The Prime Spectrum of Quantum $SL_3$ and the Poisson-prime Spectrum of its Semi-classical Limit

A bijection $\psi$ is defined between the prime spectrum of quantum $SL_3$ and the Poisson prime spectrum of $SL_3$, and we verify that $\psi$ and $\psi^{-1}$ both preserve inclusions of primes, i.e. that $\psi$ is in fact a homeomorphism between these two spaces. This is accomplished by developing a Poisson analogue of Brown and Goodearl's framework for describing the Zariski topology of spectra of quantum algebras, and then verifying directly that in the case of $SL_3$ these give rise to identical pictures on both the quantum and Poisson sides. As part of this analysis, we study the Poisson primitive spectrum of $\mathcal{O}(SL_3)$ and obtain explicit generating sets for all of the Poisson primitive ideals.


Introduction
In the study of non-commutative quantum algebras, the theory of H-stratification (due originally to Goodearl and Letzter [13]) allows us to partition the prime spectra of various algebras into strata indexed by the prime ideals that are invariant under the rational action of some algebraic torus H (hence the name). The prime ideals in an individual stratum can be far more easily understood -each stratum is homeomorphic to the prime spectrum of a commutative Laurent polynomial ring -and under fairly mild conditions on the algebra and torus action, we can even pinpoint the primitive ideals as those prime ideals that are maximal in their strata.
A corresponding theory for Poisson algebras (usually appearing as the semiclassical limits of various quantum algebras; informally, this means we let q → 1) has also been developed in [7]; we recall the main results in Section 2. For more information on semiclassical limits and the relationship between Poisson and quantum algebras, see, for example, [8]; the definitive references for H-stratification are [1] for quantum algebras and [7] for Poisson algebras.
Many tools developed originally for quantum algebras (for example, the H-stratification mentioned above, and Cauchon's deleting derivations algorithm) turn out to transfer naturally to the Poisson case with very little modification. These two pictures -prime ideals in quantum algebras A for q ∈ k × not a root of unity, and Poisson prime ideals in their semiclassical limits R -are often remarkably alike, and there have been many recent results exploring these similarities (for example, see [9,18,27]). In light of this, Goodearl makes the following conjecture in [8]. This conjecture has been verified for some types of quantum algebras, for example, q-commuting polynomial rings and quantum affine toric varieties [14,Theorem 4.2], multiparameter-quantised symplectic and Euclidean spaces [23,24] and O q (SL 2 ) [8, Example 9.7]. In addition, it is known that there are bijections prim(A) −→ pprim(R) for many algebras of the form A = O q (G) (under various conditions on G and q, see, for example, Joseph [17], Hodges, Levasseur and Toro [16], Yakimov [27]), but the topological properties of these maps remain unknown.
The problem is as follows: Although we have an excellent understanding of the individual strata in spec(A) or pspec(R), which is sufficient to obtain the bijections mentioned above, this tells us nothing about the interaction of primes from different strata. A homeomorphism ψ : spec(A) −→ pspec(R) and its inverse ψ −1 must preserve inclusions between primes; if A and R are both noetherian and char(k) = 0, the converse is also true [8,Lemma 9.4]. In O q (SL 2 ), the picture is simple enough that we can check these inclusions directly, but for larger algebras this quickly becomes impossible.
In [2], Brown and Goodearl develop a conjectural framework to tackle this problem in the case of quantum algebras by encoding the information about inclusions of primes from different strata in terms of certain commutative algebras and maps between them. This commutative data can then (in theory) be computed, providing a full picture of the topology of the space spec(A). Verifying that this framework applies to a given algebra reduces to a question of normal generation of prime ideals modulo their H-prime, which is currently unknown in general but is accomplished in [2, § 5-7] for the cases of O q (GL 2 ), O q (M 2 ) and O q (SL 3 ).
In Section 3, we develop the corresponding framework for the Poisson case and obtain a very similar picture (up to the obvious modifications of replacing 'centre' with 'Poisson centre', 'normal' with 'Poisson normal', etc.). In particular, since the quantum and Poisson pictures each reduce to understanding a finite number of commutative algebras and associated maps (commutative with respect to both the associative multiplication and the Poisson bracket), this suggests a possible approach to tackling Conjecture 1.1: We describe these commutative algebras, and the homeomorphism will follow. As proof of concept, we carry out this program in the case of O q (SL 3 ) and O q (GL 2 ) in Sections 4 and 5.
In Section 4, we verify that the results of Section 3 apply to the Poisson algebras O(SL 3 ) and O(GL 2 ) (with respect to the Poisson bracket induced from the corresponding quantum algebras); these results follow naturally from the examination of Poisson prime and Poisson primitive ideals in O(SL 3 ), which formed part of the author's PhD thesis [6]. In Section 5, we prove the following main theorem.  Let k, q be as above. There is a homeomorphism ψ : spec(O q (GL 2 )) −→ pspec(O(GL 2 )), which restricts to a homeomorphism prim(O q (GL 2 )) −→ pprim(O(GL 2 )).
Although the direct computational approach used here will not be feasible for algebras of larger dimension, it does demonstrate how the techniques of [2] and this paper for 'patching together' the topologies of individual strata in well-behaved quantum and Poisson algebras could be used to tackle Conjecture 1.1 in greater generality.

Background and definitions
Throughout, fix k to be an algebraically closed field of characteristic 0.
We will restrict our attention exclusively to commutative Poisson algebras in this paper. If R is a Poisson algebra, an ideal I is called a Poisson ideal if it is also an ideal with respect to the Poisson bracket, that is {I, R} ⊆ I; it is called Poisson prime if it is a Poisson ideal which is also a prime ideal in the usual commutative sense † , and Poisson primitive if it is the largest Poisson ideal contained in some maximal ideal of R. We write pspec(R) for the set of Poisson primes in R, and pprim(R) for the set of Poisson primitives.
The space pspec(R) naturally inherits the Zariski topology of spec(R), and in turn induces a topology on pprim(R). The closed sets of these topologies are defined to be: where we may assume that I is a Poisson ideal in R (if it is not, simply replace it with the intersection of all of the ideals in V (I); this is a Poisson ideal that defines the same closed set as V (I)). When R is noetherian, the sets {V (P ) : P ∈ pspec(R)} form a basis for this topology: this follows from the fact that I = n i=1 P i , where I is a Poisson ideal and P 1 , . . . , P n are the finitely many Poisson prime ideals minimal over I (see [7,Lemma 1.1]).
For any topological space T , we will denote the set of its closed sets by CL(T ).
Similarly, we say that r ∈ R is Poisson normal if {r, R} ⊆ rR.
If I is a Poisson ideal, then there is an induced Poisson structure on R/I given by In a similar vein, if R is a domain and X is a multiplicative set, then the Poisson bracket on R extends uniquely to a Poisson bracket on R[X −1 ] via the formula The Poisson centre of R[X −1 ] can be related to R as follows: The ⊆ direction of this equality is clear, while the ⊇ direction follows from (2.2). The Poisson structures in which we are interested arise naturally as the semiclassical limits of various quantised coordinate rings. However, since we will quickly narrow our attention to specific examples, we will not need the formal details of this relationship here and the interested reader is referred to [8]. In this context, however, it makes sense to impose the following conditions on our Poisson algebras.
Conditions 2.1. (i) R is a commutative affine k-algebra with a Poisson bracket. (ii) H = (k × ) r is an algebraic torus acting rationally on R by Poisson automorphisms (see [7, § 2]). † This is equivalent to the standard definition of a Poisson prime ideal when R is noetherian and k has characteristic 0; all algebras considered here will be of this type (see [ ). This will be the only grading we consider here, and the terms 'eigenvectors' and 'homogeneous elements' will be used interchangably.
The H-primes induce partitions on pspec(R) and pprim(R) as follows: for J ∈ H-pspec(R), define For many algebras (in particular, those satisfying Conditions 2.1), the topology of the individual strata can be understood using the following theorem. (ii) pspec J (R) is homeomorphic to pspec(R J ) via localisation and contraction, and pspec(R J ) is in turn homeomorphic to spec(PZ(R J )) via contraction and extension; (iii) if H-pspec(R) is finite and R is an affine k-algebra, then the Poisson primitive ideals in pprim J (R) correspond to maximal ideals in PZ(R J ), and the homeomorphisms above restrict to homeomorphisms pprim J (R) ≈ pprim(R J ) ≈ max(PZ(R J )).
Although this allows us to understand each stratum individually, it carries no information about interactions between different strata. In order to understand the topology of the Poisson spectrum as a whole, some additional tools will be required.
Definition 2.4. Let T be a topological space and Π a poset. A partition T = i∈Π S i is called a finite stratification of T if Π is finite and (i) each S i is non-empty and locally closed in T ; If T = i∈Π S i is a finite stratification of a topological space T , then we can equip each S i with the induced topology from T and, for any pair i, j ∈ Π, define a map CL(S i ) → CL(S j ) by (2.4) (Here, Y denotes the closure of Y in T .) By property (ii) of Definition 2.4 above this map will only be interesting when i < j, so we will often focus only on this case. Given a finite stratification of a topological space T , and its associated family of maps {ϕ ij : i, j ∈ Π, i < j}, we can deduce some information about the structure of T as follows.
Let T be a topological space with a finite stratification T = i∈Π S i , and let {ϕ ij : i < j ∈ Π} be the collection of maps defined in (2.4).
Understanding the topology of the spaces S i and the action of the maps ϕ ij is enough to completely describe the topology of T . The aim of Section 3 is to give (in certain particularly nice settings) an alternative definition of the ϕ ij , which does not rely on already knowing the topology of T itself. First, we look at how the spaces pspec(R) and pprim(R) fit into this finite stratification framework.
this is because the ⊆ in Lemma 2.5(3)(ii) becomes an equality when X = {x} for some In other words, in order to understand inclusions of primes in pspec(R) it is enough to find an equivalent definition of the ϕ JK, which does not depend on already knowing the topology of pspec(R). In [2], Brown and Goodearl examine the corresponding problem for non-commutative quantum algebras A, which admit a similar stratification by H-primes; the interested reader is referred to [1, for the background and definitions in this setting. They develop a framework for understanding the action of the maps ϕ JK in the setting of A in terms of certain commutative 'intermediate' algebras Z JK (see [2,Definition 3.8]) and maps between them of the form where A J denotes the localisation of A/J at the set of all homogeneous H-eigenvectors as before, and Z(−) denotes the centre. The maps f JK and g JK are homomorphisms of commutative k-algebras, and so they induce morphisms of schemes . In [2], Brown and Goodearl make the following general conjecture.  Although it is not immediately apparent how to check this condition in general, it certainly holds in low-dimensional examples that have been computed directly; in particular, it is verified for the algebras O q (GL 2 ), O q (M 2 ) and O q (SL 3 ) in [2, § 5-7].
Our first aim is to develop the corresponding Poisson version of this framework; we will see that it is very similar to the quantum case, and indeed many of the proofs go through with almost no modification. By making it possible to reduce both the topologies of spec(A) for quantum A and pspec(R) for R the semiclassical limit of A to questions of commutative algebra, we will then be able to compare the two topologies directly.

A framework for patching together the strata of pspec(R)
In this section, our goal is to construct intermediate commutative algebras PZ JK for each pair of H-primes J ⊂ K in Poisson algebras satisfying Conditions 2.1, and also certain homomorphisms between these algebras and the Poisson centres PZ(R J ), PZ(R K ). The results in this section closely follow the corresponding quantum results in [2].
Throughout, let R be a commutative Poisson algebra satisfying Conditions 2.1, and let J ⊂ K be Poisson H-primes in R.
Recall from Theorem 2.3 the definitions E J := {all homogeneous non-zero elements in R/J}, By Theorem 2.3, the topological spaces pspec J (R) and spec(PZ(R J )) are homeomorphic via the map P → P R J ∩ PZ(R J ); we will often identify these two spaces without comment.
Define a new multiplicative set by Unhampered by the problems of non-commutative localisation faced in [2], we can immediately define The Poisson bracket on R/J[E −1 JK ] is the unique extension of the one on R via the formulas given in (2.1) and (2.2). It is clear that (3.2) is equivalent to (a Poisson version of) the definition used in [2].
Since E JK ⊂ E J , the natural inclusion map R/J[E −1 JK ] → R J restricts to a map on the Poisson centres. This is a homomorphism of commutative k-algebras and we will make use of it below, so we name it as follows: We also require a map PZ JK −→ PZ(R K ); this is the purpose of Lemma 3.1. f ) since e f (E JK ) ⊆ E K . Since projection to a quotient algebra sends Poisson central elements to Poisson central elements, it is easy to check that the following restriction is well defined: 4) and this is exactly the homomorphism of commutative k-algebras promised in the statement of the lemma.
By rewriting PZ JK in the same form as [2, Definition 3.6], it is an easy exercise to check that this definition of f JK agrees with (the Poisson equivalent of) the map defined in [2, Lemma 3.10].
We are now ready to put everything together. Recall that whenever h : R → S is a homomorphism of commutative algebras, h • will denote the induced comorphism spec(S) → spec(R).
Definition 3.2. Let J ⊂ K be Poisson H-primes, and let PZ JK , g JK and f JK be as defined in (3.2), (3.3) and (3.4), respectively. Define a map CL(spec(PZ(R J ))) → CL(spec(PZ(R K ))) as follows: Corresponding to [2, Conjecture 3.11], we expect that for all pairs of H-primes J ⊂ K in Poisson algebras R satisfying Conditions 2.1, we will have In order to establish conditions under which (3.5) holds, we need to describe the action of both ϕ JK and f • JK |g • JK on closed sets. Since most of the action takes place in purely commutative algebras, the proofs necessarily follow those of [2, § 4] closely.
We begin by restricting to the case J = 0 and describing the action of f • 0K |g • 0K on closed sets of the form V (P ) ∩ pspec 0 (R) for some P ∈ pspec 0 (R). Since the next few proofs are quite technical, we hope that this approach will serve to illustrate the main ideas before we state the result in full generality (Proposition 3.5).
First, consider the set g • (Y ): since g is the natural embedding map PZ 0K → PZ(R 0 ), the comorphism g • is given by contraction to PZ 0K . Contraction clearly preserves inclusions and so g • (Y ) contains a minimal element, namely ( For Q ∈ pspec K (R), we can now see that , and that f JK is induced from the projection map π JK : R/J → R/K. This allows us to rewrite f (P R 0 ∩ PZ 0K ) as As in [2, Proposition 4.1], we now observe that for p ∈ P and c ∈ E 0K , we have We conclude that This motivates the following result, as in [2].
and hence Proof. The equality (3.6) is clear from Lemma 3.3 and the following discussion. To prove (3.7), first recall that since P ∈ pspec 0 (R) the action of ϕ 0K on Y is as follows: With all of the pieces in place, we can now expand our attention to arbitrary pairs of H-primes J ⊂ K and to arbitrary closed sets Y ∈ CL(pspec J (R)). The next proposition shows that the general case follows easily from the specific cases considered above.
for all P ∈ pspec J (R) and all Q ∈ pspec K (R).
Proof. We begin by considering an arbitrary closed set Y ∈ CL(pspec J (R)); this has the form Y = V (I) ∩ pspec J (R) for some Poisson ideal I. As in the quantum case, we can first observe that for any Poisson H-prime L ⊇ J and any Q ∈ V (I) ∩ pspec L (R), we have J ⊆ Q; hence that is, we may reduce to the case J = 0 without loss of generality.
Next, recall that V (I) ∩ pspec J (R) is a finite union of sets of the form V (P i ) ∩ pspec J (R), where the P i are the Poisson primes minimal over I. Both of the maps ϕ JK and f • JK |g • JK preserve finite unions, so we can focus our attention on closed sets of the form V (P ) ∩ spec J (R) for P ∈ pspec(R). Since we have reduced to the case J = 0 we know that J ⊆ P , but we need to consider the possibility that there is some L ∈ H-pspec(R) such that J L ⊆ P . In this case, however, V (P ) ∩ pspec J (R) = ∅, and it follows trivially that The proposition therefore holds if and only if it holds in the case J = 0 and Y = V (P ) ∩ pspec J (R) for any prime P in pspec J (R). The result now follows from Proposition 3.4.
Proposition 3.6. Apply the same hypotheses as Proposition 3.5, and fix P ∈ pspec J (R). If P/J has a generating set consisting of Poisson normal elements in R/J, then (3.8) holds for that choice of Poisson prime P .
Proof. Fix some Q ∈ pspec K (R). Since J ⊂ Q, we may as well assume J = 0. We prove the contrapositive of (3.8), so suppose P Q. By assumption there is some non-zero element f ∈ P \Q that is Poisson normal in R, which we can decompose as f = f 1 + . . . + f n for some H-eigenvectors {f 1 , . . . , f n } with distinct eigenvalues. It is standard that each f i will also be Poisson normal in R, and further that they are compatible with f in the following sense: for any r ∈ R, let r f ∈ R be such that {f, r} = fr f , then we must have {f i , r} = f i r f as well. (This follows from the corresponding result for homogeneous r, which can easily be seen by Finally, when applying these results we will often want to replace the set E JK with a subset e E JK such that PZ JK ]); the conditions under which this can be done are described in Lemma 3.7.
Lemma 3.7. Let E JK be as defined in (3.1), and suppose e E JK ⊂ E JK is a multiplicative set that satisfies the following condition: for all H-primes L such that J ⊂ L and L K, we have Proof. As above, it is enough to consider J = 0. its Poisson H-prime. L is non-zero (in particular c ∈ L) so by standard localisation theory it corresponds to a non-zero H-prime L in R that is disjoint from e E JK . By the definition of e E JK, it follows that L ⊂ K, but now c ∈ L ⊂ K is a contradiction since c is regular mod K. We conclude that I ∩ e E JK = ∅. As a commutative ideal, I is generated by c and all n-fold Poisson brackets In addition, it follows from the Poisson centrality of ac −1 that {a, b}{c, b} −1 = ac −1 for any homogeneous b ∈ R, and hence inductively with n, b 1 , . . . , b n as above. In particular, we can conclude from this that ac −1 · I ⊂ R.
Let c ∈ I ∩ e E JK , and let a := ac −1 c ∈ R. But now we are done, since a c −1 = ac −1 with a ∈ R, c ∈ e E JK as required.

The case of O(SL 3 )
The results of the previous section give us a way to understand the topological structure of pspec(R) in terms of certain commutative algebras and maps between them. On the other hand, we have yet to see any examples of Poisson algebras which fit into this framework. With this in mind, we now turn our attention to certain specific Poisson matrix varieties, and verify that they do indeed satisfy the conditions of Proposition 3.6. We begin by defining O(M 2 ), the coordinate ring of all 2 × 2 matrices with entries in k. As a commutative algebra, this is just the polynomial ring k[a, b, c, d]; viewed as the semiclassical limit of the corresponding quantum algebra O q (M 2 ) (defined in § 5), it acquires the structure of a Poisson algebra, with Poisson bracket defined by For more details about the semiclassical limit process, and in particular to see that As one might expect, all of the Poisson algebras defined above admit rational actions by tori H acting by Poisson automorphisms. In the case of O(M m,p ), the action is given as follows: When m = p, this extends to a rational action on O(GL m ). Meanwhile, for O(SL m ) we must instead consider the torus  use the same generating sets to obtain the Poisson H-primes of O(SL 3 ) over arbitrary fields k. Generators are depicted as follows: the dots in the 3 × 3 grids represent the indeterminates Y ij , 1 i, j 3 in the natural way; a black dot in position (i, j) means that Y ij is included in the generating set for that ideal, and a square indicates that the corresponding 2 × 2 minor is in the generating set. For example, denotes the ideal Y 31 , [12|23] .
We retain the notation of [12] for indexing the 36 H-primes by elements of S 3 × S 3 : for ω = (ω + , ω − ) ∈ S 3 × S 3 , the corresponding H-prime I ω has the generating set listed in row ω + and column ω − of Figure 1. Elements of S 3 are written in a truncated form of two-line notation, for example, 321 corresponds to the permutation 1 → 3, 2 → 2, 3 → 1.
Our aim in this section is to verify that R := O(SL 3 ) satisfies the conditions of Proposition 3.6, that is • for each J ∈ H-pspec(R), and for every Poisson prime ideal P in pspec J (R), P/J has a Poisson normal generating set in R/J.
In order to do this, we will need to understand the centres PZ(R J ) and obtain explicit generating sets for the Poisson primitive ideals in R. This formed part of the author's PhD thesis [6]; since the proofs are computational and long, we simply state the results here and refer the interested reader to [6,Chapter 5] for the details.
Blank entries indicate that the corresponding centres is trivial.
and so the Poisson primitive ideals in R/I ω are precisely those of the form A natural guess at a generating set for P λ in R/I ω would be It is quite easy to check that these generators are Poisson normal in R/I ω (for example, see the proof of Theorem 4.4) and hence that but as yet there is no general approach to proving that this ideal is actually P λ itself. For O(SL 3 ), this is again done using case-by-case analysis in [6], and we quote the relevant result below.  Table 2.
We will need one further result from [6], which is the following. Proof. [6,Proposition 5.3.19].
We are now ready to verify that , let I ω be a Poisson H-prime in R and assume that k is algebraically closed field of characteristic 0. Then for any P ∈ pspec Iω (R), the ideal P/I ω has a generating set consisting of Poisson normal elements in R/I ω .
Proof. To simplify the notation, write J for I ω . From Table 1, we see that P/J has height at most 2. If P = J, then the result is trivial. When ht(P/J) = 1, P/J must be principally generated since R/J is a UFD by Proposition 4.3; since P/J is a Poisson ideal, this generator must be Poisson normal.
This leaves only the case where ht(P/J) = 2, that is P/J is maximal in its stratum and hence it is Poisson primitive. All that remains is to show that the generating sets given in Table 2 consist of Poisson normal elements in R/J.
Observe that every generator in Table 2 has the form U − λV , where UV −1 is one of the Poisson central generators of PZ(R J ) listed in Table 1. Since UV −1 will also be Poisson central in R/J, we compute that for any a ∈ R/J, Furthermore, the generators in Table 1 Proof. Combine Propositions 3.5 and 3.6 and Theorem 4.4.

A homeomorphism between pspec(O(SL 3 )) and spec(O q (SL 3 ))
In Section 4, we have seen that we can understand the Zariski topology of pspec(O(SL 3 )) in terms of the commutative algebras PZ(R J ), PZ(R K ) and PZ JK , and homomorphisms f JK and g JK between them. As already noted, this setup is based on the corresponding results for quantum algebras in [2], which similarly reduces the study of the non-commutative prime spectrum of various quantum algebras to questions about commutative algebras and maps between them. In this section, we will show that for O(SL 3 ) and its uniparameter quantum analogue O q (SL 3 ), these commutative algebras and homomorphisms are the same in each case, and hence that these two topological spaces are homeomorphic.
Throughout this section, fix q ∈ k × to be not a root of unity. We continue to assume that k is algebraically closed field of characteristic 0.
We first need to define the algebra of quantum matrices, which quantises the Poisson structure given in (4.1). The 2 × 2 quantum matrix algebra O q (M 2 ) is defined to be the quotient of the free algebra k a, b, c, d by the relations ab = qba, ac = qca, bd = qdb, cd = qdc, bc = cb, As in the Poisson case, this extends to a definition of the m × p quantum matrices by considering the free algebra on mp generators {X ij : 1 i m, 1 j p} and imposing the condition that each set of four variables {X ij , X il , X kj , X kl } satisfies a copy of relations (5.1). The quantum determinant is given by the formula where l(π) denotes the length of π ∈ S m , and quantum minors are defined analogously to the Poisson case (but see Notation 5.1). As we might expect, D q is a central element in O q (M m ), and we obtain the algebras Finally, there are rational actions of tori H on each of these algebras that are defined in the same way as the Poisson case (see (4.2) and (4.3)). We also have a transposition map τ : X ij −→ X ji as before, which defines an automorphism on each of the algebras O q (M m ), Generating sets for H-primes in quantum matrices have been studied extensively (for example, [4,9,26]) under various restrictions on k and q. In particular, generating sets for all 230 H-primes in O q (M 3 ) are computed in [11] for fields of arbitrary characteristic and q not a root of unity. The H-primes of O q (SL 3 ) correspond to exactly those H-primes of O q (M 3 ) which do not contain D q ; as noted in Section 4, we may take the same generating sets for H-primes in O q (SL 3 ) and Poisson H-primes in O(SL 3 ) (up to the obvious modification of replacing quantum minors with minors). Recall that these generating sets are listed in Figure 1.
Notation 5.1. Since we will often want to treat the quantum and Poisson cases simultaneously in what follows, we make the following conventions. Generators of A := O q (SL 3 ) are denoted by X ij and generators of R := O(SL 3 ) by Y ij ; we will use W ij to mean 'X ij in the quantum case, Y ij in the Poisson case'. Minors and quantum minors will both be denoted by [I|J] and referred to simply as 'minors', but minors in O q (SL 3 ) will always be computed using (5.2). Any references to H-primes in O(SL 3 ) implicitly mean Poisson H-primes, and 'centre' should be taken to mean the centre with respect to the non-commutative multiplication or Poisson bracket as appropriate. Where a formula in the W ij involves terms in q, we take q to be a non-zero scalar, which is not a root of unity in the quantum case, and q = 1 in the Poisson case.
Two types of algebra appear in the definitions of the maps f JK , g JK : the commutative algebras Z JK and PZ JK defined in [2, Definition 3.8] and (3.2), respectively, and the centres of H-simple localisations Z(A J ) and PZ(R J ) appearing in the quantum (respectively, Poisson) stratification theorems. We deal with the latter algebras first.
Definition 5.2. Since we are using the same notation for quantum minors and minors but will sometimes need to emphasise that we are passing from a minor in the quantum world to the corresponding minor in the Poisson world, we make the following definition: let θ be the 'preservation of notation' map which takes a quantum minor [I|J] and maps it to the corresponding commutative minor [I|J] with the same index sets. In particular, θ(X ij ) = Y ij .
Remark 5.3. Clearly θ has no hope of extending to a well-defined map A −→ R; it exists purely as a notational convenience in order to pass from certain generating sets on the quantum side to the corresponding constructions on the Poisson side. No attempt should be made to expand out a quantum minor and apply θ as if it were a homomorphism. We will sometimes need to apply θ to products and inverses of quantum minors, so we make the following conventions: if z 1 z 2 . . . z n is a product of quantum minors, we apply θ pointwise to the minors in that order, that is Similarly, if z is an invertible quantum minor we define where each Z i is a product of quantum minors (and possibly their inverses), and there is an isomorphism of commutative algebras determined by Proof. Generators for the centres PZ(R J ) are given in Table 1; on the quantum side, the centres are computed for the corresponding algebras O q (GL 3 ) J (that is H-simple localisations of the algebras O q (GL 3 )/J) in [12, Figure 5]. These can be translated into results for O q (SL 3 ) via the Levasseur- [21]; since this isomorphism maps D q to z, and D q does not belong to any H-prime of O q (GL 3 ), it is easily verified that where by a slight abuse of notation we use J to denote both a H-prime in O q (GL 3 ) and its projection to O q (SL 3 ). In other words, as long as we can find a presentation of Z(O q (GL 3 ) J ) in which D q appears as one of the generators, we can then obtain a presentation for Z(A J ) simply by deleting D q from the list of generators. In all but three cases, namely the result now follows immediately by comparing Table 1 with [12, Figure 5]. (Note that whenever there is a product of quantum minors in [12, Figure 5], we will always use the order specified in that table.) The remaining three cases can be handled by a simple change of variables; for example, when J = I 132,132 we have D q = X 11  We start by constructing sets E K for each K; these will correspond to the case J = 0, and the general sets E JK will follow easily as projections of these.
Definition 5.5. For a H-prime K appearing in position (ω + , ω − ) of Figure 1, define E K to be the multiplicative set generated in B by where E ω+ and E ω− are defined in Figure 2.
Remark 5.6. In the quantum setting, we need to know that a multiplicative set satisfies the Ore conditions (for example, [15,Chapter 6]) before attempting to localise at it. For all of the localizations we will consider in this paper (and in particular for the sets E K in Definition 5.5) we get this condition for free, thanks to the descriptively named paper 'Every quantum minor generates an Ore set' [25].
Remark 5.7. The commutation relations in O q (SL 3 ) (respectively, the Poisson bracket in O(SL 3 )) mean that we cannot always easily identify which minors belong to a given ideal. Solving this problem has been the subject of considerable recent work, for example, [3,5,10]. The choice of generating sets in Figure 2 in particular is informed by the general theory of Cauchon diagrams and techniques from the study of totally non-negative matrices, which we do not discuss here. The interested reader is referred to the survey [19] for more detail on this.
We also highlight the following key results, which hold when k is a field of characteristic zero and q ∈ k × is not a root of unity. (ii) All prime ideals in O q (SL 3 ) are completely prime (that is prime in the commutative sense) [1, Corollary II.6.10].
Lemma 5.8. For each H-prime K in B and its corresponding multiplicative set E K , we have Proof. This can be verified directly (for example, by computer) using the results of [10, Definition 2.6] to construct a complete list of the minors belonging to a given H-prime.
For those familiar with Cauchon diagrams and their relationship to H-primes, there is also a more elegant proof: given a H-prime K and its Cauchon diagram C K , observe that the generating minors of E K are chosen to be exactly those minors defined by lacunary sequences (see [20,  We can now verify that the sets E K have the property (M2) in the case where J = 0.
Proof. Fix a H-prime L = 0. For any K with L K, there must be at least one element u ∈ L\K; in fact, we can take u to be a generator of L, that is u ∈ {W 12 , W 13 , W 21 , W 23 , W 31 , W 32 , [23|12], [12|23]}. We will consider each of these elements in turn and show that if u ∈ L and K is a H-prime with u ∈ K, then L ∩ E K = ∅.
First suppose that u ∈ {W 13 , W 23 , W 31 , W 32 }, and let K be a H-prime with u ∈ K. Comparing Figures 1 and 2, we see that whenever u does not appear as a generator of K, it always appears as a generator of the corresponding set E K ; applying Lemma 5.8, it follows that u ∈ E K if and only if u ∈ K. Since u ∈ L\K by assumption, we have u ∈ L ∩ E K = ∅ as required.
Now suppose that u = [23|12] ∈ L. If K is a H-prime such that [23|12] ∈ K, then it must lie somewhere in rows 321 or 231 of Figure 1 (since these are the only two rows whose H-primes do not feature [23|12] as either a generator or an obvious corollary of the generators). But [23|12] appears as a generator for E K whenever ω + = 321 or 231, and so u ∈ L ∩ E K as required. A similar argument handles u = [12|23] (with ω − = 321 or 312).
Finally, suppose that u = W 21 ∈ L. Using the relation X 21 X 32 − X 32 X 21 = (q − q −1 )X 22 X 31 in the quantum case and {Y 21 , Y 32 } = 2Y 22 Y 31 in the Poisson case, we can conclude that W 31 ∈ L as well: this follows from the complete primality of K and the fact that W 22 ∈ K would imply 1 ∈ K (for example, [6, Lemma 5.1.5]). It follows that Since K ∩ E K = ∅ by Lemma 5.8, it makes sense to consider the image of the sets E K modulo another H-prime J ⊂ K.
Definition 5.10. Let J ⊂ K be H-primes, and let E K be as in Definition 5.5. Define E JK to be the image of We are now in a position to describe the algebras Z JK and PZ JK in the setting of SL 3 .
Proof. By Lemma 5.9, we have E 0K ∩ L = ∅ in B, that is, there exists some non-zero v ∈ L ∩ E 0K = L ∩ E K . This element v continues to be non-zero in B/J, since v ∈ K and J ⊂ K. We therefore have v + J ∈ L/J ∩ E JK = ∅, as required.
, let Z JK be defined as in [2,Definition 3.8] and PZ JK defined as in (3.2). Then for the set E JK in Definition 5.10, we have Proof. In the Poisson case, this follows directly from Lemmas 3.7 and 5.11. For the quantum case, combine [2, Lemma 3.9], Lemma 5.11 and [25].
A standard tactic with algebras of this type is to invert certain elements and then make strategic changes of variables in order to simplify the commutation (respectively, Poisson bracket) relations. In particular, our aim will be to reduce as many of the relations as possible to skew-commutation relations (or their Poisson equivalents).
Definition 5.13. Let q = (a ij ) be an additively skew-symmetric n × n matrix for some integer n 2, and r an integer such that 0 r n. Define the skew-commuting k-algebra associated to q, r and n by When r = 0, this is a uniparameter quantum affine space on n variables. We define the corresponding Poisson algebra by In some cases, it will be convenient to list the generators of these algebras in a different order (for example, to list the uninverted variables before the inverted ones); for notational simplicity, we will treat reorderings of the same algebra as equal. Given a pair of algebras A r,n q and P r,n q , if there is a choice of reordering of the variables in each algebra such that q = q , then we will say that the two algebras are compatible.
The centres of these algebras are particularly easy to compute, as we now describe.
Lemma 5.14. Let q be an additively skew-symmetric n × n matrix, and A r,n q , P r,n q the corresponding quantum and Poisson algebras for some 0 r n (see Definition 5.13). Then, Z(A r,n q ), PZ(P r,n q ) are each generated by their (Poisson) central monomials, and if Proof. The first statement is an easy generalisation of [8, § 9.6], and the second statement is proved in [22, § 2] when r = n. The result for r < n now follows easily by observing Z A r,n q = Z A n,n q ∩ A r,n q and PZ P r,n q = PZ P n,n q ∩ P r,n q , and the fact that the (Poisson) centre is generated by the (Poisson) central monomials.
The fact that we obtain genuine isomorphisms rather than quotients of A r,n q and P r,n q follows from a GK dimension argument as in [12,Lemma 4.3(b)]. , respectively (using the same idea as in Example 5.16 above); the advantage of this is that these new elements are (Poisson-)normal. We have obtained the compatible algebras Remark 5.18. If we compute the matrix q for the algebras in (5.5) or (5.6) and apply Remark 5.15, we find that the centres of these algebras are k in each case. This observation will be useful in Theorem 5.19, since any algebra that embeds into one of the algebras (5.5) or (5.6) will also have trivial centre.
We will not usually compute the centres PZ JK and Z JK explicitly; for our purposes, it will suffice to show that for each pair of H-primes J ⊂ K the centres PZ JK and Z JK are isomorphic and that we can move between them in an intuitive way. We are now in a position to make this idea precise and to prove our first main theorem. (Recall that θ is the preservation of notation map from Definition 5.2, which replaces each quantum minor with the corresponding minor.) Proof. The proof is by case-by-case analysis. In each case, we fix some H-prime J and consider all H-primes K such that J ⊂ K. Recall that when referring to a H-prime in terms of the generating sets in Figure 1, I ω+,ω− will denote the H-prime listed in position (ω + , ω − ).
Recall also that there is a transposition automorphism τ : W ij −→ W ji defined on both O q (SL 3 ) and O(SL 3 ). From Figures 1 and 2, we can easily check that τ (I ω+,ω− ) = I ω −1 , so that if I ω ⊂ I ω , then τ induces an isomorphism of k-algebras or Poisson algebras where we use ω −1 as a shorthand for (ω −1 − , ω −1 + ). This symmetry will allow us to reduce the number of examples we need to compute. Recall from [2, equation (3.7)] that Z JK is a subalgebra of Z(A J ), and similarly from Section 3 that PZ JK is a subalgebra of PZ(R J ). In each of the 12 cases listed here, we see immediately from [12, Figure 5] and Table 1 that Z(A J ) and PZ(R J ) are both trivial, and hence Z JK = PZ JK = k for all K ⊃ J as well.
Case II. J = I ω , for ω one of: In each of these cases, we will see that O q (SL 3 )/J and O(SL 3 )/J are already compatible algebras of form A r,n q , P r,n q , and that localising at the set E JK just corresponds to inverting some generators of these algebras. Consider, for example, J = I 132,123 , where it is easily computed that we have Only one H-prime lies above J, namely K = I 123,123 , the maximal H-prime.
JK ] = P 2,3 q (with q as in (5.7)). We can now apply Lemma 5.14 to complete the proof in this case, and the other three cases follow by similar analysis.
There are seven H-primes lying above J, but in each case inverting the remaining elements of E JK corresponds to inverting some of the generators in (   This allows us to identify each pair of algebras Z(A J ) and PZ(R J ), and similarly Z JK and PZ JK ; this induces a natural homeomorphism between the spectra of each pair of algebras as well, which by an abuse of notation we will also denote by Θ J (respectively, Θ JK ). The following corollary now follows immediately from the definitions of the maps f JK and g JK .
Corollary 5.20. Let J ⊂ K be a pair of H-primes in A (respectively, R), and let Y be a closed set in spec(Z(A J )). Then We introduce a few final pieces of notation in order to make the statement of the final theorem precise. For each H-prime J ∈ H-spec(A), let h q J denote the homeomorphism spec J (A) ≈ spec(Z(A J )) from the Stratification Theorem, and similarly for each Poisson H-prime J ∈ H-pspec(R), let h p J denote the homeomorphism pspec J (R) ≈ spec(PZ(R J )). Since there are now many different maps to keep track of, ranging from homomorphisms to homeomorphisms to maps on closed sets, Figure 3 may help to highlight the most relevant ones and how they fit together. Then, ψ is a bijection, and both ψ and ψ −1 preserve inclusions, that is ψ is a homeomorphism with respect to the Zariski topology.
Proof. It is clear that ψ is a bijection, since it is patched together from the homeomorphisms ψ J . To prove that ψ and ψ −1 preserve inclusions, it is enough to check that ψ(V (P )) = V (ψ(P )) for each P ∈ spec(A). So we fix some P ∈ spec(A), and let J be the H-prime such that P ∈ spec J (A). Now for any H-prime K ⊇ J, we have ψ (V (P ) ∩ spec K (A)) = ψ K • ϕ JK (V (P ) ∩ spec J (A)) = ϕ JK • ψ J (V (P ) ∩ spec J (A)) , where the first equality follows from the definition of ϕ JK and the second from the fact that if we view all of the maps in Figure 3 as maps between spaces of closed sets (so we are = V (ψ(P )) ∩ pspec K (R), and the result follows since V (P ) is the disjoint union of the V (P ) ∩ spec K (A) for K ⊇ J, and similarly V (ψ(P )) is the disjoint union of the V (ψ(P )) ∩ pspec K (A).
We may also restrict our attention to just the primitive ideals, where we can give an explicit description of the action of this homeomorphism as follows. obtained by restricting the homeomorphism of Theorem 5.21 to the space of primitive (respectively, Poisson primitive) ideals. Furthermore, for each primitive ideal P ∈ prim(O q (SL 3 )), we can find a generating set with each U i and V i products of quantum minors (we take the empty product of minors to be 1), such that ψ(P ) = θ(U 1 ) − λ 1 θ(V 1 ), . . . , θ(U d ) − λ d θ(V d ) .
Proof. By [8,Lemma 9.4(c)], the homeomorphism from Theorem 5.21 restricts to a homeomorphism (which we also denote by ψ) on the primitive level. Generating sets for the primitive (respectively, Poisson primitive) ideals of O q (SL 3 ) (respectively, O(SL 3 )) are given in [12, Figure 7] (respectively, Table 2), and the second part of the result now follows easily.