The nonstandard quantum plane

Ivo Herzog^{∗} Department of Mathematics The Ohio State University at Lima Lima, OH 45804 USA

           herzog.23@osu.edu

Sonia L’Innocente Department of Mathematics and Computer Science University of Camerino Camerino, Italy

                                      sonia.linnocente@unicam.it

1 Introduction

Let k be a field of char(k) ≠ 2, and suppose that q ∈ k is not a root of unity. The associated quantum plane [5, §IV.1], denoted by k_{q}[x, y], is defined to be the free k-algebra k{x, y} generated by x and y, modulo the relation yx = qxy. The set of monomials {x^{i}y^{j}}_{i, j≥0} is a basis for the underlying k-vector space, and for every pair (i, j) of nonnegative integers, we have

y^{j}x^{i} = q^{ij}x^{i}y^{j}.

There is a natural action on the quantum plane by the quantum group U_{q}, which is defined to be the k-algebra generated by the four variables E, F, K, K^{−1}, modulo the relations:

KK^{−1} = K^{−1}K = 1 ,
KEK^{−1} = q^{2}E ,  KF K^{−1} = q^{−2}F ,  (1)
EF − F E = \frac{K − K^{−1}}{q − q^{−1}} .

Indeed, the quantum plane k_{q}[x, y] acquires the structure of a left U_{q}-module where the action of the generators is given by

Kx^{i}y^{j} = q^{i−j}x^{i}y^{j}, Ex^{i}y^{j} = [i] x^{i−1}y^{j+1}, F x^{i}y^{j} = [j] x^{i+1}y^{j−1} ,  (2)

and extended linearly; the coefficients are given by [a] := \frac{q^{a} − q^{−a}}{q − q^{−1}} . To appreciate the

significance of the quantum plane and the representation theory of U_{q}, consult [4, 5]. This article is devoted to the model-theoretic study of the quantum plane, regarded as a U_{q}- module. One consequence of the main result Theorem 1.1 is Theorem 5.1, which states that, in the language of left U_{q}-modules, the ring of definable scalars of the quantum plane is a von Neumann regular epimorphic ring extension of the quantum group U_{q}.

The action of U_{q} given by (2) preserves the total degree i + j of the monomial cx^{i}y^{j}, c ∈ k, so the quantum plane k_{q}[x, y] decomposes as a U_{q}-module into a direct sum

k_{q}[x, y] = ⨁_{n≥0} k_{q}[x, y]_{n},

∗The first author was partially supported by INDAM-GNSAGA and the NSF.

where k_{q}[x, y]_{n} denotes the k-vector space of all homogenous elements in the quantum plane of degree n. Each of the components k_{q}[x, y]_{n} is a simple module, i.e., irreducible representation, whose dimension over k is n + 1. In general, a simple U_{q}-module V is called finite dimensional if the dimension of V as a k-vector space is finite. Every finite dimensional representation of U_{q} admits a decomposition as a direct sum of simple modules, and for every whole number n there exist (up to isomorphism) exactly two simple representations of dimension n + 1, denoted V_{+, n} and V_{−, n} [4, Thm 2.6]. In this regard, the representation theory of the quantum group U_{q} resembles the representation theory of the Lie algebra sl_{2}(k) where k is an algebraically closed field of characteristic 0. The principal aim of this paper is to prove for U_{q} the quantum analogues of results obtained by the first author [3] for the universal enveloping algebra U(L) of L = sl_{2}(k).

The simple representation V_{+, n} is isomorphic to k_{q}[x, y]_{n}. The other simple representation V_{−, n} of dimension n + 1 is obtained by composing the action of U_{q} on V_{+, n} with the automorphism σ (see [4, §5.2]) of U_{q} determined by

σ(E) = −E, σ(F) = F, σ(K) = −K.

We will also refer to the module V_{−, n} as k_{q}^{σ}[x, y]_{n}; and to k_{q}^{σ}[x, y] as the direct sum of one copy of each k_{q}^{σ}[x, y]_{n}, n ≥ 0. This module k_{q}^{σ}[x, y] is the U_{q}-module obtained by composing the action of U_{q} on the quantum plane with the automorphism σ. Throughout the paper, we will denote by M the module

M = k_{q}[x, y] ⊕ k_{q}^{σ}[x, y],

obtained by taking the direct sum of one copy of each simple representation of U_{q}, up to isomorphism. Rather than working with the quantum plane directly, it is easier to prove the following theorem for M, and then specialize to the quantum plane in the last section of the paper. To prove the theorem, we follow the general strategy used in [3]. It is shown in [6] that a large portion of this procedure can be carried out effectively.

Theorem 1.1 The lattice Latt(M) of pp-definable subspaces of M is complemented.

The third section of this article is devoted to an account of this general strategy, while the second and fourth sections describe how this strategy needs to be amended to suit the case of a quantum group.

The usual model-theoretic language L(U_{q}) for U_{q}-modules has symbols with which to express addition (in a module) and scalar multiplication (of each element of U_{q} on the module), as well as a constant symbol 0 for the zero element of the module. So the basic (atomic) formulas are just linear equations r_{1}u_{1} + … + r_{n}u_{n} ≐ 0 with scalars from U_{q} acting on the left. A system of linear equations can be expressed by a finite conjuction of linear equations, which will be abbreviated as (A, B)\binom{u}{v} ≐ 0, where u denotes the tuple (u_{1}, …, u_{n}), v the tuple (v_{1}, …, v_{k}), A denotes an m × n matrix and B an m × k matrix with entries from U_{q}. A pp (“positive primitive”) formula is obtained by existentially quantifying a system of linear equations, say over the variable v. Formally, it has the shape

φ(u) = ∃v (A, B) ( \begin{matrix}u \\ v \end{matrix} ) ≐ 0

If V is a U_{q}-module, the set of the solutions in V to the formula φ(v) is a k-subspace of V^{n}. If φ(v) is a pp-formula in one free variable v, then this solution set of φ in V is denoted φ(V) and is a typical pp-definable subspace of V. The collection of pp-definable subspaces of V has the structure of a modular lattice (with respect to inclusion), which is the subject of Theorem 1.1.

Let φ(v) be a pp-formula in one free variable and consider a finite dimensional simple representation V_{ϵ, n} of U_{q}, where ϵ is + or −. By virtue of the fact that V_{ϵ, n} is finite dimensional, it is easy to find a pp-formula ψ_{ϵ, n}(v) such that

φ(V_{ϵ, n}) ⊕_{k} ψ_{ϵ, n}(V_{ϵ, n}) = V_{ϵ, n}.

Theorem 1.1 states that a complementary pp-formula ψ(v) may be found, which is independent of ϵ and n.

Let V be a representation of U_{q}. A definable scalar of V is a k-linear transformation ρ_{V} : V → V, whose graph is definable in V by a pp-formula ρ(u_{1}, u_{2}) in two variables,

V ⊧ ∀u_{1}∃!u_{2} ρ(u_{1}, u_{2}).

The collection of definable scalars of V has the structure of a ring, denoted by U_{V}; the operations in U_{V} are composition and pointwise addition. There is a canonical morphism from the ring U_{q} to the ring of definable scalars U_{V} , which sends the element r to its action on V, defined by the pp-formula

u_{2} = ru_{1}.

It follows from a general fact [3, Prop 7] that if the lattice of pp-definable subspaces of the U_{q}-module V is complemented, then the ring U_{V} of definable scalars of V is von Neumann regular and that the canonical ring morphism is an epimorphism. Because the quantum plane k_{q}[x, y] is a direct summand of M, its lattice of pp-definable subspaces is also complemented. The ring of definable scalars of k_{q}[x, y] is obtained as the quotient ring of the definable scalars of M, modulo the ideal of definable scalars that vanish in k_{q}^{σ}[x, y]. As the quantum plane is a faithful U_{q}-module, we may identify the quantum group U_{q} with a subring of its ring of definable scalars.

In general, a left module N over an associative ring R is called pure injective or algebraically compact if it is pp-saturated: every pp-type p^{+}(v, A) (consisting only of ppformulas) over A ⊆ N which is consistent in N has a realization in N. The Ziegler spectrum of R is a topological space Zg(R) whose points are the pure-injective indecomposable left R-modules; a basis of open subsets is indexed by ordered pairs φ(v), ψ(v) of pp-formulas in one variable:

O_{φ, ψ} = { U ∈ Zg(R) : U ⊧ ∃v (φ(v) ∧ ¬ψ(v)) }.

If V a finite dimensional representation of U_{q}, then it is a pure-injective U_{q}-module and because each V_{ϵ,n} is simple, it is an indecomposable pure-injective U_{q}-module and therefore a point in Zg(U_{q}).

If N is a left R-module, then the closed subset of N in Zg(R) is defined to be

Cℓ(N) := ⋂_{N⊧φ→ψ}(O_{φ, ψ})^{c}.

For example, the closed subset of the quantum plane k_{q}[x, y] in Zg(U_{q}) is the closure of the points k_{q}[x, y]_{n} = V_{+, n}, while the closed subset of

M = k_{q}[x, y] ⊕ k_{q}^{σ}[x, y]

is the closure of all the simple finite dimensional representations V_{ϵ,n} where ϵ = ±. It is shown in this article that all but one of the points of the closed set Cℓ(k_{q}[x, y]) associated to the quamtum plane is pseudo finite. These points represent nonstandard homogeneous components of the quantum plane.

2 The lattice of K-invariant pp-definable subspaces

A pp-definable subspace φ(V) of a U_{q}-module V is called K-invariant if Kφ(V) ⊆ φ(V). Thus a pp-definable subspace of M = k_{q}[x, y] ⊕ k_{q}^{σ}[x, y] is K-invariant if for every simple representation V_{ϵ,n},

Kφ(V_{ϵ,n}) ⊆ φ(V_{ϵ,n}).

Since the vector space φ(V_{ϵ,n}) is finite dimensional, and K is an invertible element of U_{q}, we have that Kφ(V_{ϵ,n}) = φ(V_{ϵ,n}), and hence that Kφ(M) = φ(M). The collection of K-invariant pp-definable subspacess of M forms a sublattice of the pp-definable subspaces of M. In this section, we will prove that this sublattice is complemented.

According to Equation (2), the simple representation V_{+,n} ≅ k_{q}[x, y]_{n} has a basis of K-eigenvectors given by the monomials x^{n−i}y^{i}. These monomials also form a basis of Keigenvectors for the representation k_{q}^{σ}[x, y]_{n}. The 1-dimensional vector spaces generated by these monomials are called the weight spaces of k_{q}[x, y] (resp., k_{q}^{σ}[x, y]_{n}). If the ppdefinable subspace φ(V_{ϵ,n}) is a direct sum of weight spaces, for every V_{ϵ,n}, then the ppdefinable subspace φ(M) is clearly K-invariant. Conversely, if φ(M) is a K-invariant ppdefinable subspace, then we may multiply each of the φ(V_{ϵ,n}) by various of the K − ϵq^{n−2i} to see that it must be a direct sum of weight spaces.

Let U_{q}^{opp} denote the opposite ring of U_{q}, defined by reversing the multiplication in U_{q}. The formulas in the language L(U_{q}^{opp}) of left U_{q}^{opp}-modules, or, equivalently, of the right U_{q}-modules, are expressed with the scalars from U_{q} acting on the right. Given a pp-formula φ(u) = ∃v (A, B) \binom{u}{v} ≐ 0 (with u = (u_{1}, …, u_{n}) and v = (v_{1}, …, v_{k})) of the language L(U_{q}), we may associate to it as in [8] a pp-formula φ^{∗}(u) of L(U_{q}^{opp}) in the same variable u, called dual of φ(u):

φ^{∗}(u) = ∃w (u, w) ( \begin{matrix}I_{n} & 0 \\ A & B \end{matrix} ) ≐ 0,

where I_{n} denotes the n × n identity matrix. If V is a left U_{q}-module, then the space V^{∗} := Hom_{k}(V, k) of functionals acquires the structure of a right U_{q}-module, given by (ηr)(v) = η(rv), for every r ∈ U_{q}. If φ(V) is a pp-definable subspace of V, then φ^{∗}(V^{∗}) is the subspace of V^{∗} consisting of functionals that vanish on φ(V). This association yields an anti-isomorphism of the lattice of pp-definable subspaces of V and that of V^{∗}.

There is another way of associating to a pp-formula φ(v) in the language L(U_{q}) a ppformula formula in the language of right U_{q}-modules. It depends on the existence of the anti-automorphism Tr of U_{q} determined by the values E ↦ F, F ↦ E, and K ↦ K. This anti-automorphism is obtained by composing the anti-automorphism τ and automorphism ω given in [4, Lemma 1.2]. Quite generally, let α : R → S be an isomorphism of rings. It induces an isomorphism of languages L(α) : L(R) → L(S), which sends a pp-formula φ in L(R) to the pp-formula α(φ) in L(S) obtained by replacing every occurence of the unary function symbol r with α(r). If N is a left R-module, then N^{α} is defined as the left S-module whose underlying abelian group is given by N, and the action of S by sn := α^{−1}(s)(n). It is readily verified that for n ∈ N, we have that

N ⊧ φ(n) iff N^{α} ⊧ α(φ)(n).

The key operation on pp-formulae that turns out to best suit our needs is the composition of the operation φ ↦ φ^{∗} with φ ↦ Tr(φ). It will be denoted by φ ↦ φ^{−}. The importance of this operation stems from the well-known fact that for every finite dimensional simple representation V_{ϵ,n}, we have that

V_{ϵ,n}^{∗} ≅ V Tr_{ϵ,n}.

We provide a proof for lack of a reference. Recall that the most important element in U_{q} is the quantum Casimir element [4, §2.7]:

C_{q} = EF + \frac{q^{−1}K + K^{−1}q^{−1}}{(q − q^{−1})^{2}} .

The fundamental property of C_{q} is that it acts on every simple finite dimensional representation V_{ϵ,n} of U_{q} as multiplication by the scalar

C_{ϵ,n} = \frac{q^{−1}(ϵq^{n}) + q(ϵq^{n})^{−1}}{(q − q^{−1})^{2}} .

Furthermore, the action of the Casimir element permits us to decide when two simple finite dimensional U_{q}-modules are isomorphic.

Lemma 2.1 [4, Lemma 2.8] Let V and V′_{ϵ,n} be some simple finite dimensional representations of U_{q}. If C_{q} acts on V by the same scalar as on V′_{ϵ,n}, then V ≅ V′_{ϵ,n}.

By the lemma, all that needs to be verified is that C_{q} acts by the same scalar on V_{ϵ,n}^{∗} as it does on V_{ϵ,n}^{Tr}. According to the action of U_{q} on V_{ϵ,n}^{∗}, it holds that for every v ∈ V_{ϵ,n},

(ηC_{q})(v) = η(C_{q}v) = η(C_{ϵ,n}v) = (ηC_{ϵ,n})(v),

and therefore ηC_{q} = ηC_{ϵ,n} for every η ∈ V_{ϵ,n}^{∗}. On the other hand, considering the action of U_{q} on V_{ϵ,n}^{Tr}, we have that

vC_{q} = Tr(C_{q})v = C_{q}v = C_{ϵ,n}v = Tr(C_{ϵ,n})v = vC_{ϵ,n},

for every v ∈ V; the equality Tr(C) = C follows from Equation (3) on [4, p. 21].

Theorem 2.2 The rule φ(M) → φ^{−}(M) is an anti-isomorphism of the lattice Latt(M) of pp-definable subspaces of M.

Proof. Let V_{ϵ,n}^{∗} and V_{ϵ,n}^{Tr} as above. By definition of M, it is enough to restrict our attention to a finite dimensional simple representation V_{ϵ,n} of U_{q}. Indeed, if V ⊧ φ(v) → ψ, then

V_{ϵ,n}^{Tr} = V_{ϵ,n}^{∗} ⊧ ψ^{∗}(v) → φ^{∗}(v),

which is equivalent to

V_{ϵ,n} ⊧ ψ^{−}(v) → φ^{−}(v).

□

The following proposition implies that the sublattice of K-invariant pp-definable subspaces of the U_{q}-module M is complemented. To prove it, the argument used in [3, Lemma 13] may be adapted in the quantum case.

Proposition 2.3 If φ is a K-invariant pp-formula, then φ^{−} is also K-invariant and for every simple finite dimensional representation V_{ϵ,n},

φ(V_{ϵ,n}) ⊕ φ^{−}(V_{ϵ,n}) = V_{ϵ,n}.

3 The general strategy

Like the universal enveloping algebra U(L) of L = sl_{2}(k), the quantum group U_{q} is a left Ore domain and so admits a field of fractions, denoted by Q = Q(U_{q}). The U_{q}-module Q is indecomposable and pp-simple: for every pp-formula φ(v) in one variable, the pp-definable subspace φ(Q) equals either Q or 0. Thus the lattice of pp-formulas in the language of U_{q}-modules may be partitioned into the filter F of high formulas - those formulas for which φ(Q) = Q - and the ideal I of low formulas - those formulas for which φ(Q) = 0. It is a general fact about left Ore domains that a pp-formula is high if and only if it is implied by some divisibility pp-formula r|u, with r ∈ U_{q} nonzero. On the other hand, every annihilator pp-formula sv ≐ 0, with s ∈ U_{q} nonzero, is necessarily low, because it defines in Q the zero subspace. Because U_{q} is also a right Ore domain, every low formula in the language of left U_{q}-modules implies some annihilator pp-formula sv ≐ 0.

A pp-formula φ(v) is called uniformly bounded, with bound n_{φ} ≥ 0, if for every finite dimensional simple representation V_{ϵ,n} of U_{q},

dim_{k} φ(V_{ϵ,n}) ≤ n_{φ}.

Dually, we define a pp-formula ψ(v) to be uniformly cobounded, if there is a bound on the k-dimension of the quotient V_{ϵ,n}/ψ(V_{ϵ,n}), for every V_{ϵ,n}. The next section of the article is devoted to a proof of the following fundamental result.

Theorem 3.1 If s in U_{q} is nonzero, and φ(v) is the annihilator formula sv ≐ 0, then there is a uniformly cobounded formula ψ(v) such that the pp-definable subspace ψ(M) is K-invariant, and

φ(M) ∩ ψ(M) = 0.

Let us note, assuming the theorem, how every low pp-formula φ(v) is uniformly bounded. Since any pp-definable subspace φ(M) defined by a low formula is contained in a pp-definable subspace defined by an annihilator pp-formula, we may assume that φ(v) is an annihilator pp-formula. The bound n_{φ} is then obtained by taking the bound on the k-dimension of V_{ϵ,n}/ψ(V_{ϵ,n}), where ψ(v) is the formula given by Theorem 3.1. This implies that every high formula φ(v) is uniformly cobounded, because φ(v) is implied by a divisibility pp-formula s|v, and the k-dimension of V_{ϵ,n}/sV_{ϵ,n} is bounded by n_{ψ}, where ψ(v) is the annihilator pp-formula sv ≐ 0.

Because the anti-isomorphism φ ↦ φ^{−} associates to a divisibility pp-formula s|v the annihilator pp-formula Tr(s)v ≐ 0 - and vice versa - we see that φ(v) is high if and only if φ^{−}(v) is low. The following proposition is proved mutatis mutandis as in [3, Props 16-18], using a highest pseudoweight argument.

Proposition 3.2 If φ(v) is a low pp-formula for which the pp-definable subspace φ(M) is K-invariant, then the interval [0, φ(M)] of the lattice Latt(M) is complemented.

Given Theorem 3.1 and Proposition 3.2, we may now proceed with the proof of Theorem 1.1 as in [3]. If φ(M) is defined by a high formula, then φ^{−}(v) is low, so that we may use Theorem 3.1 to obtain a high formula ψ(v) such that ψ(M) is K-invariant and

φ^{−}(M) ∩ ψ(M) = 0.

Applying the anti-automorphism φ ↦ φ^{−} once more gives that

φ(M) + ψ^{−}(M) = M.

Now ψ^{−}(M) is a K-invariant subspace defined by a low pp-formula, so Proposition 3.2 implies that the interval [0, ψ^{−}(M)] is complemented. A complement of φ(M) ∩ ψ^{−}(M) in ψ^{−}(M) then serves as a complement of φ(M) in M.

If, on the other hand, the pp-definable subspace φ(M) is defined by a low pp-formula, then we may apply the preceding argument to obtain a complement ψ(M) of φ^{−}(M) in M, and apply the anti-automorphism φ ↦ φ^{−} to see that ψ^{−}(M) is then a complement of φ(M) in M.

4 Homogeneous elements of degree 0

The quantum group U_{q} may be equipped with the structure of a ℤ-graded algebra by assigning degrees as follows: deg(E) = 1, deg(F) = −1, and deg(K) = deg(K^{−1}) = 0. Then the relations given in Equation (1) are all homogenous, so that U_{q} becomes a ℤgraded algebra such that for every m ∈ ℤ, the m-th homogenous component of U_{q}, denoted U_{q}^{m}, is the vector space spanned by {E^{i}K^{l}F^{j} : i, j ∈ , l ∈ ℤ, i − j = m} . This follows from the quantum version of the Boincaré -Birkhoff-Witt Theorem [4, Thm 1.5]. In fact, we have

K (E^{i}K^{l}F^{j}) K^{−1} = q^{2(i−j)}(E^{i}K^{l}F^{j}),  (3)

so that U_{q} = ⨁_{m∈ℤ} U_{q}^{m} and the grading is the same as the eigenspace decomposition for the action of K on U_{q} by conjugation.

Remark 4.1 (see [5, Lemma VI.4.2]) Let u be an element of U_{q}. Then u ∈ U_{q}^{0} if and only if u commutes with K.

Proof. If u ∈ U_{q}^{0}, then Equation (3) implies that u commutes with K. For the converse, write u = ∑ c_{i,j}E^{i}K^{l}F^{j} (for some i, j ∈ ℕ and l ∈ ℤ). If u commutes with K, then by the quantum PBK Theorem the coefficient q^{2(i−j)} in Equation (3) must equal 1: as q is not a root of unity, i − j = 0. □

The remark implies that U_{q}^{0} contains the center of U_{q}. Repeated application of the Relations (1) shows that every element in U_{q}^{0} is a polynomial in EF, K, and K^{−1} with coefficients in k. Since these elements of U_{q} commute we see that the subring U_{q}^{0} is a finitely generated commutative k-algebra U_{q}^{0} = k[C_{q}, K, K^{−1}].

Lemma 4.2 If r ∈ U_{q}^{0}, then the annihilator formula φ(v) = (rv ≐ 0) is uniformly bounded.

Proof. Express r = r(C_{q}, K, K^{−1}) as a polynomial in C_{q}, K, and K^{−1} over k. By factoring out a suitably large power of K^{−1}, we have that r = K^{−t}r̄(C_{q}, K), where r̄ is a polynomial in C_{q} and K over k. Since the axioms for a left U_{q}-module imply that the pp-formulae rv ≐ 0 and r̄v ≐ 0 are equivalent, we may assume, without loss of generality, that r = r̄ is a polynomial in C_{q} and K over k.

Since r commutes with K, the pp-definable subspace φ(V_{ϵ,n}) is K-invariant for every finite dimensional simple representation V_{ϵ,n}. The pp-definable subspace φ(V_{ϵ,n}) is therefore a sum of weight spaces,

V_{ϵ, n}^{p} = {v_{p} ∈ V_{ϵ,n} Kv_{p} = ϵq^{n−2p}v_{p}},

p = 0, …, n, of V_{ϵ, n}. By considering the action of C_{q} and K on V_{ϵ, n}^{p}, we see that the pp-definable subspace φ(V_{ϵ,n}) will contain V_{ϵ, n}^{p} if and only if

r(C_{ϵ,n}, ϵq^{n−2p}) = 0.

Because q is not a root of unity, the exponential map p ↦ q^{p} is a monomorphism as is the linear function p ↦ n − 2p. We may deduce that the number of solutions p is bounded by the degree of K in the polynomial r, that is, dim_{k} φ(V_{ϵ,n}) ≤ deg_{K}(r). □

To generalize Lemma 4.2 for every element in U_{q}, we may apply the following property of the m-th homogeneous component of U_{q} :

∀m > 0, U_{q}^{m} = E^{m} U_{q}^{0},
∀m < 0, U_{q}^{m} = U_{q}^{0} F^{m}.

It may established by repeated application of Equations (1). The following result then follows using a quantum version of the argument used in [3, Lemma 21].

Proposition 4.3 Let s ∈ U_{q} and consider the corresponding annihilator formula φ(v) = (sv ≐ 0). There are an element r in U_{q}^{0} and a natural number m such that for every simple finite dimensional representation V_{ϵ,n} of U_{q},

φ(V_{ϵ,n}) ∩ F^{m}V_{ϵ,n} ∩ rV_{ϵ,n} ∩ E^{m}V_{ϵ,n} = 0.

An immediate consequence of Proposition 4.3 is the proof of Theorem 3.1. The uniformly cobounded formula ψ(v) is given by the conjunction of F^{m}|v, r|v, and E^{m}|v. It clearly defines a K-invariant subspace of M.

5 Ideals of definable scalars

Before discussing the ring of definable scalars of the quantum plane, let us review some of the consequences of Theorem 1.1 for the representation theory of U_{q}. These results are recounted without proof, which in all cases is entirely analogous to that given in [3] for the universal enveloping algebra U(L) of L = sl_{2}(k), when k is an algebraically closed field of characteristic 0.

Let U′_{q} be the ring of definable scalars of the U_{q}-module M. If r belongs to U′_{q}, then the pp-definable subspace rM is complemented by some pp-definable subspace ψ(M) of M

rM ⊕_{k} ψ(M) = M.

If e ∈ U′_{q} is the idempotent projection onto rM with respect to this decomposition, then

M ⊧ ∀v(ψ(v) ↔ (ev ≐ 0)),

and rU′_{q} = eU′_{q}. Similarly, define e_{0} ∈ U′_{q} to be the idempotent projection onto the ppdefinable subspace φ(M) defined by φ(v) = (Ev ≐ 0), with respect to the decomposition

φ(M) ⊕_{k} F M = M.

For every simple finite dimensional representation V_{ϵ,n} of U_{q}, e_{0}V_{ϵ,n} is the highest weight space. If I_{0} ⊆ U′_{q} denotes the two-sided ideal generated by e_{0}, then, as in [3], I_{0} consists of all the elements r ∈ U′_{q} for which the formula r|v is is uniformly bounded, and U′_{q}/I_{0} is isomorphic to the field of fractions Q of U_{q}.

As in the general case of a von Neumann regular ring, the Ziegler spectrum Zg(U′_{q}) of U′_{q} consists of the injective indecomposable U′_{q}-modules where the open subsets in Zg(U′_{q}) are in bijective correspondence with the two-sided ideals of U′_{q} according to the rule

I ↦ O(I) := {E ∈ Zg(U′_{q}) : IE ≠ 0}.

If φ(v) is a pp-formula in L(U′_{q}), then there is a complementary pp-formula ψ(v) such that that

φ(M) ⊕_{k} ψ(M) = M.

If e ∈ U′_{q} is the projection onto φ(M) with respect to this decomposition, then it may be easily checked that in Zg(U′_{q}),

O_{φ,v ._{=}0} = O(I),

where I is the two-sided ideal of U′_{q} generated by e.

As in the case of the universal enveloping algebra of sl_{2}(k),

Zg(U′_{q}) = O(I_{0}) ∪^{.} {Q}.

The open subset O(I_{0}) forms a compact totally disconnected subspace of Zg(U′_{q}), and the subset of finite dimensional simple representations V_{ϵ,n} is a dense and discrete open subset of Zg(U′_{q}). It follows that if I_{1} and I_{2} are two-sided ideals of U′_{q} contained in I_{0}, then I_{1} = I_{2} if and only if the corresponding open subsets O(I_{1}) and O(I_{2}) contain the same finite dimensional representations.

If V ∈ Zg(U′_{q}) is not Q, then I_{0}V ≺ V is a simple U′_{q}-module which is an elementary substructure of V regardless of whether V is considered as a structure for the language of U_{q}-modules or U′_{q}-modules. An indecomposable representation V in Zg(U′_{q}) is finite dimensional if and only if I_{0}V = V.

A U_{q}-module V is said to be pseudo-finite dimensional if it satisfies all the first order sentences of the language of U_{q}-modules satisfied by every finite dimensional module. A U_{q}-module V is pseudo-finite if and only if it is a U′_{q}-module and I_{0}V ≺ V. Every pseudofinite dimensional representation V is elementary equivalent to a direct sum

V ≡ ⨁_{W ∈Cℓ(V)} I_{0}W,

where every I_{0}W is a pseudo-finite dimensional simple representation of U′_{q}. This is an elementary version of [4, Theorem 2.9].

Let φ_{+} be the sum of the following pp-formulae

Kv = qv, Kv = v;

the pp-definable subspace φ_{+}(M) of M is K-invariant and is complemented by the ppdefinable subspace φ_{+}^{−}(M) of M

φ_{+}(M) ⊕_{k} φ_{+}^{−}(M) = M.

So, define e_{+} to be the idempotent projection onto φ_{+}(M), with respect to this decomposition. Then e_{+}V_{ϵ,n} ≠ 0 if and only if ϵ = +. If we denote by I_{+} the ideal generated by the idempotent e_{+}, then O(I_{+}) is the unique open subset of Zg(U′_{q}) contained in O(I_{0}) that contains all the indecomposable summands V_{+,n} of the quantum plane, and none of the V_{−,n}.

Similarly, let φ_{−} = σ(φ_{+}) be the pp-formula gotten by applying the automorphism σ to the scalar in φ_{+}. It is the sum of the pp-formulae

Kv = −qv, Kv = −v.

If we define e_{−} = σ(e_{+}) to be the idempotent projection onto φ_{−}(M), with respect to the corresponding K-invariant decomposition of M, then e_{−}V_{ϵ,n} ≠ 0 if and only if ϵ = −. Then the ideal I_{−} generated by e_{−} is nothing more than σ(I). Here we are making tacit use of the fact that M^{σ} ≅ M, and if r ∈ U′_{q} is a definable scalar of M, represented say by the pp-formula ρ(u, v), then the pp-formula σ(ρ) also defines an element σ(r) ∈ U′_{q}. As both of the ideals I_{−} and I_{+} are contained in I_{0}, and the open subsets associated to I_{0} and I_{−} + I_{+} both contain all the finite dimensional points of Zg(U′_{q}), we conclude that

I_{−} + I_{+} = I_{0}.

Since O(I_{−}) ∩ O(I_{+}) = O(I_{−} ∩ I_{+}) contains no finite dimensional points, the sum must be direct.

Theorem 5.1 The lattice Latt(k_{q}[x, y]) of pp-definable subspaces of the quantum plane is complemented. The ring of definable scalars of k_{q}[x, y] may be identified with the von Neumann regular ring U′_{q}/I_{−}. The canonical morphism ρ : U_{q} → U′_{q}/I_{−} is an epimorphism of rings with 0 kernel.

Proof. The first statement is clear, because if φ(v) is a pp-formula, then any ppformula ψ(v) that defines a complementary subspace of φ(M) in M, will also define a complementary subspace of φ(k_{q}[x, y]) in k_{q}[x, y].

To prove the second statement, let us first note that any definable scalar r of M that vanishes on k_{q}[x, y] must belong to I_{0}. This is because k_{q}[x, y] contains finite dimensional indecomposable summands of arbitrarily large k-dimension. The formula r|v cannot therefore be uniformly cobounded. It is therefore uniformly bounded, and r ∈ I_{0}. Since I_{−} consists of the elements of I_{0} that vanish on k_{q}[x, y], our claim is established.

Since every element of U′_{q}, acts definably on k_{q}[x, y], there is a canonical morphism of rings from U′_{q} to the ring U′′_{q} of definable scalars of k_{q}[x, y]. The lattice of pp-definable subspaces of k_{q}[x, y] is the same whether we consider k_{q}[x, y] as a U_{q}-module or a U′_{q}- module. Since it is complemented, the canonical morphism is an epimorphism of rings. If we can show that this ring epimorphism is onto, it will follow that U′′_{q} is isomorphic in the obvious way to the quotient ring U′_{q}/I_{−}. That the kernel of ρ is 0 follows from the fact that the quantum plane is a faithful representation of U_{q}. This follows from Proposition 4.3, because if s ∈ U_{q} is nonzero, then the annihilator formula sv ≐ 0 is uniformly bounded.

Let us note, quite generally, that if ρ : R → S is a ring epimorphism, with R von Neumann regular, then it must be onto. Replacing R with the von Neumann regular ring R/Ker ρ, we may assume without loss of generality that Ker ρ = 0. As ρ is an epimorphism, the morphism μ : S ⊗_{R} S → S of abelian groups, given by s ⊗ r ↦ sr, is an isomorphism. Consider the commutative diagram

[figure: 0; R ⊗_{R} S; ρ ⊗ S; S ⊗_{R} S; S/R ⊗_{R} S; 0; μ; S.]

The top row is exact, because every short exact sequence of R-modules is pure exact. Since the vertical and diagonal arrows are both isomorphisms, it follows that the monomorphism ρ⊗S in the top row is also an isomorphism, and hence that S/R⊗_{R} S = 0. But this implies that S/R = 0, because the morphism

S/R ⊗ ρ : S/R ⊗_{R} R → S/R ⊗_{R} S

is a monomorphism. □

References

[1] K. Brown and K. Goodearl, Lectures on Algebraic Quantum Groups, Birkhauser, 2002.

[2] K. Burke and M. Prest, Rings of definable scalars and biendomorphism rings, pp. 188-201, in Model Theory of Groups and Isomorphism Groups, London Math. Soc. Lect. Note Ser. 244, Cambridge University Press, 1997.

[3] I. Herzog, The pseudo-finite dimensional representations of sl_{2}(k), Selecta Mathematica 7 (2001), 241-290.

[4] J.C. Jantzen, Lectures on quantum groups, Grad. Stud. in Mathematics 6, American Math. Soc., 1995.

[5] C. Kassel, Quantum Groups, Grad. Texts in Math. 155, Springer-Verlag, 1995.

[6] S. L’Innocente and A. Macintyre, Towards decidability of the theory of pseudofinite dimensional representations of sl_{2}(k), Part I, Submitted.

[7] S. L’Innocente and M. Prest, Rings of definable scalars of Verma modules, Journal of Algebra and its applications, to appear.

[8] M. Prest, Model theory and modules, London Math. Soc. Lect. Note Ser. 130, Cambridge University Press, 1988.