Weight polynomial modules over the 𝔤𝔩_{2} Yangian

V. M. Futorny and A. I. Molev

Abstract

We study representations of the Yangian Y(𝔤𝔩_{2}) which admit a weight space decomposition with respect to a natural commutative subalgebra. We describe irreducible generic weight modules satisfying a polynomiality condition. We also determine the representation type of categories of modules with finitedimensional weight spaces and classify indecomposable modules in the categories of tame representation type.

Instituto de Matematica e Estatistica Universidade de São Paulo Caixa Postal 66281 CEP 05315-970 São Paulo, Brazil futorny@ime.usp.br

School of Mathematics and Statistics University of Sydney, NSW 2006, Australia alexm@maths.usyd.edu.au

1 Introduction

The Yangian for 𝔤𝔩_{n} is a unital associative algebra Y(𝔤𝔩_{n}) with countably many generators t_{ij}^{(1)}, t_{ij}^{(2)}, … where 1 ≤ i, j ≤ n, and the defining relations

(u − v) [t_{ij}(u), t_{kl}(v)] = t_{kj}(u) t_{il}(v) − t_{kj}(v) t_{il}(u),  (1.1)

where

t_{ij}(u) = δ_{ij} + t_{ij}^{(1)}u^{−1} + t_{ij}^{(2)}u^{−2} + ⋯  (1.2)

and u, v are formal variables. This algebra originally appeared in the works on the quantum inverse scattering method; see e.g. Takhtajan–Faddeev [16], Kulish– Sklyanin [12]. The term “Yangian” and generalizations of Y(𝔤𝔩_{n}) to an arbitrary simple Lie algebra were introduced by Drinfeld [7]. He then classified finite-dimensional irreducible modules over the Yangians in [8] using earlier results of Tarasov [17, 18] for the 𝔰𝔩_{2} case. An explicit construction of all such modules over Y(𝔰𝔩_{2}) is given in those papers by Tarasov and also in the work by Chari and Pressley [3]. Apart from this case, the structure of a general Yangian representation remains unknown. In the case of Y(𝔤𝔩_{n}) a description of “generic” modules was given in [13] via Gelfand–Tsetlin bases. A more general class of “tame” representations of Y(𝔤𝔩_{n}) was introduced and explicitly constructed by Nazarov and Tarasov [15]. An important role in these works is played by the Drinfeld generators [8]

a_{i}(u), i = 1, …, n,  b_{i}(u), c_{i}(u), i = 1, …, n − 1  (1.3)

of the algebra Y(𝔤𝔩_{n}) which are defined as certain quantum minors of the matrix T (u) = (t_{ij}(u)). The coefficients of the series a_{i}(u), i = 1, …, n form a commutative subalgebra of Y(𝔤𝔩_{n}) which can be regarded as an analog of a Cartan subalgebra for a semisimple Lie algebra. We shall call a representation of Y(𝔤𝔩_{n}) weight if it is a direct sum of weight spaces with respect to this subalgebra. In particular, finite-dimensional weight modules are precisely the tame modules of [15]. We, however, prefer the weight terminology to avoid an ambiguity with the tame categories discussed below in Section 6.

In this paper we are concerned with weight representations of the Yangian Y(𝔤𝔩_{2}). Recall that every irreducible finite-dimensional Y(𝔤𝔩_{2})-module contains a unique vector ξ annihilated by t_{12}(u) and which is an eigenvector for the Drinfeld generators a_{1}(u) and a_{2}(u) defined by

a_{1}(u) = t_{11}(u) t_{22}(u − 1) − t_{21}(u) t_{12}(u − 1),  a_{2}(u) = t_{22}(u);  (1.4)

see [17, 18] and [3]. Moreover, there exists an automorphism t_{ij}(u) ↦ c(u) t_{ij}(u) of Y(𝔤𝔩_{2}), where c(u) ∈ 1 + u^{−1} ℂ[[u^{−1}]], such that the eigenvalues of ξ become polynomials in u^{−1} under the twisted action of the Yangian. This prompts the introduction of the class of weight polynomial modules over Y(𝔤𝔩_{2}), i.e., such weight modules where the operators a_{1}(u) and a_{2}(u) are polynomials. More precisely, by (1.4) it is natural to require that for some positive integer p the polynomials a_{1}(u) and a_{2}(u) have degrees 2p and p, respectively. Note that a_{1}(u) is the quantum determinant of the matrix T (u) [11], [12]. Its coefficients are algebraically independent generators of the center of Y(𝔤𝔩_{2}).

We can interpret the definition of weight polynomial modules using the algebra Y_{p}(𝔤𝔩_{2}) called the Yangian of level p; see Cherednik [4, 5]. It is defined as the quotient of Y(𝔤𝔩_{2}) by the ideal generated by the elements t_{ij}^{(r)} with r ≥ p + 1. A weight polynomial module over Y(𝔤𝔩_{2}) is just a weight module over Y_{p}(𝔤𝔩_{2}) for some positive integer p.

For another interpretation consider the Yangian for 𝔰𝔩_{2} which is the subalgebra Y(𝔰𝔩_{2}) of Y(𝔤𝔩_{2}) generated by the coefficients of the series e(u), f(u) and h(u) [8] defined by

\begin{aligned}e(u) & = t_{22}(u)^{−1}t_{12}(u), \\ f(u) & = t_{21}(u) t_{22}(u)^{−1}, \\ h(u) & = t_{11}(u) t_{22}(u)^{−1} − t_{21}(u) t_{22}(u)^{−1}t_{12}(u) t_{22}(u)^{−1}. \end{aligned}  (1.5)

Note that the series h(u) can also be given by

h(u) = a_{1}(u) a_{2}(u)^{−1}a_{2}(u − 1)^{−1}  (1.6)

so that the coefficients of h(u) form a commutative subalgebra of Y(𝔰𝔩_{2}). Therefore, the restriction of a Y(𝔤𝔩_{2})-weight module to Y(𝔰𝔩_{2}) admits an eigenbasis for this subalgebra. We also point out that both the above interpretations extend to an arbitrary Yangian Y(𝔤𝔩_{n}).

In this paper we study generic weight polynomial modules over Y(𝔤𝔩_{2}). This imposes a certain integrability condition on the eigenvalues of a_{2}(u) while those of a_{1}(u) can be arbitrary; see Section 2 for the details. We consider the class of modules admitting a central character so that the coefficients of a_{1}(u) act as scalars. This class contains all irreducible weight modules. Our main results are the complete descriptions of all irreducible and indecomposable tame weight polynomial modules from this class. In particular, we show that every irreducible module is a quotient of the corresponding universal weight polynomial module. We also describe the supports (weight diagrams) of the irreducible modules.

2 Generic weight modules over Y_{p}(𝔤𝔩_{2})

Let p be a positive integer. The level p Yangian Y_{p}(𝔤𝔩_{2}) for the Lie algebra 𝔤𝔩_{2} [5] can be defined as the algebra with generators t_{ij}^{(1)}, …, t_{ij}^{(p)}, i, j = 1, 2, subject to the relations

[T_{ij}(u), T_{kl}(v)] = \frac{1}{u − v} (T_{kj}(u)T_{il}(v) − T_{kj}(v)T_{il}(u)),  (2.1)

where u, v are formal variables and

T_{ij}(u) = δ_{ij} u^{p} + ∑_{k=1}^{p} t_{ij}^{(k)} u^{p−k} ∈ Y_{p}(𝔤𝔩_{2})[u].  (2.2)

Explicitly, (2.1) reads

[t_{ij}^{(r)}, t_{kl}^{(s)}] = ∑_{a=1}^{min(r,s)} (t_{kj}^{(a−1)}t_{il}^{(r+s−a)} − t_{kj}^{(r+s−a)}t_{il}^{(a−1)}),  (2.3)

where t_{ij}^{(0)} = δ_{ij} and t_{ij}^{(r)} = 0 for r ≥ p + 1. Note that the level 1 Yangian Y_{1}(𝔤𝔩_{2}) coincides with the universal enveloping algebra U(𝔤𝔩_{2}). We have the following analog of the Poincaré–Birkhoff–Witt theorem for the algebra Y_{p}(𝔤𝔩_{2}) [5]; see also [14].

Theorem 2.1. Given an arbitrary linear ordering on the set of the generators t_{ij}^{(k)}, any element of the algebra Y_{p}(𝔤𝔩_{2}) is uniquely written as a linear combination of ordered monomials in the generators.

Denote by D(u) the quantum determinant

\begin{aligned}D(u) & = T_{11}(u)T_{22}(u − 1) − T_{21}(u)T_{12}(u − 1) \\ & = T_{11}(u − 1)T_{22}(u) − T_{12}(u − 1)T_{21}(u). \end{aligned}  (2.4)

It was shown in [4, 5] (see also [14] for a different proof) that the coefficients of the polynomial D(u) are algebraically independent generators of the center of the algebra Y_{p}(𝔤𝔩_{2}).

Let V be a Y_{p}(𝔤𝔩_{2})-module. Given two monic polynomials β = β(u) and γ = γ(u) in u of degrees p and 2p, respectively, introduce the corresponding weight subspace of V by

V_{β,γ} = {η ∈ V | T_{22}(u)η = β(u)η, D(u)η = γ(u)η}.  (2.5)

Definition 2.2. A Y_{p}(𝔤𝔩_{2})-module V is said to be weight if V is a direct sum of its weight subspaces:

V = ⊕_{β,γ} V_{β,γ}.  (2.6)

As we pointed out in the Introduction, if V is irreducible then γ(u) is the same for all nonzero weight subspaces. So if γ(u) is given we shall simply write V_{β} instead of V_{β,γ}.

From now on we shall assume that all our modules over Y_{p}(𝔤𝔩_{2}) admit a central character γ = γ(u). A standard argument [6] shows that any non-trivial submodule of a weight module V is also a weight module.

Write

β(u) = (u + β_{1}) ⋯ (u + β_{p}).  (2.7)

The collection (β_{1}, …, β_{p}) uniquely determines β(u). We shall also denote this collection by β if this does not lead to an ambiguity. We denote by Supp V the support of a weight Y_{p}(𝔤𝔩_{2})-module V, i.e., the set of all β such that the subspace V_{β} is nonzero.

We shall say that a weight Y_{p}(𝔤𝔩_{2})-module V is generic if β_{i} − β_{j} ∉ ℤ for all i ≠ j and all β = (β_{1}, …, β_{p}) ∈ Supp V.

Lemma 2.3. Let V be a generic weight Y_{p}(𝔤𝔩_{2})-module and let β = (β_{1}, …, β_{p}) ∈ Supp V. Then

T_{21}(u)V_{β} ⊆ ∑_{i=1}^{p} V_{β+δ_{i}}  and  T_{12}(u)V_{β} ⊆ ∑_{i=1}^{p} V_{β−δ_{i}}  (2.8)

where β ± δ_{i} = (β_{1}, …, β_{i} ± 1, …, β_{p}).

Proof. First we show that T_{21}(−β_{i})V_{β} ⊆ V_{β+δ_{i}} for all i = 1, …, p. Since

T_{22}(u − 1)T_{21}(u) = T_{21}(u − 1)T_{22}(u)

we have

T_{22}(−β_{i} − 1)T_{21}(−β_{i}) η = T_{21}(−β_{i} − 1)T_{22}(−β_{i}) η = 0

for all η ∈ V_{β}. Also,

\begin{aligned}T_{22}(−β_{j})T_{21}(−β_{i}) η & = (β_{i} − β_{j})^{−1}(T_{21}(−β_{i})T_{22}(−β_{j}) − T_{21}(−β_{j})T_{22}(−β_{i})) η \\ & + T_{21}(−β_{i})T_{22}(−β_{j}) η = 0 \end{aligned}

since T_{22}(−β_{k}) η = 0 for all k = 1, …, p. Using the fact that β_{i} − β_{j} ∉ ℤ we conclude that T_{21}(−β_{i})V_{β} ⊆ V_{β+δ_{i}} for all i = 1, …, p. Since T_{21}(u) is a polynomial of degree p−1 in u and β_{i} ≠ β_{j} if i ≠ j, we have that T_{21}(u)V_{β} ⊆ ∑_{i=1}^{p} V_{β+δ_{i}}. The case of T_{12}(u) is treated analogously using the identity T_{22}(u)T_{12}(u − 1) = T_{12}(u)T_{22}(u − 1).

Corollary 2.4. If V is indecomposable generic weight module over Y_{p}(𝔤𝔩_{2}) and β ∈ Supp V then Supp V ⊆ β + ℤ^{p}.

Lemma 2.5. If V is a generic weight Y_{p}(𝔤𝔩_{2})-module with central character γ(u) then for any β = (β_{1}, …, β_{p}) ∈ Supp V and any η ∈ V_{β} we have

T_{12}(−β_{r})T_{21}(−β_{s}) η = T_{21}(−β_{s})T_{12}(−β_{r}) η,

if s ≠ r, and

\begin{aligned}T_{12}(−β_{i} − 1)T_{21}(−β_{i}) η & = −γ(−β_{i}) η, \\ T_{21}(−β_{i} + 1)T_{12}(−β_{i}) η & = −γ(−β_{i} + 1) η. \end{aligned}

Proof. The first equality follows from the defining relations (1.1). The others follow from (2.4).

Corollary 2.6. Let V be a generic weight Y_{p}(𝔤𝔩_{2})-module with a central character γ(u) and let β = (β_{1}, …, β_{p}) ∈ Supp V.

(i) If γ(−β_{i}) ≠ 0 then Ker T_{21}(−β_{i}) ∩ V_{β} = 0.

(ii) If γ(−β_{i} + 1) ≠ 0 then Ker T_{12}(−β_{i}) ∩ V_{β} = 0.

(iii) If V is indecomposable and γ(−β_{i} + k) ≠ 0 for all k ∈ ℤ then

Ker T_{21}(−ψ_{i}) ∩ V_{ψ} = Ker T_{12}(−ψ_{i}) ∩ V_{ψ} = 0

for all ψ = (ψ_{1}, …, ψ_{p}) ∈ Supp V.

3 Irreducible weight Y_{p}(𝔤𝔩_{2})-modules

In this section we give a description of all irreducible generic weight Y_{p}(𝔤𝔩_{2})-modules. Let β = (β_{1}, …, β_{p}) ∈ ℂ^{p}, β_{i} − β_{j} ∉ ℤ for i ≠ j and let γ = (γ_{1}, …, γ_{2p}) ∈ ℂ^{2p}. Set

\begin{aligned}β(u) & = (u + β_{1}) ⋯ (u + β_{p}), \\ γ(u) & = (u + γ_{1}) ⋯ (u + γ_{2p}). \end{aligned}  (3.1)

Denote by R_{β,γ} the category of generic weight modules V with central character γ(u) and with Supp V ⊆ β +ℤ^{p}. Now we construct a universal module in R_{β,γ}. Let I_{β,γ} be the left ideal of Y_{p}(𝔤𝔩_{2}) generated by the coefficients of the polynomials T_{22}(u) − β(u) and D(u) − γ(u), where D(u) is defined in (2.4). Define the corresponding quotient module over Y_{p}(𝔤𝔩_{2}) by

M(β, γ) = Y_{p}(𝔤𝔩_{2})/I_{β,γ}.

It is clear from Theorem 2.1 that I_{β,γ} is a proper ideal of Y_{p}(𝔤𝔩_{2}) and so M(β, γ) is a non-trivial module. Therefore, the image of 1 in M(β, γ) is nonzero. We shall denote it by ξ. The next proposition shows the universality of the module M(β, γ).

Proposition 3.1. Let V be a weight Y_{p}(𝔤𝔩_{2})-module with a central character γ generated by a nonzero η ∈ V_{β}. Then V is a homomorphic image of M(β, γ).

Proof. Indeed, there is a homomorphism f : M(β, γ) → V which maps ξ to η. Since η generates V the statement follows.

Given (k) = (k_{1}, …, k_{p}) ∈ ℤ^{p} define the corresponding vector of the module M(β, γ) by

\begin{matrix}ξ^{(k)} = & ∏_{i, k_{i}>0} T_{21}(−β_{i} − k_{i} + 1) ⋯ T_{21}(−β_{i} − 1) T_{21}(−β_{i}) \\ × & ∏_{i, k_{i}<0} T_{12}(−β_{i} − k_{i} − 1) ⋯ T_{12}(−β_{i} + 1) T_{12}(−β_{i}) ξ. \end{matrix}

Theorem 3.2. The vectors ξ^{(k)}, (k) ∈ ℤ^{p} form a basis of M(β, γ). Moreover, we have the formulas

T_{22}(u) ξ^{(k)} = ∏_{i=1}^{p}(u + β_{i} + k_{i}) ξ^{(k)},  (3.2)

\begin{aligned}T_{21}(u) ξ^{(k)} & = ∑_{i=1}^{p} A_{i}(k) \frac{(u + β_{1} + k_{1}) ⋯ ∧_{i} ⋯ (u + β_{p} + k_{p})}{(β_{1} − β_{i} + k_{1} − k_{i}) ⋯ ∧_{i} ⋯ (β_{p} − β_{i} + k_{p} − k_{i})} ξ^{(k+δ_{i})}, \\ T_{12}(u) ξ^{(k)} & = ∑_{i=1}^{p} B_{i}(k) \frac{(u + β_{1} + k_{1}) ⋯ ∧_{i} ⋯ (u + β_{p} + k_{p})}{(β_{1} − β_{i} + k_{1} − k_{i}) ⋯ ∧_{i} ⋯ (β_{p} − β_{i} + k_{p} − k_{i})} ξ^{(k−δ_{i})}, \end{aligned} (3.3)

where

A_{i}(k) = \begin{cases}1 & if k_{i} ≥ 0 \\ −γ(−β_{i} − k_{i}) & if k_{i} < 0 \end{cases}

and

B_{i}(k) = \begin{cases}−γ(−β_{i} − k_{i} + 1) & if k_{i} > 0 \\ 1 & if k_{i} ≤ 0. \end{cases}

The action of T_{11}(u) is found from the relation

(T_{11}(u)T_{22}(u − 1) − T_{21}(u)T_{12}(u − 1)) ξ^{(k)} = γ(u) ξ^{(k)}.  (3.4)

Proof. We start by proving the formulas for the action of the generators of Y_{p}(𝔤𝔩_{2}). Relation (3.2) follows by induction from the defining relations (1.1). By Lemma 2.5 we have: if k_{i} > 0 then

\begin{aligned}T_{21}(−β_{i} − k_{i}) ξ^{(k)} & = ξ^{(k+δ_{i})}, \\ T_{12}(−β_{i} − k_{i}) ξ^{(k)} & = −γ(−β_{i} − k_{i} + 1) ξ^{(k−δ_{i})}; \end{aligned}  (3.5)

if k_{i} < 0 then

\begin{aligned}T_{12}(−β_{i} − k_{i}) ξ^{(k)} & = ξ^{(k−δ_{i})}, \\ T_{21}(−β_{i} − k_{i}) ξ^{(k)} & = −γ(−β_{i} − k_{i}) ξ^{(k+δ_{i})}; \end{aligned}  (3.6)

and if k_{i} = 0 then

\begin{aligned}T_{12}(−β_{i}) ξ^{(k)} & = ξ^{(k−δ_{i})}, \\ T_{21}(−β_{i}) ξ^{(k)} & = ξ^{(k+δ_{i})}. \end{aligned}  (3.7)

Applying the Lagrange interpolation formula we obtain the remaining formulas.

To show that the vectors ξ^{(k)} form a basis of M(β, γ), denote by T_{β} the subspace of Y_{p}(𝔤𝔩_{2}) spanned by the elements

\begin{matrix}τ^{(k)} = & ∏_{i, k_{i}>0} T_{21}(−β_{i} − k_{i} + 1) ⋯ T_{21}(−β_{i} − 1) T_{21}(−β_{i}) \\ × & ∏_{i, k_{i}<0} T_{12}(−β_{i} − k_{i} − 1) ⋯ T_{12}(−β_{i} + 1) T_{12}(−β_{i}), \end{matrix}

where (k) runs over ℤ^{p}. It suffices to prove the vector space decomposition

Y_{p}(𝔤𝔩_{2}) = T_{β} ⊕ I_{β,γ}.  (3.8)

Due to the formulas proved above, Y_{p}(𝔤𝔩_{2}) = T_{β} + I_{β,γ}. We now need to show that the vectors τ^{(k)} are linearly independent modulo the left ideal I_{β,γ}. By (3.2) and the genericity assumption, the elements τ^{(k)} mod I_{β,γ} are eigenvectors for T_{22}(u) with distinct eigenvalues. So the claim will follow if we demonstrate that each τ^{(k)} is nonzero modulo I_{β,γ}. Suppose first that γ(u) is generic: γ(−β_{i} − k) ≠ 0 for all k ∈ ℤ and all i. Then we deduce from (3.5)–(3.7) that τ^{(k)} ≠ 0 mod I_{β,γ} since 1 ≠ 0 mod I_{β,γ} which gives (3.8) for generic γ(u).

Let now γ(u) be arbitrary. Suppose that a nonzero element τ belongs to the intersection T_{β} ∩ I_{β,γ}. Then

τ = ∑_{i=1}^{p} a_{i} (t_{22}^{(i)} − β^{(i)}) + ∑_{i=1}^{2p} b_{i} (D^{(i)} − γ^{(i)}),  (3.9)

where D^{(i)}, β^{(i)} and γ^{(i)} are the coefficients of the polynomials D(u), β(u) and γ(u), respectively, while a_{i}, b_{i} ∈ Y_{p}(𝔤𝔩_{2}). Let γ̃(u) be generic. Then we can rewrite (3.9) as

τ = ∑_{i=1}^{p} a_{i} (t_{22}^{(i)} − β^{(i)}) + ∑_{i=1}^{2p} b_{i} (D^{(i)} − γ̃^{(i)}) + ∑_{i=1}^{2p} b_{i} (γ̃^{(i)} − γ^{(i)}).  (3.10)

Consider the unique decompositions of the elements b_{j} in accordance with (3.8) where γ(u) is taken to be γ̃(u):

b_{j} = τ_{j} + ∑_{i=1}^{p} a_{ij} (t_{22}^{(i)} − β^{(i)}) + ∑_{i=1}^{2p} b_{ij} (D^{(i)} − γ̃^{(i)})  (3.11)

for some a_{ij}, b_{ij} ∈ Y_{p}(𝔤𝔩_{2}). Using the decomposition (3.8) for generic γ̃(u) we must have

τ = ∑_{j=1}^{2p} τ_{j} (γ̃^{(j)} − γ^{(j)}).  (3.12)

for all such γ̃(u). This means that the T_{β}-component of each element b_{j} (γ̃^{(j)} −γ^{(j)}) is independent of γ̃(u). However, due to the formulas (3.2)–(3.4), this is only possible if all b_{j} are zero. Finally, the elements a_{i} must be zero too by the decomposition (3.9) with generic γ(u). So, (3.8) holds for all γ(u).

Given two monic polynomials α = α(u) and β = β(u) of degree p define the corresponding Verma module V (α, β) as the quotient of Y_{p}(𝔤𝔩_{2}) by the left ideal generated by the coefficients of the polynomials T_{11}(u) − α(u), T_{22}(u) − β(u) and T_{12}(u); cf. [17, 18]. Then the same argument as above shows that V (α, β) has a basis {ξ^{(k)}} parametrized by p-tuples of nonnegative integers (k) = (k_{1}, …, k_{p}). The formulas of Theorem 3.2 hold for the basis vectors ξ^{(k)}, where γ(u) should be taken to be α(u) β(u − 1) which is the central character of V (α, β). In fact, V (α, β) is isomorphic to the quotient of the corresponding universal module M(β, γ) by the submodule spanned by the vectors {ξ^{(k)}} containing at least one negative component k_{i}. Corollary 3.3. With the notation (3.1) we have

(i) The module M(β, γ) is a generic weight Y_{p}(𝔤𝔩_{2})-module with central character γ(u), Supp M(β, γ) = ℤ^{p} and all weight spaces are 1-dimensional.

(ii) The module M(β, γ) has a unique maximal submodule and hence a unique ir-

reducible quotient. Proof. Statement (i) follows immediately from Theorem 3.2. By Proposition 3.1 the sum of all proper submodules is again a proper submodule. Thus M(β, γ) has a unique maximal submodule which implies (ii).

We will denote the unique irreducible quotient of M(β, γ) by L(β, γ). It follows from Corollary 3.3 that all weight spaces of L(β, γ) are 1-dimensional. Using Proposition 3.1 we can now describe all irreducible generic weight Y_{p}(𝔤𝔩_{2})-modules. Theorem 3.4. (i) For any β = (β_{1}, …, β_{p}) ∈ ℂ^{p}, such that β_{i} − β_{j} ∉ ℤ for all

i ≠ j, and for any γ = (γ_{1}, …, γ_{2p}) ∈ ℂ^{2p} there exists an irreducible generic weight Y_{p}(𝔤𝔩_{2})-module L(β, γ) with L(β, γ)_{β} ≠ 0 and with central character γ. Moreover, dim L(β, γ)_{ψ} = 1 for all ψ ∈ Supp L(β, γ).

(ii) Any irreducible weight module over Y_{p}(𝔤𝔩_{2}) with central character γ generated by a nonzero vector of generic weight β is isomorphic to L(β, γ).

4 Category K_{β,γ}

Fix β = (β_{1}, …, β_{p}) ∈ ℂ^{p} and γ = (γ_{1}, …, γ_{2p}) ∈ ℂ^{2p} such that β_{i} − β_{j} ∉ ℤ for all i, j = 1, …, p, i ≠ j. Set

β(u) = ∏_{i=1}^{p}(u + β_{i}) and γ(u) = ∏_{i=1}^{2p}(u + γ_{i}).  (4.1)

Consider the following category K_{β,γ}: Ob (K_{β,γ}) = ℤ^{p} and the morphisms are generated by

f_{i}(k) : (k) ↦ (k + δ_{i}) and e_{i}(k) : (k) ↦ (k − δ_{i}),  (4.2)

where i = 1, …, p and (k) = (k_{1}, …, k_{p}) ∈ ℤ^{p} with the following relations:

\begin{aligned}f_{j}(k + δ_{i}) f_{i}(k) & = f_{i}(k + δ_{j}) f_{j}(k), \\ e_{j}(k − δ_{i}) e_{i}(k) & = e_{i}(k − δ_{j}) e_{j}(k), \\ e_{i}(k + δ_{j}) f_{j}(k) & = f_{j}(k − δ_{i}) e_{i}(k) for i ≠ j, \\ e_{i}(k + δ_{i}) f_{i}(k) & = −γ(−β_{i} − k_{i}) 1_{(k)}, \\ f_{i}(k − δ_{i}) e_{i}(k) & = −γ(−β_{i} − k_{i} + 1) 1_{(k)}. \end{aligned}

It follows immediately from Lemmas 2.3 and 2.5 that any module in the category R_{β,γ} can naturally be viewed as a module over the category K_{β,γ}. Hence there exists a monic functor F : R_{β,γ} → K_{β,γ}−mod. Consider the cyclic subalgebra C_{β,γ}(a) = Hom_{K_{β,γ}}(a, a) for any a ∈ ℤ^{p}. Clearly, C_{β,γ}(a) ≃ ℂ for any a ∈ ℤ^{p} due to the defining relations of K_{β,γ}. For any a = (k_{1}, …, k_{p}) ∈ ℤ^{p} we can construct a universal module M(β, γ, a) ∈ K_{β,γ}−mod. Consider ℂ as a C_{β,γ}(a)-module with

\begin{aligned}e_{i}(k + δ_{i}) f_{i}(k) 1 & = −γ(−β_{i} − k_{i}), \\ f_{i}(k − δ_{i}) e_{i}(k) 1 & = −γ(−β_{i} − k_{i} + 1). \end{aligned}

Let A_{β,γ,a} be an algebra of paths in K_{β,γ} originating in a. Now construct a ℤ^{p}-graded K_{β,γ}-module

M(β, γ, a) = A_{β,γ,a} ⊗_{C_{β,γ}(a)} ℂ.

Clearly, all graded components of M(β, γ, a) are 1-dimensional and M(β, γ, a)_{a} = 1_{a} ⊗ℂ. A module M(β, γ, a) contains a unique maximal ℤ^{p}-graded submodule which intersects M(β, γ, a)_{a} trivially and hence has a unique irreducible quotient L(β, γ, a) with L(β, γ, a)_{a} ≃ ℂ and dim L(β, γ, a)_{b} ≤ 1 for all b ∈ ℤ^{p}. If V is another irreducible K_{β,γ}-module with V_{a} ≠ 0 then there exists a non-trivial C_{β,γ}(a)-homomorphism from ℂ to V_{a} which can be extended to an epimorphism from M(β, γ, a) to V. Since V is irreducible we conclude that V ≃ L(β, γ, a).

Obviously, we can view M(β, γ) as a module over the category K_{β,γ} with a natural action of the morphisms of K_{β,γ} and F (M(β, γ)) = M(β, γ, β). Thus a K_{β,γ}-module M(β, γ, β) can be extended to a Y_{p}(𝔤𝔩_{2})-module M(β, γ). Moreover, the functor F preserves the submodule structure of M(β, γ). In particular, F (L(β, γ)) = L(β, γ, β).

Proposition 4.1. Categories K_{β,γ}−mod and R_{β,γ} are equivalent.

Proof. We already have a monic functor F : R_{β,γ} → K_{β,γ}−mod. Suppose that V ∈ K_{β,γ}−mod. We want to show that V can be extended to a Y_{p}(𝔤𝔩_{2})-module. Fix v ∈ V_{(k)} \ {0}. Let W ⊆ V be a submodule generated by v. Then W_{(k)} = ℂv and there is an epimorphism from M(β, γ, a) to W, where a = (k_{1}, …, k_{p}), which maps 1_{a} ⊗ 1 to v. Since F (M(β + a, γ)) = M(β, γ, a), β + a = (β_{1} + a_{1}, …, β_{p} + a_{p}), then W can be extended to a corresponding quotient of M(β + a, γ). We conclude that V can be extended to a Y_{p}(𝔤𝔩_{2})-module and will denote that module by G(V). Clearly, G defines a functor from K_{β,γ}−mod to R_{β,γ} (action on morphisms is obvious). One can easily see that the functors F and G define an equivalence between the categories K_{β,γ}−mod and R_{β,γ}.

5 Support of irreducible generic weight modules

To complete the classification of irreducible modules we have to know when two irreducible modules L(β, γ) and L(β′, γ) are isomorphic. For that we need to describe the support Supp L(β, γ).

We shall say that the weight subspaces M(β, γ)_{ψ} and M(β, γ)_{ψ+δ_{i}} are strongly isomorphic if γ(−ψ_{i}) ≠ 0 where ψ = (ψ_{1}, …, ψ_{p}). This implies

f_{i}(ψ_{1}, …, ψ_{p}) M(β, γ)_{ψ} ≠ 0 and e_{i}(ψ_{1}, …, ψ_{i} + 1, …, ψ_{p}) M(β, γ)_{ψ+δ_{i}} ≠ 0.

The statement below follows immediately from the relations in K_{β,γ} (cf. also Corollary 2.6).

Lemma 5.1. If M(β, γ)_{ψ} and M(β, γ)_{ψ+δ_{i}} are strongly isomorphic then M(β, γ)_{ψ±δ_{j}} and M(β, γ)_{ψ+δ_{i}±δ_{j}} are strongly isomorphic for all i, j = 1, …, p, i ≠ j. Moreover, if

f_{i}(ψ_{1}, …, ψ_{p}) M(β, γ)_{ψ} = 0  or  e_{i}(ψ_{1}, …, ψ_{p}) M(β, γ)_{ψ} = 0

then

\begin{aligned}f_{i}(ψ_{1}, …, ψ_{j} ± 1, …, ψ_{p}) M(β, γ)_{ψ±δ_{j}} & = 0 or \\ e_{i}(ψ_{1}, …, ψ_{j} + 1, …, ψ_{p}) M(β, γ)_{ψ±δ_{j}} & = 0, \end{aligned}

respectively, for all j ≠ i.

Let a_{i}, b_{i} ∈ ℤ ∪ {±∞}, a_{i} ≤ b_{i}, i ∈ {1, …, p}. Denote

P (a_{1}, …, a_{p}, b_{1}, …, b_{p}) = {(x_{1}, …, x_{p}) ∈ ℤ^{p} | a_{i} ≤ x_{i} ≤ b_{i}, i = 1, …, p},

a parallelepiped in ℤ^{p}. Note that some faces of the parallelepiped can be infinite in some directions. In particular, in the case a_{i} = −∞, b_{i} = ∞ for all i, the parallelepiped coincides with ℤ^{p}.

Theorem 5.2. For any irreducible weight module L(β, γ) over Y_{p}(𝔤𝔩_{2}) there exist elements a_{i}, b_{i} ∈ ℤ ∪ {±∞}, a_{i} ≤ b_{i}, i ∈ {1, …, p} such that

Supp L(β, γ) = P (a_{1}, …, a_{p}, b_{1}, …, b_{p}).

Proof. Fix i ∈ {1, …, p}. If γ(−β_{i} + k) ≠ 0 for all k ∈ ℤ then (k_{1}, …, k_{i} + m, …, k_{p}) ∈ Supp L(β, γ) as soon as (k_{1}, …, k_{p}) ∈ Supp L(β, γ). This follows immediately from Lemma 5.1. In this case we set a_{i} = −∞ and b_{i} = ∞. Let now γ(−β_{i} + k) = 0 for some k ∈ ℤ. Let m ≥ 0 be the smallest integer (if exists) such that γ(−β_{i} − m) = 0 and let n ≤ 0 be the largest integer (if exists) such that γ(−β_{i} − n + 1) = 0. It follows from Lemma 5.1 that

Supp L(β, γ) ∩ {β + kδ_{i} | k ∈ ℤ} = {β + nδ_{i}, …, β, …, β + mδ_{i}}.

If β + sδ_{j} ∈ Supp L(β, γ), j ≠ i then

Supp L(β, γ)∩{β +sδ_{j} +kδ_{i} | k ∈ ℤ} = {β +sδ_{j} +nδ_{i}, …, β +sδ_{j}, …, β +sδ_{j} +mδ_{i}}.

In this case we set a_{i} = β_{i} + n and b_{i} = β_{i} + m. The statement of the theorem now follows.

Example 5.3. Suppose that p = 2. Then we have the following possibilities.

(i) Supp L(β, γ) = ℤ^{2} if and only if γ(−β_{i} + k) ≠ 0 for all i = 1, …, p and all k ∈ ℤ.

(ii) Let γ(−β_{1}+k) ≠ 0 for all k ∈ ℤ, γ(−β_{2}−m) = 0, m ≥ 0 and let γ(−β_{2}−n) ≠ 0 for all integer n ≤ m. Then Supp L(β, γ) = {β + kδ_{1} + nδ_{2} | k, n ∈ ℤ, n ≤ m}. Hence the support in this case is a half-plane.

(iii) Let γ(−β_{1}+k) ≠ 0 for all k ∈ ℤ, γ(−β_{2}−m+1) = 0, m ≤ 0 and let γ(−β_{2}−n) ≠ 0 for all integer n ≥ m. Then Supp L(β, γ) = {β +kδ_{1} +nδ_{2} | k, n ∈ ℤ, n ≥ m}.

(iv) Let γ(−β_{1} + k) ≠ 0 for all k ∈ ℤ, γ(−β_{2} − m) = 0, m ≥ 0, γ(−β_{2} − n + 1) = 0, n ≤ 0 and let γ(−β_{2} − s) ≠ 0 for all integer n ≤ s ≤ m. Then Supp L(β, γ) = {β + kδ_{1} + sδ_{2} | k, s ∈ ℤ, n ≤ s ≤ m}. Hence the support in this case is a lane of width m − n.

(v) Let γ(−β_{i} − m_{i}) = 0 for some m_{i} ≥ 0, γ(−β_{i} − n_{i} + 1) = 0 for some n_{i} ≤ 0 and let γ(−β_{i} − s_{i}) ≠ 0 for all n_{i} ≤ s_{i} ≤ m_{i}, i = 1, 2. Then Supp L(β, γ) = {β + s_{1}δ_{1} + s_{2}δ_{2} | s_{i} ∈ ℤ, n_{i} ≤ s_{i} ≤ m_{i}, i = 1, 2}.

(vi) Let γ(−β_{i} − m_{i}) = 0, m_{i} ≥ 0 and let γ(−β_{i} − k_{i}) ≠ 0 for all integer k_{i} ≤ m_{i}, i = 1, 2. Then Supp L(β, γ) = {β + k_{1}δ_{1} + k_{2}δ_{2} | k_{i} ∈ ℤ, k_{i} ≤ m_{i}}.

(vii) Let γ(−β_{i} − m_{i} + 1) = 0, m_{i} ≤ 0 and let γ(−β_{i} − k_{i}) ≠ 0 for all integer k_{i} ≥ m_{i}, i = 1, 2. Then Supp L(β, γ) = {β + k_{1}δ_{1} + k_{2}δ_{2} | k_{i} ∈ ℤ, k_{i} ≥ m_{i}}.

(viii) Let γ(−β_{1} −m_{1}) = 0, m_{1} ≥ 0, γ(−β_{2} −m_{2} +1) = 0, m_{2} ≤ 0, γ(−β_{i} −k_{i}) ≠ 0 for all integer k_{1} ≤ m_{1} and k_{2} ≥ m_{2}. Then Supp L(β, γ) = {β + k_{1}δ_{1} + k_{2}δ_{2} | k_{i} ∈ ℤ, k_{1} ≤ m_{1}, k_{2} ≥ m_{2}}.

(ix) Interchange β_{1} and β_{2} in (ii), (iii), (iv) and (viii).

6 Indecomposable generic weight modules

A full subcategory S ⊆ K_{β,γ} is called a skeleton of K_{β,γ} provided the objects of S are pairwise non-isomorphic and any object of K_{β,γ} is isomorphic to some object of S. In this case the categories of K_{β,γ}−mod and S−mod are equivalent.

For each i ∈ {1, …, p} consider a set I_{i} = {k ∈ ℤ | γ(−β_{i} − k) = 0}. Define a category S_{β,γ} as a ℂ-category with the set of objects

S_{0} = {0, …, |I_{1}|} × … × {0, …, |I_{p}|}

and with morphisms generated by

\begin{matrix}α_{(i_{1},...,i_{p})}^{k} : (i_{1}, …, i_{p}) & ↦ (i_{1}, …, i_{k} + 1, …, i_{p}), \\ β_{(j_{1},...,j_{p})}^{k} : (j_{1}, …, j_{p}) & ↦ (j_{1}, …, j_{k} − 1, …, j_{p}), \end{matrix}

where k ∈ {1, …, p} is such that I_{k} ≠ ∅, i_{k} < |I_{k}|, j_{k} > 0, subject to the relations:

β_{(i_{1},...,i_{k}+1,...,i_{p})}^{k}α_{(i_{1},...,i_{p})}^{k} = α_{(i_{1},...,i_{p})}^{k}β_{(i_{1},...,i_{k}+1,...,i_{p})}^{k} = 0

and

x_{(a_{1},...,a_{p})}^{k}y_{(b_{1},...,b_{p})}^{r} = y_{(c_{1},...,c_{p})}^{r}x_{(b_{1},...,b_{p})}^{k}

for all k ≠ r and all possible x, y ∈ {α, β}, a_{i}, b_{i}, c_{i}, 1 ≤ i ≤ p for which this equality makes sense.

It follows from the construction that S_{β,γ} is the skeleton of the category K_{β,γ}. Note that the corresponding algebra is finite-dimensional. In particular, S_{β,γ} is semisimple when I_{k} = ∅ for all 1 ≤ k ≤ p, i.e. when γ(−β_{k} + r) ≠ 0 for all k ∈ ℤ and all i = 1, …, p. Hence it is enough to describe all indecomposable modules over S_{β,γ}.

Fix a ∈ S_{0} and define a simple S_{β,γ}-module S_{a} such that S_{a}(b) = δ_{a,b}ℂ for all b ∈ S_{0} and all morphisms are trivial. Since S_{β,γ} defines a finite-dimensional algebra we have the following

Proposition 6.1. Any simple module over S_{β,γ} is isomorphic to S_{a} for some a ∈ S_{0}.

This is another confirmation of the fact that all weight spaces in any irreducible generic weight Y_{p}(𝔤𝔩_{2})-module are 1-dimensional. But this need not to be the case for indecomposable modules. We restrict ourselves to a full subcategory R_{β,γ}^{f} ⊆ R_{β,γ} which consists of weight modules V with dim V_{ψ} < ∞ for all ψ ∈ Supp V.

We shall employ the following definition from [9].

Definition 6.2. (i) A category C is called wild if there exists a C × ℂ ⟨x, y⟩- bimodule M, free and of finite rank as right ℂ ⟨x, y⟩-module, and such that the functor M ⊗_{ℂ⟨x,y⟩} {·} preserves indecomposability and isomorphism classes.

(ii) C is called tame if for each dimension vector d there is a finite number of C × ℂ[x]-bimodules C_{1}, ⋯, C_{n} such that each C_{j} is free and of finite rank as right ℂ[x]-module and such that every indecomposable X ∈ C − mod with dim X(i) = d_{i} is isomorphic to C_{j} ⊗_{ℂ[x]} S for some j and some simple ℂ[x]- module S.

We will also say that a category is of finite type if it has only finitely many non-isomorphic indecomposable modules.

To establish the representation type of the category R_{β,γ}^{f} we need to find the representation type of the category S_{β,γ}−mod^{f} due to the category equivalence, where S_{β,γ}−mod^{f} denotes a category of modules over the category S_{β,γ} with finitedimensional weight spaces. Denote X_{β,γ} = {k ∈ {1, …, p} | I_{k} ≠ ∅}.

6.1 Indecomposable modules in the case |X_{β,γ}| = 1

In this section we describe all indecomposable modules over S_{β,γ} in the case |X_{β,γ}| = 1. Let X_{β,γ} = {i} and let |I_{i}| = r > 0. In this case the category S_{β,γ} has the following quiver A with relations:

[figure: a_{1}; b_{1}; 1; 2; a_{r}; b_{r}; r + 1; ∘; a_{i} b_{i} = b_{i} a_{i} = 0; r; ∘; ∘; ∘ …]

We denote by S_{i}, i ∈ {1, …, r +1}, the simple module corresponding to the point i. These modules correspond to all irreducible modules in R_{β,γ}^{f} by Proposition 6.1.

Now describe remaining indecomposable modules for a quiver above. Fix integers 1 ≤ k_{1} < k_{2} ≤ r + 1 and a function ξ_{k_{1},k_{2}} : {k_{1}, k_{1} + 1, …, k_{2}} → {0, 1}. Define a module M = M(k_{1}, k_{2}, ξ_{k_{1},k_{2}}) as follows: M(i) = ℂe_{i}, k_{1} ≤ i ≤ k_{2}, M(j) = 0 otherwise, a_{i}e_{i} = e_{i+1}, b_{i}e_{i+1} = 0 if ξ_{k_{1},k_{2}}(i) = 1 and a_{i}e_{i} = 0, b_{i}e_{i+1} = e_{i} if ξ_{k_{1},k_{2}}(i) = 0 for all 1 ≤ i < k_{2}.

The proof of the following proposition is standard; see e.g. [10].

Proposition 6.3. The modules S_{i}, 1 ≤ i ≤ r + 1 and M(k_{1}, k_{2}, ξ_{k_{1},k_{2}}) with 1 ≤ k_{1} < k_{2} ≤ r + 1 and

ξ_{k_{1},k_{2}} : {k_{1}, k_{1} + 1, …, k_{2}} → {0, 1},

exhaust all non-isomorphic indecomposable modules for A.

6.2 Indecomposable modules in the case |X| = 2

In this section we describe the indecomposable modules for S_{β,γ} when |X| = 2 and |I_{k}| = 1 for each k ∈ X. Then S_{β,γ} is isomorphic to the following category B considered in [2, Section 6.1.2]:

[figure: B :; a_{2}; b_{2}; b_{0}; a_{0}; a_{1}; b_{1}; b_{3}; a_{3}; 1; 2; 0; 3; a_{i}b_{i} = b_{i}a_{i} = 0,  i = 0, …, 3,; a_{i}a_{j} = b_{l}b_{m} for any i, j, l, m ∈ {0, 1, 2, 3},; where possible; ∘; ∘; ∘; ∘]

We will consider the objects 0, 1, 2, 3 as the elements of ℤ_{4} = ℤ/4ℤ. We have four simple modules S_{i}, 0 ≤ i ≤ 3, with a support in a chosen point i by Proposition 6.1.

Fix an 0 ≤ i ≤ 3 and define the B-module M_{i} such that M_{i}(j) = ℂe_{j} for each j = 0, …, 3 and a_{i}e_{i} = e_{i+1}, a_{i+1}e_{i+1} = e_{i+2}, b_{i−1}e_{i} = e_{i−2}, b_{i−2}e_{i−1} = e_{i−2} and u_{j}e_{k} = 0 for all other cases of u ∈ {a, b} and j, k = 0, …, 3. Obviously, M_{i} is indecomposable module for any i.

Next we define a discrete family of indecomposable B-modules. Let n ∈ ℕ, n > 1, and j ∈ ℤ_{4}. Define a B-module M_{n,j,1} (resp., M_{n,j,2}) as follows. Consider n elements e_{1}, …, e_{n}. A ℂ-basis of the vector space M_{n,j,1}(l) (resp., M_{n,j,2}(l)) is the set of e_{k} such that j + k − 1 ≡ l(mod 4). The elements a_{l} and b_{l−1} act as follows:

a_{l}e_{k} = \{ \begin{matrix}e_{k+1}, & if l is even (resp., odd), k < n and j + k − 1 ≡ l(mod 4); \\ 0, & otherwise. \end{matrix}

b_{l−1}e_{k} = \{ \begin{matrix}e_{k−1}, & if l is even (resp., odd), k > 1 and j + k − 1 ≡ l(mod 4); \\ 0, & otherwise. \end{matrix}

All modules M_{n,j,1} and M_{n,j,2}, n > 1, 0 ≤ j ≤ 3 are non-isomorphic indecomposable B-modules.

Finally, we construct a continuous family of indecomposable B-modules. For each λ ∈ ℂ and d ∈ ℤ, d > 0 define the B-modules M_{d,λ,1} and M_{d,λ,2} as follows. Set

\begin{aligned}M_{d,λ,1}(i) & = ℂ^{d}, \\ M_{d,λ,1}(a_{0}) & = M_{d,λ,1}(a_{2}) = M_{d,λ,1}(b_{1}) = I_{d}, \\ M_{d,λ,1}(b_{0}) & = M_{d,λ,1}(b_{2}) = M_{d,λ,1}(a_{1}) = M_{d,λ,1}(a_{3}) = 0, \\ M_{d,λ,1}(b_{3}) & = J_{d,λ} \end{aligned}

and

\begin{aligned}M_{d,λ,2}(i) & = ℂ^{d}, \\ M_{d,λ,2}(b_{0}) & = M_{d,λ,2}(b_{2}) = M_{d,λ,2}(a_{1}) = I_{d}, \\ M_{d,λ,2}(a_{0}) & = M_{d,λ,2}(a_{2}) = M_{d,λ,2}(b_{1}) = M_{d,λ,2}(b_{3}) = 0, \\ M_{d,λ,2}(a_{3}) & = J_{d,λ}, \end{aligned}

where J_{d,λ} is the Jordan cell of dimension d with the eigenvalue λ.

All modules M_{d,λ,k}, k = 1, 2 are indecomposable and corresponding indecomposable modules in R_{β,γ}^{f} have all weight spaces of dimension d.

The next result follows from Proposition 3.3.1 in [1] (cf. [2], Proposition 6).

Proposition 6.4. The modules S_{i}, M_{i}, M_{n,i,1}, M_{n,i,2}, M_{d,λ,1}, M_{d,λ,2} where 0 ≤ i ≤ 3, d is a positive integer, λ ∈ ℂ, λ ≠ 0, and n ≥ 2 is an integer, constitute an exhaustive list of pairwise non-isomorphic indecomposable B-modules.

6.3 Representation type

The main result in this section is the following theorem which describes the representation type of R_{β,γ}^{f}.

Theorem 6.5. (i) If |X_{β,γ}| = 0 then R_{β,γ}^{f} is a semisimple category with a unique indecomposable (=irreducible) module;

(ii) If |X_{β,γ}| = 1 then R_{β,γ}^{f} has finite representation type;

(iii) If |X_{β,γ}| = 2 then R_{β,γ}^{f} has tame representation type if and only if |I_{k}| = 1 for all k ∈ X. Otherwise, R_{β,γ}^{f} has wild representation type;

(iv) If |X_{β,γ}| > 2 then R_{β,γ}^{f} has wild representation type.

Proof. In the case when |X_{β,γ}| = 1 all indecomposable modules for S_{β,γ} are described in Proposition 6.3. Hence R_{β,γ}^{f} has finite representation type. If |X_{β,γ}| = 2 and |I_{k}| = 1 for each k ∈ X then all indecomposable modules for S_{β,γ} are described in Proposition 6.4. It follows from the definition that R_{β,γ}^{f} has tame representation type in this case. If |I_{k}| > 1 for at least one k then it is easy to construct a family of indecomposable modules that depends on two continuous parameters. Hence, in this case R_{β,γ}^{f} has wild representation type. Suppose now that |X_{β,γ}| > 2. Then S_{β,γ} contains a full subcategory of wild representation type considered in [1], Theorem 1. We immediately conclude that R_{β,γ}^{f} has wild representation type. This completes the proof.

Acknowledgment

The first author is grateful to the University of Sydney for support and hospitality.

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