y of all partitions of all nonnegative integers is the prototypical example.

A partition μ of a nonnegative integer m can be regarded as a descending sequence μ = (μ_{1} ≥ μ_{2} ≥ …) of parts whose sum |μ| = ∑_{i} μ_{i} equals m. If ν = (ν_{1} ≥ ν_{2} ≥ …) is a second partition, then μ ≤ ν when μ_{i} ≤ ν_{i} for all i.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 16S15, 16S30, 17B35, 17B10 Secondary 81R50, 06A07.

The author gratefully acknowledges support from National Science Foundation Grant #DMS– 9622447.

The partition ν covers μ (written here as μ ≺ ν) if μ < ν and |ν| = 1 + |μ|. Thus, μ ≺ ν if the partition μ is obtained from ν by subtracting 1 from exactly one of the parts of ν, and d(ν) is the sum of all such μ. Analogously, u(ν) is the sum of all partitions π obtained from ν by adding 1 to one part of ν. Young’s lattice Y is a 1-differential poset, and Y^{r} is r-differential ([St2, Cor. 1.4]).

The down and up operators on Young’s lattice have a representation theoretic significance. The simple modules of the symmetric group S_{n} are indexed by the partitions ν of n. Upon restriction to S_{n−1}, the representation labelled by ν decomposes into a direct sum of simple S_{n−1}-modules indexed by the partitions μ ≺ ν, so it is given by d(ν). When the simple module labelled by ν is induced to a representation of S_{n+1}, it decomposes into a sum of simple S_{n+1}-modules indexed by the partitions π of n + 1 such that ν ≺ π, which is just u(ν).

In his study [T1] of uniform posets, Terwilliger considered finite ranked posets P whose down and up operators satisfy the following relation

d_{i}d_{i+1}u_{i} = α_{i}d_{i}u_{i−1}d_{i} + β_{i}u_{i−2}d_{i−1}d_{i} + γ_{i}d_{i},

where d_{i} and u_{i} denote the restriction of d and u to the elements of rank i. (There is an analogous second relation,

d_{i+1}u_{i}u_{i−1} = α_{i}u_{i−1}d_{i}u_{i−1} + β_{i}u_{i−1}u_{i−2}d_{i−1} + γ_{i}u_{i−1},

which holds automatically in this case because d_{i+1} and u_{i} are adjoint operators relative to a certain bilinear form.) In many examples the constants in these relations do not depend on the rank i. In particular, a poset whose down and up operators satisfy

\begin{aligned}d^{2}u & = q(q + 1)dud − q^{3}ud^{2} + rd \\ du^{2} & = q(q + 1)udu − q^{3}u^{2}d + ru \end{aligned}

where q and r are fixed complex numbers is said to be (q, r)-differential. Many interesting examples of (q, r)-differential posets in [T1] arise from considering certain subspaces of a vector space over the field GF (q) of q elements:

(1) Assume W is an n-dimensional vector space over GF (q) and consider the set of pairs P = {(U, f) | U is a subspace of W and f is an alternating bilinear form on U} with the ordering: (U, f) ≤ (V, g) if U is a subspace of V and g|_{U} = f. Then P is a (q, r)-differential poset with r = −q^{n}(q + 1).

(2) In example (1), replace an alternating bilinear form” with a quadratic form”. The resulting poset P is (q, −q^{n+1}(q + 1))-differential.

(3) In this example assume W is an n-dimensional space over GF (q^{2}) and the bilinear forms are Hermitian. The poset P is (q^{2}, −q^{2n+1}(q^{2} +1))-differential in this case.

Down-up algebras.

To better understand the algebra generated by the down and up operators of a poset and its action on the poset, we introduced the notion of a down-up algebra in our joint work with Roby (see [BR]). Although the initial motivation for our investigations came from posets, we made no assumptions about the existence of posets whose down and up operators satisfy our relations. However, when such a poset exists, it affords a representation of the down-up algebra, so our primary focus in [BR] was on determining explicit information about the representations of down-up algebras.

Definition 1. Let α, β, γ be fixed but arbitrary complex numbers. The unital associative algebra A(α, β, γ) over C with generators d, u and defining relations

(R1) d^{2}u = αdud + βud^{2} + γd,

(R2) du^{2} = αudu + βu^{2}d + γu,

is a down-up algebra.

It is easy to see that when γ ≠ 0 the down-up algebra A(α, β, γ) is isomorphic to A(α, β, 1) by the map, d ↦ d′, u ↦ γu′. Therefore, it would suffice to treat just two cases γ = 0, 1, but to avoid dividing considerations into these two cases, we retain the notation γ.

Examples of down-up algebras.

Example (i). If B is the associative algebra generated by the down and up operators d, u of a (q, r)-differential poset, then relations (R1) and (R2) hold with α = q(q + 1), β = −q^{3}, and γ = r. Thus, B is a homomorphic image of the algebra A(α, β, γ) with these parameters, and the action of B on the poset gives a representation of A(α, β, γ).

Example (ii). The relation du − ud = rI of an r-differential poset can be multiplied on the left by d and on the right by d and the resulting equations can be added to get the relation d^{2}u − ud^{2} = 2rd of a (−1, 2r)-differential poset. Thus, the Weyl algebra is a homomorphic image (by the ideal generated by du − ud − r1) of the algebra A(0, 1, 2r). Similarly, the q-Weyl algebra is a homomorphic image of the algebra A(0, q^{2}, (q+1)) by the ideal generated by du−qud−1. The skew polynomial algebra C_{q}[d, u], or quantum plane (see [M]), is the associative algebra with generators d, u which satisfy the relation du = qud. Therefore, C_{q}[d, u] is a homomorphic image (by the ideal generated by du − qud) of the algebra A(2q, −q^{2}, 0).

Example (iii). Consider the poset L(2, 2) = \{\underline{a} = (a_{1}, a_{2}) | 2 ≥ a_{1} ≥ a_{2} and a_{1}, a_{2} ∈ Z_{≥0}\} with the order relation \underline{a} ≤ \underline{b} if a_{i} ≤ b_{i} for i = 1, 2. This is just the set of partitions which fit into a 2 × 2 box. By direct calculation it is easy to verify that the down and up operators on this poset satisfy d^{2}u = dud − ud^{2} + d and du^{2} = udu − u^{2}d + u, so the algebra they generate is a homomorphic image of the down-up algebra A(1, −1, 1).

Example (iv). Suppose 𝔤 is a 3-dimensional Lie algebra over C with basis x, y, [x, y] such that [x[x, y]] = γx and [[x, y], y] = γy. In the universal enveloping algebra U(𝔤) of 𝔤 where [x, y] = xy − yx, these relations become

\begin{matrix}x^{2}y − 2xyx + yx^{2} = γx \\ xy^{2} − 2yxy + y^{2}x = γy. \end{matrix}

Thus, U(𝔤) is a homomorphic image of the down-up algebra A(2, −1, γ) via the mapping ϕ : A(2, −1, γ) → U(𝔤) with ϕ : d ↦ x, ϕ : u ↦ y. The mapping ψ : 𝔤 → A(2, −1, γ) with ψ : x ↦ d, ψ : y ↦ u, and ψ : [x, y] ↦ du − ud extends, by the universal property of U(𝔤), to an algebra homomorphism ψ : U(𝔤) → A(2, −1, γ) which is the inverse of ϕ. Consequently, U(𝔤) is isomorphic to A(2, −1, γ).

The Lie algebra 𝔰𝔩_{2} of 2 × 2 complex matrices of trace zero has a standard basis e = E_{1,2}, f = E_{2,1}, and h = E_{1,1} − E_{2,2} of matrix units, which satisfies [e, f] = h, [h, e] = 2e, and [h, f] = −2f. From this we see that U(𝔰𝔩_{2}) ≅ A(2, −1, −2). The Heisenberg Lie algebra ℌ has a basis x, y, z where [x, y] = z, and [z, ℌ] = 0, so U(ℌ) ≅ A(2, −1, 0).

Example (v). The 2 × 2 complex matrices y = ( \begin{matrix}y_{1} & y_{2} \\ y_{3} & y_{4} \end{matrix} ) with supertrace

y_{1} − y_{4} = 0 is the special linear Lie superalgebra L = 𝔰𝔩(1, 1) = L\overline{0} ⊕ L\overline{1} under the supercommutator [x, y] = xy − (−1)^{ab}yx for x ∈ L\overline{a}, y ∈ L\overline{b}. It has a presentation by generators e, f (which belong to L\overline{1} and can be identified with the matrix units e = E_{1,2}, f = E_{2,1}) and relations [e, [e, f]] = 0, [[e, f], f] = 0, [e, e] = 0, [f, f] =

0. The universal enveloping algebra U(𝔰𝔩(1, 1)) of 𝔰𝔩(1, 1) has generators e, f and relations e^{2}f − fe^{2} = 0, ef^{2} − f^{2}e = 0, e^{2} = 0, f^{2} = 0. Thus, U(𝔰𝔩(1, 1)) is a homomorphic image of the down-up algebra A(0, 1, 0) by the ideal generated by the elements e^{2} and f^{2}, which are central in A(0, 1, 0).

Example (vi). The orthosymplectic Lie superalgebra 𝔬𝔰𝔭(1, 2) = L\overline{0} ⊕ L\overline{1} has generators x, y ∈ L\overline{1} which satisfy

xy + yx = t ∈ L\overline{0}  tx − xt = x  yt − ty = y.

By combining these relations, we see that its universal enveloping algebra U(𝔬𝔰𝔭(1, 2)) is a homomorphic image of A(0, 1, 1).

Example (vii). Consider the field C(q) of rational functions in the indeterminate q over the complex numbers, and let U_{q}(𝔤) be the quantized enveloping algebra (quantum group) of a finite-dimensional simple complex Lie algebra 𝔤 corresponding to the Cartan matrix 𝔄 = (a_{i,j})_{i,j=1}^{n}. There are relatively prime integers ℓ_{i} so that the matrix (ℓ_{i}a_{i,j}) is symmetric. Let

q_{i} = q^{ℓ_{i}},  and  [m]_{i} = \frac{q_{i}^{m} − q_{i}^{−m}}{q_{i} − q_{i}^{−1}}

for all m ∈ Z_{≥0}. When m ≥ 1, let [m]_{i}! = ∏_{j=1}^{m}[j]_{i}. Set [0]_{i}! = 1 and define

[ \begin{matrix}m \\ n \end{matrix} ]_{i} = \frac{[m]_{i}!}{[n]_{i}![m − n]_{i}!}.

Then U = U_{q}(𝔤) is the unital associative algebra over C(q) with generators E_{i}, F_{i}, K_{i}, K_{i}^{−1} (i = 1, …, n) subject to the relations

(Q1) K_{i}K_{i}^{−1} = K_{i}^{−1}K_{i}, K_{i}K_{j} = K_{j}K_{i}

(Q2) K_{i}E_{j}K_{i}^{−1} = q_{i}^{a_{i,j}}E_{j} K_{i}F_{j}K_{i}^{−1} = q_{i}^{−a_{i,j}}F_{j}

(Q3) E_{i}F_{j} − F_{j}E_{i} = δ_{i,j} \frac{K_{i} − K_{i}^{−1}}{q_{i} − q_{i}^{−1}}

(Q4) ∑_{k=0}^{1−a_{i,j}}(−1)^{k} [ \begin{matrix}1 − a_{i,j} \\ k \end{matrix} ]_{i} E_{i}^{1−a_{i,j}−k}E_{j}E_{i}^{k} = 0 for i ≠ j

(Q5) ∑_{k=0}^{1−a_{i,j}}(−1)^{k} [ \begin{matrix}1 − a_{i,j} \\ k \end{matrix} ]_{i} F_{i}^{1−a_{i,j}−k}F_{j}F_{i}^{k} = 0 for i ≠ j.

Suppose a_{i,j} = −1 = a_{j,i} for some i ≠ j, and consider the subalgebra U_{i,j} generated by E_{i}, E_{j}. In this special case, the quantum Serre relation (Q4) reduces to

\begin{matrix}E_{i}^{2}E_{j} − [2]_{i}E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0 and \\ E_{j}^{2}E_{i} − [2]_{j}E_{j}E_{i}E_{j} + E_{i}E_{j}^{2} = 0. \end{matrix}

Since −ℓ_{i} = ℓ_{i}a_{i,j} = ℓ_{j}a_{j,i} = −ℓ_{j}, the coefficients [2]_{i} and [2]_{j} are equal. The algebra U_{i,j} (with q specialized to a complex number which is not a root of unity) is isomorphic to A([2]_{i}, −1, 0) by the mapping E_{i} ↦ d, E_{j} ↦ u. The same result is true if the corresponding F ’s are used in place of the E’s. In particular, when 𝔤 = 𝔰𝔩_{3} (3 × 3 matrices of trace 0), the algebra U_{i,j} is just the subalgebra of U_{q}(𝔰𝔩_{3}) generated by the E’s.

Example (viii). To provide an explanation of the existence of quantum groups, Witten ([W1], [W2]) introduced a 7-parameter deformation of the universal enveloping algebra U(𝔰𝔩_{2}). Witten’s deformation is a unital associative algebra over a field K (which is algebraically closed of characteristic zero and which could be assumed to be C) and depends on a 7-tuple \underline{ξ} = (ξ_{1}, …, ξ_{7}) of elements of K. It has a presentation by generators x, y, z and defining relations

xz − ξ_{1}zx = ξ_{2}x (2) zy − ξ_{3}yz = ξ_{4}y (3)

yx − ξ_{5}xy = ξ_{6}z^{2} + ξ_{7}z. (4)

We denote the resulting algebra by 𝔚(\underline{ξ}). In applications of these deformation algebras, the parameters depend on the coupling constant of the particular physical theory, and Witten [W2] gives an evaluation of them in the special case of the threedimensional Chern-Simons gauge theory.

Let us assume ξ_{6} = 0 and ξ_{7} ≠ 0. Then substituting expression (4) into (2) and (3) and rearranging shows that

\begin{matrix}− ξ_{5}x^{2}y + (1 + ξ_{1}ξ_{5})xyx − ξ_{1}yx^{2} = ξ_{2}ξ_{7}x \\ − ξ_{5}xy^{2} + (1 + ξ_{3}ξ_{5})yxy − ξ_{3}y^{2}x = ξ_{4}ξ_{7}y. \end{matrix}

In particular, when ξ_{5} ≠ 0, ξ_{1} = ξ_{3}, and ξ_{2} = ξ_{4} we obtain

\begin{matrix}x^{2}y = \frac{1 + ξ_{1}ξ_{5}}{ξ_{5}} xyx − \frac{ξ_{1}}{ξ_{5}} yx^{2} − \frac{ξ_{2}ξ_{7}}{ξ_{5}} x \\ xy^{2} = \frac{1 + ξ_{1}ξ_{5}}{ξ_{5}} yxy − \frac{ξ_{1}}{ξ_{5}} y^{2}x − \frac{ξ_{2}ξ_{7}}{ξ_{5}} y. \end{matrix}

From this it is easy to see that a Witten deformation algebra 𝔚(\underline{ξ}) with ξ_{6} = 0, ξ_{5}ξ_{7} ≠ 0, ξ_{1} = ξ_{3}, and ξ_{2} = ξ_{4} is a homomorphic image of the down-up algebra A(α, β, γ) with

(5)  α = \frac{1 + ξ_{1}ξ_{5}}{ξ_{5}} ,  β = −\frac{ξ_{1}}{ξ_{5}} ,  γ = −\frac{ξ_{2}ξ_{7}}{ξ_{5}} .

In fact, in [B, Thm. 2.6] we proved the following Proposition 6. A Witten deformation algebra 𝔚(\underline{ξ}) with

(7)  ξ_{6} = 0, ξ_{5}ξ_{7} ≠ 0, ξ_{1} = ξ_{3}, and ξ_{2} = ξ_{4}

is isomorphic to the down-up algebra A(α, β, γ) with α, β, γ given by (5). Conversely, any down-up algebra A(α, β, γ) with not both α and β equal to 0 is isomorphic to a Witten deformation algebra 𝔚(\underline{ξ}) whose parameters satisfy (7).

A deformation algebra 𝔚(\underline{ξ}) has a filtration, and Le Bruyn ([L1], [L2]) investigated the algebras 𝔚(\underline{ξ}) whose associated graded algebras are Auslander regular. They determine a 3-parameter family of deformation algebras which are called conformal 𝔰𝔩_{2} algebras and whose defining relations are

(8)  xz − azx = x,  zy − ayz = y,  yx − cxy = bz^{2} + z

When c ≠ 0 and b = 0, the conformal 𝔰𝔩_{2} algebra with defining relations given by (8) is isomorphic to the down-up algebra A(α, β, γ) with α = c^{−1}(1 + ac), β = −ac^{−1} and γ = −c^{−1}. If b = c = 0 and a ≠ 0, then the conformal 𝔰𝔩_{2} algebra is isomorphic to the down-up algebra A(α, β, γ) with α = a^{−1}, β = 0 and γ = −a^{−1}.

In a recent paper [K1], Kulkarni has shown that under certain assumptions on the parameters a Witten deformation algebra is isomorphic to a conformal 𝔰𝔩_{2} algebra or to a double skew polynomial extension. Kulkarni studies the simple modules of the conformal sl_{2} algebras and of the skew polynomial algebras. Critical to the investigations in [K1] is the observation that the conformal 𝔰𝔩_{2} algebra of (8) can be realized as a hyperbolic ring. Kulkarni then applies results of Rosenberg [R] on noncommutative algebraic geometry to describe the left ideals in the left spectrum of the algebra and to determine the maximal left ideals for the conformal sl_{2} algebras.

Example (ix). The quadratic Askey-Wilson algebras studied in [GLZ] can be regarded as having generators a, b and defining relations which depend on fixed parameters (α, γ, δ, ϵ, ζ, η, μ, ν) according to:

\begin{aligned}a^{2}b & = αaba − ba^{2} + ζ(ab + ba) + ηa^{2} + γa + δb + μ1 \\ ab^{2} & = αbab − b^{2}a + η(ab + ba) + ζb^{2} + γb + ϵa + ν1. \end{aligned}

It is apparent that when δ = ϵ = ζ = η = μ = ν = 0, this algebra is just the downup algebra A(α, −1, γ). Askey-Wilson algebras are related to the Leonard systems introduced in [T2] as abstract algebraic generalizations of q-Racah polynomials and of families of orthogonal polynomials that include the quantum q-Krawtchouk, Racah, Hahn, dual Hahn, and Krawtchouk polynomials.

Highest weight modules.

Down-up algebras have a rich representation theory (see [BR, Sec. 2]). In particular, they have highest weight modules and weight modules which mimic those of 𝔰𝔩_{2}.

A module V for A = A(α, β, γ) is said to be a highest weight module of weight λ if V has a vector y_{0} such that d · y_{0} = 0, (du) · y_{0} = λy_{0}, and V = Ay_{0}. The vector y_{0} is a maximal vector or highest weight vector of V.

Proposition 9. (See [BR, Sec. 2]) Set λ_{−1} = 0 and let λ_{0} = λ ∈ C be arbitrary. For n ≥ 1, define λ_{n} inductively by the recurrence relation,

(10)  λ_{n} = αλ_{n−1} + βλ_{n−2} + γ.

The C-vector space V (λ) with basis {v_{n} | n = 0, 1, 2, … } and with A(α, β, γ)-action given by

(11)  \begin{aligned}d · v_{n} & = λ_{n−1}v_{n−1}, n ≥ 1, and d · v_{0} = 0 \\ u · v_{n} & = v_{n+1}. \end{aligned}

is a highest weight module for A(α, β, γ). Every A(α, β, γ)-module of highest weight λ is a homomorphic image of V (λ). The module V (λ) is simple if and only if λ_{n} ≠ 0 for any n.

Because it shares the same universal property and many of the same features as Verma modules for finite-dimensional semisimple complex Lie algebras, the module V (λ) is said to be a Verma module for A(α, β, γ).

Weight modules.

If we multiply the relation d^{2}u − αdud − βud^{2} = γd on the left by u and the relation du^{2} − αudu − βu^{2}d = γu on the right by d and subtract the second from the first, the resulting equation is

0 = ud^{2}u − du^{2}d  or  (du)(ud) = (ud)(du).

Therefore, the elements du and ud commute in A = A(α, β, γ). For any basis element v_{n} ∈ V (λ), we have du · v_{n} = λ_{n}v_{n} and ud · v_{n} = λ_{n−1}v_{n}. Using that with n = 0 and λ ≠ 0, it is easy to see that du and ud are linearly independent. Let 𝔥 = Cdu ⊕ Cud.

We say an A-module V is a weight module if V = ⨁_{ν∈𝔥^{∗}} V_{ν}, where V_{ν} = {v ∈ V | h · v = ν(h)v for all h ∈ 𝔥}, and the sum is over elements in the dual space 𝔥^{∗} of 𝔥. Any submodule of a weight module is a weight module. If V_{ν} ≠ (0), then ν is a weight and V_{ν} is the corresponding weight space. Each weight ν is determined by the pair (ν′, ν′′) of complex numbers, ν′ = ν(du) and ν′′ = ν(ud). In particular, highest weight modules are weight modules in this sense. The basis vector v_{n} of V (λ) is a weight vector whose weight is given by the pair (λ_{n}, λ_{n−1}). Finding these weights explicitly involves solving the linear recurrence relation in (10), which can be done by standard methods as in [Br, Chap.7] for example.

Proposition 12. Assume λ_{−1} = 0, λ_{0} = λ ∈ C, and λ_{n} for n ≥ 1 is given by the recurrence relation λ_{n} − αλ_{n−1} − βλ_{n−2} = γ. Fix t ∈ C such that

t^{2} = \frac{α^{2} + 4β}{4} .

(i) If α^{2} + 4β ≠ 0, then

λ_{n} = c_{1}r_{1}^{n} + c_{2}r_{2}^{n} + x_{n}, where

r_{1} = \frac{α}{2} + t, r_{2} = \frac{α}{2} − t,

x_{n} = \{ \begin{matrix}(1 − α − β)^{−1}γ if α + β ≠ 1 \\ (2 − α)^{−1}γn if α + β = 1 (necessarily α ≠ 2), \end{matrix}

and ( \begin{matrix}c_{1} \\ c_{2} \end{matrix} ) = \frac{1}{r_{2} − r_{1}} ( \begin{matrix}r_{2} & −1 \\ −r_{1} & 1 \end{matrix} ) ( \begin{matrix}λ − x_{0} \\ αλ + γ − x_{1} \end{matrix} ) .

(ii) If α^{2} + 4β = 0 and α ≠ 0, then

λ_{n} = c_{1}s^{n} + c_{2}ns^{n} + x_{n} where

s = \frac{α}{2}

x_{n} = \{ \begin{matrix}(1 − α − β)^{−1}γ if α + β ≠ 1 \\ 2^{−1}n^{2}γ if α + β = 1 i.e. if α = 2, β = −1, \end{matrix}

and ( \begin{matrix}c_{1} \\ c_{2} \end{matrix} ) = ( \begin{matrix}1 & 0 \\ −1 & 2α^{−1} \end{matrix} ) ( \begin{matrix}λ − x_{0} \\ αλ + γ − x_{1} \end{matrix} ) .

(iii) If α^{2} + 4β = 0 and α = 0, then β = 0 and λ_{n} = γ for all n ≥ 1.

If α, β are real, then it is natural to take t = \frac{\sqrt{α^{2} + 4β}}{2} in the above calculations.

Let us consider several special cases.

Example (a). Recall that the universal enveloping algebra U(𝔰𝔩_{2}) of 𝔰𝔩_{2} is isomorphic to the algebra A(2, −1, −2), and the universal enveloping algebra U(ℌ) of the Heisenberg Lie algebra ℌ is isomorphic to A(2, −1, 0). Applying (ii) with s = α/2 = 1 and x_{n} = n^{2}γ/2 for any algebra A(2, −1, γ), we have that

λ_{n} = λ + (λ + \frac{γ}{2} )n + \frac{γn^{2}}{2} = (n + 1)(λ + \frac{γn}{2}).

In the 𝔰𝔩_{2}-case, it is customary to use the operator h = du − ud rather than du. The eigenvalues of h are λ_{n} − λ_{n−1} = λ + nγ = λ − 2n, n = 0, 1, . . .. The analogous computation in the Heisenberg Lie algebra shows that the central element z = du − ud has constant eigenvalue λ_{n} = λ.

Example (b). Recall that the quantum case discussed earlier involves the downup algebra A([2]_{i}, −1, 0). In the particular case of U_{q}(𝔰𝔩_{3}), the subalgebra generated

by the E_{i}’s is isomorphic to A([2], −1, 0) where [2] = \frac{q^{2} − q^{−2}}{q − q^{−1}} , and λ_{n} = [n+1]λ =

(\frac{q^{n+1} − q^{−(n+1)}}{q − q^{−1}} )λ for all n ≥ 0 in that case.

Example (c). For the algebra A(1, 1, 0), the solutions to the associated linear recurrence λ_{n} = λ_{n−1} + λ_{n−2}, λ_{0} = λ, λ_{−1} = 0, (hence the eigenvalues of du and ud on V (λ)) are given by the Fibonacci sequence λ_{0} = λ, λ_{1} = λ, λ_{2} = 2λ, λ_{3} = 3λ, λ_{4} = 5λ, . . .. In this case, the equations in Proposition 12 reduce to

λ_{n} = λ\frac{\sqrt{5}}{5} ((\frac{1 + \sqrt{5}}{2})^{n+1} − (\frac{1 − \sqrt{5}}{2})^{n+1}) .

In [BR] we investigated in detail the weight space and submodule structure of the Verma module V (λ). Roots of unity play a critical role in determining the dimension of a weight space. We introduced category O” modules in the spirit of [BGG] and showed the simple objects were highest weight modules, and we explored a more general category O′ of modules for down-up algebras. We briefly summarize some of the main results.

Proposition 13. ([BR, Secs. 2, 4, 5])

(a) In V (λ) each weight space is either one-dimensional or infinite-dimensional. If an infinite-dimensional weight space occurs, there are only finitely many weights.

(b) If each weight space of V (λ) is one-dimensional, then the proper submodules of V (λ) have the form N = span_{C}{v_{j} | j ≥ n + 1} for some n ≥ 0 with λ_{n} = 0. Hence they are contained in M(λ) = span_{C}{v_{j} | j ≥ m + 1}, where λ_{m} = 0 and m is minimal with that property.

(c) If γ = 0 = λ, then V (λ) has infinitely many maximal proper submodules, each of the form N^{(τ)} = span_{C}{v_{n} −τv_{n−1} | n = 1, 2, … } for some τ ∈ C, and infinitely many one-dimensional simple modules, L(0, τ) = V (0)/N^{(τ)}. In all other cases, M(λ) is the unique maximal submodule of V (λ), and there is a unique simple highest weight module, L(λ) = V (λ)/M(λ), of weight λ up to isomorphism.

Example. Recall that the poset L(2, 2) affords a representation of the downup algebra A(1, −1, 1). It is easy to see that the down and up operators satisfy d(\underline{0}) = 0 and du(\underline{0}) = \underline{0}, where \underline{0} = (0, 0). Thus, the element \underline{0} ∈ L(2, 2) generates a highest weight module with λ = 1. If we solve the corresponding recurrence relation in Proposition 12, we get from (i) that r_{1} = 1/2(1 + \sqrt{−3}), r_{2} = 1/2(1 − \sqrt{−3}), and λ_{n} = 1 + (r_{2}^{n} − r_{1}^{n})/(r_{2} − r_{1}). Since r_{1}^{3} = −1 = r_{2}^{3}, (and hence r_{1}, r_{2} are 6th roots of unity), we see that the sequence λ_{0} = λ, λ_{1}, λ_{2}, …, is given by 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, . . .. Thus, in the Verma module V (1), the maximal submodule M(1) = span_{C}{v_{j} | j ≥ 5}. The irreducible quotient L(1) = V (1)/M(1) is 5-dimensional, and it is spanned modulo M(1) by v_{0}, v_{1}, v_{2}, v_{3}, v_{4}. As an A(1, −1, 1)-module, L(2, 2) decomposes as L(1)⊕L(0), where we identify the copy of L(1) with the span of the vectors v_{0} = (0, 0), v_{1} = (1, 0), v_{2} = (2, 0)+(1, 1), v_{3} = 2 · (2, 1), v_{4} = 2 · (2, 2), and L(0) with the span of (2, 0) − (1, 1).

The structure of down-up algebras.

¿From a ring theoretic viewpoint, down-up algebras exhibit many interesting features. For example, it is apparent from the defining relations that the monomials u^{i}(du)^{j}d^{k}, i, j, k = 0, 1, … in a down-up algebra A = A(α, β, γ) determine a spanning set. In [BR, Thm. 3.1] we applied the Diamond Lemma (see [Be]) to prove a Poincaré-Birkhoff-Witt type result for down-up algebras. There is one essential ambiguity, (d^{2}u)u = d(du^{2}), and the result of resolving the ambiguity in the two possible ways is the same.

Theorem 14. (Poincaré-Birkhoff-Witt Theorem) Assume A = A(α, β, γ) is a down-up algebra over C. Then {u^{i}(du)^{j}d^{k} | i, j, k = 0, 1, … } is a basis of A.

The Gelfand-Kirillov dimension is a natural dimension to assign to an algebra A, and in many cases (such as when A is a domain), it provides important structural information. Theorem 14 enables us to compute the GK-dimension of any down-up algebra A = A(α, β, γ). The spaces A^{(n)} = span_{C}{u^{i}(du)^{j}d^{k} | i+2j +k ≤ n} afford a filtration (0) ⊂ A^{(0)} ⊂ A^{(1)} ⊂ ⋯ ⊂ ∪_{n}A^{(n)} = A(α, β, γ) of the down-up algebra, and A^{(m)}A^{(n)} ⊆ A^{(m+n)} since the defining relations replace the words d^{2}u and du^{2} by words of the same or lower total degree. The number of monomials u^{i}(du)^{j}d^{k} with i + 2j + k = ℓ is (m + 1)(m + 1) if ℓ = 2m and is (m + 1)(m + 2) if ℓ = 2m + 1. Thus, dimA^{(n)} is a polynomial in n with positive coefficients of degree 3, and the Gelfand-Kirillov dimension is given by

GKdim(A(α, β, γ)) = lim sup_{n→∞} log_{n}(dim A^{(n)}) = lim_{n→∞} \frac{ln ( dim A^{(n)})}{ln n} = 3.

Proposition 15. ([BR, Sec. 3]) If A(α, β, γ) has infinitely many simple Verma modules V (λ), then the intersection of the annihilators of the simple Verma modules is zero.

It follows immediately that for such a down-up algebra A(α, β, γ) the Jacobson radical, which is the intersection of the annihilators of all the simple modules, is zero.

When β = 0, then d(du − αud − γ) = 0 so that A(α, β, γ) has zero divisors for any choice of α, γ ∈ C. Necessary and sufficient conditions for A(α, β, γ) to be a domain or to be Noetherian have been proven recently using Theorem 14:

Proposition 16. ([KMP], and compare also [K2].) For a down-up algebra A = A(α, β, γ), the following are equivalent:

(i) β ≠ 0.

(ii) A is a domain.

(iii) A is right and left Noetherian.

(iv) C[du, ud] is a polynomial ring in the two variables, (i.e. du and ud are algebraically independent).

A down-up algebra A = A(α, β, γ) is Z-graded by assigning deg(d) = −1 and deg(u) = 1 and extending to all of A by setting deg(ab) = deg(a)+deg(b). Then A = ⨁_{n∈Z} A_{n} where A_{n} = {a ∈ A | deg(a) = n}, and A_{0} is a commutative subalgebra. It is shown in [BR] that if A has infinitely many simple Verma modules, the center lies in A_{0}. The center of A has been completely described in recent work ([K2] and [Z]).

References

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[BGG] J. Bernstein, I.M. Gelfand, and S.I. Gelfand, A category of 𝔤-modules, Func. Anal. Appl. 10 (1976), 87–92.

DIFFERENTIAL POSETS AND DOWN-UP ALGEBRAS 11

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Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 USA benkart@math.wisc.edu