Quantum solvable algebras. Ideals and representations at roots of 1

A. N. Panov.

January 3, 2002

Abstract. There proved that every prime stable with respect to quantum adjoint action ideal I is completely prime and Fract(R/I) is isomorphic to the skew field of fractions of an algebra of twisted polynomials. We study correspondence between symplectic leaves and irreducible representations. The Conjecture of De Concini-Kac-Procesi on dimension of irreducible representations is proved for sufficiently great l.

1 Introduction.

A quantum solvable algebra is an iterated q-skew extension, constructed by derivations and diagonal automorphisms. Among examples are Quantum Matrices, Quantum Weyl algebra, Quantum Heisenberg algebra, U_{q}(n) and also some their subalgebras and factor algebras. Representations of these algebras were treated separately ([DC-K-P],[DC-P],[JZ]). The main approach is correspondence between irreducible representations of a quantum algebra and symplectic leaves of Poisson manifold of the center.

In the paper we study quantum solvable algebras in general setting, obeying some natural and easily checkable conditions (see Conditions 2.2-2.4). We consider the specializations of these algebras at primitive l-root of unity ε.

We make some assumptions on l. First, we claim that ε is rather great: ε is a point of good reduction of stratification of prime spectra for intederminate q (see Definition 2.8, Theorem 2.9). Second, we claim that ε is relatively prime with all minors of integer matrix 𝕊 (Definition 2.1). We say that ε is admissible, if it obeys the above conditions. There are infinitely many admissible roots of unity.

We study prime ideals stable with respect to quantum adjoint action. We prove that these ideals have prime intersections with components of filtration of quantum solvable algebras (Theorem 1.13). All prime and stable with respect to quantum adjoint action ideals are completely prime (Theorem 2.20). We get stratification of these ideals. This helps us to study irreducible representations and symplectic leaves (Theorem 3.3). In particular, we prove that dimension of any irreducible representation equals to l raising to the power of one-half of dimension of corresponding symplectic leaf (Conjecture of De Concini-Kac-Procesi).

2 On quantum adjoint action and stable ideals

In this section we study q-skew extensions E = R[x; τ, δ]. The main statement of this section is Theorem 1.13 on reduction of prime ideals.

Throughout the paper we use the following notations. Let K be a field of zero characteristic and q is an indeterminate. We denote by C a localization of K[q, q^{−1}] over some finitely generated denominator set and F = Fract(C). Let Γ be the cyclic subgroup {q^{n}}_{n∈ℤ} in the group of invertible elements C^{∗} of C. Let ε be a primitive l-root of unity. Denote by K(ε) the field extension of K (if ε ∈ K, we put K(ε) = K). Suppose further that C admits specialization via q ↦ ε in the field K(ε).

Throughout this section R is a C-algebra and a free C-module. We consider specialization ρ_{ε} : R ↦ R_{ε} = Rmod(q − ε).

Further we differ the elements in R with their images in R_{ε}, using the following notations. For x ∈ R we denote x_{ε} = xmod(q − ε)R or for y ∈ R_{ε} we denote by \underline{y} an element in its preimage in R.

Let u be an element in R such that u_{ε} = umod(q − ε) lies in the center Z_{ε} of R_{ε}. For any \underline{a} ∈ R the element ad_{u}(\underline{a}) = u\underline{a} − \underline{a}u lies in (q − ε)R. Whence \frac{ad_{u}(\underline{a})}{(q−ε)} ∈ R. We denote

D_{u}(a) = \frac{ad_{u}(\underline{a})}{(q − ε)}mod(q − ε)  (1.1)

for a = \underline{a}mod(q − ε) ∈ R_{ε}. Definition 1.1. The map D_{u} : R_{ε} ↦ R_{ε} is called the quantum adjoin action of u ∈ R. We recall some properties of the quantum adjoin action [DC-K-P]. Property 1.2. D_{u} is a derivation of R_{ε}. That is D_{u}(ab) = D_{u}(a)b + aD_{u}(b) for all a, b ∈ R_{ε}. Property 1.3. If umod(q − ε) = u_{1}mod(q − ε), then u_{1} − u = (q − ε)r, r ∈ R and D_{u_{1}} = D_{u} + ad_{r}. This implies that an ideal P is D_{u_{1}}-stable whenever it is D_{u}-stable. Property 1.4. If u_{ε}, v_{ε} ∈ Z_{ε} for u, v ∈ R, then

D_{uv}(a) = D_{u}(a)v_{ε} + u_{ε}D_{v}(a)

Property 1.5. The bracket

{u_{ε}, v_{ε}} = D_{u}(v_{ε}) = −D_{v}(u_{ε})

is a Poisson bracket on Z_{ε}.

Consider a skew polynomial extension E = R[\underline{x}; τ, δ] where τ is an automorphism and δ is τ-derivation of R. That is δ(ab) = δ(a)b + τ(a)δ(b). We assume that τ|_{C} = id, δ|_{C} = 0 and suppose that E is q^{s}-skew extension of R. That is τδ = q^{s}δτ for some s ∈ ℤ. One can extend the automorphism τ upto an automorphism of E, putting τ(\underline{x}) = q^{−s}\underline{x}. We save the notation τ for this extension.

All prime ideals in q-skew iterated extension of C are completely prime ([GL1], Theorem 2.3). It follows that the intersection with R of a prime ideal of E is a prime (indeed, completely prime) ideal in R. But in the case of roots of unity the intersection with R_{ε} of a prime ideal of E_{ε} may be not prime (even not semiprime) (see Example 2.6). The main goal of this section is to prove that intersection remains to be prime in the case of roots of unity if we claim that a given ideal in stable with respect to the quantum adjoint action. The following assumptions are valid till the end of this section. Assumptions 1.6 1) As above ε is a primitive l-root of unity. We suppose that GCD(s, l) = 1 if s ≠ 0.

2) Suppose that x^{l}, x = \underline{x}mod(q − ε) lie in the center Z_{ε} of E_{ε} = E mod(q − ε). We shall denote by D_{x} the derivation D_{\underline{x}}_{l} defined by (1.1).

3) We assume that τ Γ-diagonalizable automorphism of R. That is, there exists a C-basis {a_{α}} of R such that τ(a_{α}) = q^{m_{α}}a_{α} for some m_{α} ∈ ℤ. It follows that the similar basis also exists in E.

One can extend τ and δ upto automorphism and τ-derivation of R_{ε}. We save the same notations for this extensions. Further we use usial notations

1) (n) = (n)_{q^{s}} = \frac{q^{sn}−1}{q−1};

2) (n)! = (n)_{q^{s}} = (1)(2) ⋯ (n);

3) \binom{n}{k} = \binom{n}{k}_{q^{s}} = \frac{(n)!}{(k)!(n−k)!}.

Lemma 1.7. Under the above assumptions δ^{l} = 0mod(q − ε), τ^{l} = id mod(q − ε). Proof. One can prove ([G],6.2;[GL],2.5)

\underline{x}^{l}a = ∑_{i=0}^{n} \binom{l}{i}τ^{l−i}δ^{i}(a)\underline{x}^{l−i} = τ^{l}(a)\underline{x}^{l} + ∑_{i=1}^{n−1} \binom{l}{i}τ^{l−i}δ^{i}(a)\underline{x}^{l−i} + δ^{l}(a)  (1.2)

Since the element x^{l} = \underline{x}^{l}mod(q − ε) lie in the center Z_{ε}, then τ^{l}(a) = amod(q − ε) and δ^{l}(a) = 0mod(q − ε).□

We shall use the following notations

Δ = \frac{δ^{l}}{q − ε} : R_{ε} ↦ R_{ε}

θ = \frac{τ^{l} − id}{q − ε} : E_{ε} ↦ E_{ε}.

Remark that Δ is a linear operator on R_{ε} and θ is a derivation of E_{ε}. For any integer m we denote

\underline{m} = (\frac{q^{ml} − 1}{q − ε})_{q−ε} = mlε^{−1}.

Notice that, if τ(a) = q^{m}a for some a ∈ E and m ∈ ℤ, then θ(a_{ε}) = \underline{m}a_{ε}.

Consider reduction of C-basis a_{α} modulo q−ε. We get a ℂ-basis a_{αε} = a_{α}mod(q−ε). The derivation θ is diagonalizable θ(a_{αε}) = \underline{m}_{α}a_{αε}. Lemma 1.8 Let R be a free C-algebra. Let τ be a diagolizable automorphism of R, obeying Assumption 1.6(3). Consider an ideal I in R_{ε}. By I_{θ}, we denote the largest θ-stable ideal in I. We assert that the ideal I is prime whenever I_{θ} is prime. Proof. If I is prime, then I_{θ} is also prime ([G],6.4). Suppose that I_{θ} is prime and I is not prime. There exist the elements a, b ∈ R_{ε} such that aR_{ε}b ⊂ I and a, b ∉ I. Decompose a and b into the sums of θ-eigenvectors with different eigenvalues: a = a_{1} + ⋯ + a_{k}, b = b_{1} + ⋯ + b_{p}, θ(a_{i}) = \underline{m}_{i}a_{i}, θ(b_{j}) = \underline{n}_{j}b_{j}, m_{1} > ⋯ > m_{k}, n_{1} > ⋯ > n_{p}. We consider that all eigenvectors a_{1}, …, a_{k}, b_{1}, …, b_{p} don’t lie in I. Let c be θ-eigenvector: θ(c) = \underline{z}(c), z ∈ ℤ. The element acb lies in I and have decomposition

acb = (a_{1} + ⋯ + a_{k})c(b_{1} + ⋯ + b_{p}) = a_{1}cb_{1} + ⋯ + a_{k}cb_{p}

into a sum of θ-eigenvectors. The term a_{1}cb_{1} have the greatest eigenvalue m_{1} + z + n_{1}. Since acb ∈ I, then a_{1}cb_{1} ∈ I_{θ}. Whence a_{1}R_{ε}b_{1} ⊂ I_{θ}. As far as I_{θ} is prime, we obtain that a_{1} ∈ I or b_{1} ∈ I. A contradiction. The ideal I is prime. □

Consider the localization R′ of R over the set

{f(q) ∈ C : f(ε) ≠ 0}.

Lemma 1.9. Let Assumption 1.6(1) holds. Let k be a positive integer. Suppose that k = ξl + η with positive integers ξ and 0 ≤ η < l. As above δ^{l}(R) ⊂ (q − ε)R. Then \frac{δ^{l}}{(k)!}(R′) ⊂ R′ Proof. For s = 0 the statement is trivial. Let s ≠ 0. For a in R we have

δ^{k}(a) = δ^{ξl+η}(a) = δ^{η}(δ^{l})^{ξ}(a) ⊂ (q − ε)^{ξ}δ^{η}(R) ⊂ (q − ε)^{ξ}R

On the other hand, (q − ε) divides (i) = \frac{q^{is}−1}{q−1} if and only if l divides i. It follows (k)! = (q − ε)^{ξ}c(q) where c(ε) ≠ 0. This implies the assertion.□

For all g ∈ R we denote

Π_{n}(g) = τ^{n−1}(g) ⋯ τ(g)g

Lemma 1.10. Let τ, δ, be as above. Suppose that s ≠ 0. Let a ∈ R and let I_{a} be a minimal τ-ideal generated by a in R′. Let n, m be positive integers. We assert that

\frac{δ^{n}(a^{m})}{(n)!} = \{ \begin{matrix}Π_{n}(δ(a))modI_{a}, & for m = n, \\ 0 modI_{a} & for n < m. \end{matrix}

Proof. We shall prove be induction on n. The case n = 1 is easy. Suppose that statement is true for all positive integers less then n. We are going to prove for n. We need the formula ([G],6.2):

δ^{n}(ab) = ∑_{i=0} \binom{n}{i}τ^{n−i}δ^{i}(a)δ^{n−i}(b).  (1.3)

We get

δ^{n}(a^{m}) = δ(a^{m−1}a) = ∑_{i=0} \binom{n}{i}τ^{n−i}δ^{i}(a^{m−1})δ^{n−i}(a).

We obtain

\frac{δ^{n}(a^{m})}{(n)!} = ∑_{i=0}^{n−2} τ^{n−i} \frac{δ^{i}(a^{m−1})}{(i)!} \frac{δ^{n−i}(a)}{(n − i)!} + \frac{τδ^{n−1}(a^{m−1})}{(n − 1)!}δ(a) + \frac{δ^{n}(a^{m−1})}{(n)!}a  (1.4)

Let n < m. By assumption of induction, \frac{δ^{i}(a^{m−1})}{(i)!} ∈ I_{a} for i ≤ n − 1. By Lemma 1.9,

\frac{δ^{n−i}(a)}{(n−i)!} ∈ R′. This proves that all terms in (1.4), besides the last one, lie in I_{a}. As about the last term, it obviously lie in I_{a}. This concludes case n < m.

Let m = n. Similarly to previous case, we get

\frac{δ^{n}(a^{n})}{(n)!} = \frac{τδ^{n−1}(a^{n−1})}{(n − 1)!}δ(a)modI_{a} = τ(Π_{n−1}(δ(a)))δ(a)modI_{a} = Π_{n}(δ(a))modI_{a}

□

Let I be an ideal in E_{ε}. Denote J = I ⋂ R_{ε}. Lemma 1.11. Suppose that I is θ-stable . Then

1) I is τ-stable;

2) The ideal J is (τ, δ)-stable. If in addition I is D_{x}-stable, then J is Δ-stable ideal in R_{ε}. Proof.

1) The automorphism τ : E ↦ E is diagonalizable. This implies that τ, θ : E_{ε} ↦ E_{ε} are diagonalizable with the common system of basic eigenvectors {a_{α}}. Recall that τ(a_{α}) = ε^{m_{α}}a_{α} and δ(a_{α}) = \underline{m}_{α}a_{α}. Every θ-eigenvector lies in a span of basis elements a_{αε} corresponding to a common θ-eigevalue \underline{m}. Then all these a_{αε} have the common τeigenvalue ε^{m}. This proves that every θ-eigenvector is also τ-eigenvector (the converse is not true).

Let a be an element in I. Decompose a into a sum of θ-eigenvectors with different eigenvalues (call θ-components). Since I is θ-stable, all θ-components also lie in I. We see above that this θ-components are τ-eigenvectors. This proves that τ(a) ∈ I. We conclude that I is τ-stable.

2) As far as I is (τ, θ)-stable, the ideal J is also (τ, θ)-stable. Since xa = τ(a)x + δ(a) and a ∈ J, then xa, τ(a)x ∈ I. Whence δ(a) ∈ J. This proves that J is δ-stable.

Suppose that I is D_{x}-stable. stableLet a ∈ J. By direct calculations,

D_{x}(a) = θ(a)x^{l} + ∑_{i=1}^{l−1} \frac{\binom{l}{i}q^{s}}{q − ε}τ^{l−i}(δ^{i}(a))x^{l−i} + Δ(a)  (1.5)

Since D_{x}(a) ∈ I, θ(a) ∈ J, we obtain Δ(a) ∈ J. □ Lemma 1.12.

1) δ(θ(a)) = (θ + \underline{s})δ(a) for all a ∈ R_{ε};

2) D_{x}(τ(a)) = τD_{x}(a) and D_{x}(θ(a)) = (θ + \underline{sl})D_{x}(a) for all a ∈ E_{ε}. Here \underline{s} = slε^{−1} and \underline{sl} = sl^{2}ε^{−1}. Proof. Direct calculations. Recall that we denote D_{x} = D\underline{x}_{l}. Theorem 1.13. Let R be a Noetherian C-algebra and a free C-module. Given q^{s}- skew extension E = R[\underline{x}; τ, δ], obeying Assumptions 1.6. Let I be a prime D_{x}-stable ideal in E_{ε}. Then the ideal J = I ⋂ R_{ε} is also prime. Proof. The proof divides in several steps. In the points 1 and 2 we shall prove the statement in the case I is θ-stable ideal. In the last point 3 we shall consider the general case.

1) Suppose that the ideal I is θ-stable. In this point we are going to prove J is semiprime. In the case s = 0 this follows from ([G],6.5). Let further s ≠ 0.

By Lemma 1.11, the ideal J is (τ, δ)-stable Then EJ is a two-sided ideal in E_{ε}. The ideal J is (τ, δ)-prime ideal in R_{ε}. ([GL2],2.1).

Let N be an ideal in R_{ε} such that N/J = Rad(R/J). Hence, N^{n} ∈ J for some positive integer n. We are going to prove that N = J.

Radical N is stable under actions of automorphisms and derivations (see [MC-R],14.2.3). In particular N is (τ, θ)-stable.

Suppose that N is δ-stable. Then EN is a two-sided ideal in E_{ε}. We see that (EN)^{n} ⊂ EN^{n} ⊂ EJ ⊂ I. Since I is prime, we have EN ⊂ I, whence N ⊂ J. This proves that J is semiprime.

The above observation shows that it remains to prove that N is δ-stable. Consider the image L = δ(N)mod(N) in R_{ε}/N. Since

δ(N) = δ(R_{ε}NR_{ε}) = δ(R_{ε})NR_{ε} + τ(R_{ε})δ(N)R_{ε} + τ(R_{ε})τ(N)R_{ε}

we see that L is an ideal in B = R/N. Moreover, by Lemma 1.12(1), the ideal L is (τ, θ)-stable. The algebra B is semiprime. Its Goldie quotient algebra A = Goldie(B) is a direct sum of matrix algebras over division rings A = ⊕_{i=1}^{N}A_{i}. Automorphism τ acts on components A_{i} by permutations. The ideal ALA is the sum A′ = ⊕_{i=1}^{s}A_{i} which is τ-stable. There exist the elements r_{1}, …, r_{k}, p_{1}, …, p_{k} ∈ A and h_{1}, …, h_{k} ∈ L such that

1_{A′} = r_{1}h_{1}p_{1} + ⋯ r_{k}h_{k}p_{k}.

Multiplying this equality by suitible regular elements of B, we show that L contains a regular element of A′, say h. This implies that Π_{n}(h) ≠ 0 for all n.

Denote by a an element of R_{ε} such that a ∈ N, δ(a) = h. Denote by \underline{a} ∈ R a representative of a ∈ R_{ε}.

By Lemma 1.10, we have

\frac{δ^{n}(\underline{a}^{n})}{(n)!} = Π(h) + \underline{b}

with some \underline{b} ∈ I\underline{a}. As in Lemma 1.9 decompose n = ξl + η, 0 ≤ η < l and ξ,η are positive integers. Denote

\underline{d} = \frac{δ^{n}(\underline{a}^{n})}{(q − ε)^{ξ}} = c(q)(Π(h) + \underline{b})

with c(ε) ≠ 0. We have proved that the element d = \underline{d}mod(q − ε) equals to A′-regular element c(ε)Π_{n}(h) modulo N.

On the other hand,

\underline{d} = \frac{δ^{n}(\underline{a}^{n})}{(q − ε)^{ξ}} = (\frac{δ^{l}}{q − ε)})^{ξ} δ^{η}(\underline{a}^{n}) = Δ^{ξ}δ^{η}(a^{n})mod(q − ε),

d = Δ^{ξ}δ^{η}(a^{n}).

Recall that a^{n} ∈ N^{n} ∈ J and J is (δ, Δ)-stable. This follows d ∈ J. Whence d = 0modN. A contradiction. The ideal N is δ-stable and J is semiprime.

2) In this point we shall prove that J is prime in the case I is θ-stable.

Since J is semiprime, one can present it as an intersection of prime ideals J = Q_{1} ⋂ ⋯ ⋂ Q_{N}. If a derivation save J, then it save all Q_{i} ([MC-R], 14.2.3). This implies that all ideals Q_{i} are θ and δ stable. Whence Q_{i} is τ-stable. Automorphism τ-acts on the system of central minimal idempotents e_{1}, …, e_{N} by permutations. Decompose A = ⊕_{i=1}^{N}A_{i}, A_{i} = Ae_{i}. Since τ(A_{i}) ⊂ A_{i}, then τ(e_{i}) = e_{i} and

δ(e_{i}) = δ(e_{i}^{2}) = δ(e_{i})e_{i} + τ(e_{i})δ(e_{i}) = 2e_{i}δ(e_{i}).

This proves δ(e_{i}) = 0. Whence δ(Q_{i}) ⊂ Q_{i} for all i. Since J is (τ, δ)-prime, J = Q_{i} . Finally, J is prime.

3) In this point we conclude the proof. Let I be a prime D_{x}-stable ideal in E_{ε}. By Lemma 1.8, I_{θ} is also prime. Let us prove that I_{θ} is D_{x}-stable. Note

I_{θ} = {a ∈ E_{ε} : θ^{k}(a) ∈ I for all k = 0, 1, 2 …}.

Given a ∈ I_{θ}. By Lemma 1.12, we have

θ^{k}(D_{x}(a)) = D_{x}(θ − \underline{sl})^{k}(a) ∈ D(I) ⊂ I

This proves that D_{x}(a) ∈ I_{θ} and I_{θ} is also D_{x}-stable and prime.

By points 1-2 of the proof, the ideal I_{θ} ⋂ R_{ε} is prime. The last ideal coinsides with J_{θ}. By Lemma 1.8, J is prime. □

3 Stratification of D-stable ideals

In this section we are going to get stratification of prime D-stable ideals at roots of unity. This provides a reduction to the case of algebra of twisted Laurent polynomials. We prove that all prime D-stable ideals are completely prime.

Recall K is a field of zero characteristic, q is an indeterminate and C is a localization of K[q, q^{−1}] over some finitely generated denominator set. Let 𝕊 = (s_{ij}) be a n × n integer skew-symmetric matrix. Denote q_{ij} = q^{s_{ij}} and form the matrix ℚ = (q_{ij}). Definition 2.1([P1],[P2]). We say that a ring R is quantum solvable over C, if R is generated by the elements x_{1}, x_{2}, …, x_{n}, k_{1}^{±1}, …, k_{m}^{±1} such that the monomials x_{1}^{t_{1}} ⋯ x_{n}^{t_{n}}k_{1}^{s_{1}} ⋯ k_{m}^{s_{m}} with t_{i} ∈ ℕ,s_{i} ∈ ℤ form a free C-basis with the relations

1) the elements k_{1}, …, k_{m} q-commute with all generators x_{1}, x_{2}, …, x_{n}, k_{1}^{±1}, …, k_{m}^{±1} ;

2) for i < j the holds

x_{i}x_{j} = q_{ij}x_{j}x_{i} + r_{ij},  (2.1)

where r_{ij} is the element of subalgebra R_{i+1} generated by x_{i+1}, …, x_{n}, k_{1}^{±1}, …, k_{m}^{±1} .

There exists the chain of subalgebras R = R_{1} ⊃ R_{2} ⊃ ⋯ ⊃ R_{n} ⊃ R_{n+1}. The last algebra R_{n+1} is an algebra of twisted Laurent polynomials generated by k_{1}^{±1}, …, k_{m}^{±1}. One can prove that a quantum solvable algebra is an iterated skew extension of C (see [MC-R],[GL1]). This means that for all i ∈ [1, n] the map τ_{i} : x_{j} ↦ q_{ij}x_{j}, i > j is extended to automorphism of R_{i+1} and the map δ_{i} : x_{j} ↦ r_{ij} is extended to τ_{i}-derivation of R_{i+1}. Each algebra R_{i} is the skew extension R_{i+1}[x_{i}; τ_{i}.δ_{i}]. All automorphisms τ_{i} are identical on C and all τ_{i}-derivations δ_{i} are equal to zero on C. A quantum solvable algebra is a Noetherian domain ([MC-R],1.2.9). We put some more conditions on R.

This conditions are comparable with more general conditions Q1-Q4 of ([P2]) Condition 2.2. We claim that R is an iterated q-skew extension in terms of ([GL1,2]). This means that δ_{i}τ_{i} = q_{i}τ_{i}δ_{i} for some q_{i} = q^{s_{i}}, s_{i} ∈ ℤ. Condition 2.3. All automorphisms τ_{i} are extended to diagonal automorphisms of R (i.e the monomials x_{1}^{t_{1}} ⋯ x_{n}^{t_{n}}k_{1}^{s_{1}} ⋯ k_{m}^{s_{m}}, t_{i} ∈ ℕ,s_{i} ∈ ℤ are eigenvectors) with eigenvalues in Γ. Denote by H the group of diagonal automorphisms of R generated by τ_{1}, …, τ_{n}. Condition 2.4. The exists an infinite set of positive integers with the property. For any element l of this set the elements x_{1}^{l}, …, x_{n}^{l} modulo q − ε lie in the center of R_{ε} (ε is a primitive l-root of unity).

The last Condition claims that R_{ε} is finite over its center for almost all prime ideals of C. Such algebras R are called pure C-algebras in the paper [P1]. The Conditions 2.2-2.4 imply the Conditions Q1-Q4 of the paper [P2] (in the case C is a localization of K[q, q^{−1}] over some finitely generated denominator set). See [P2] for examples. One get stratification of prime ideals with zero intersection with C. Theorem 2.5([P2], Theorem 3.4). Let R be a quantum solvable algebra obeying Conditions 2.2-2.4.

1) There exists the finite set 𝕄 = {P_{μ}}, μ = (i_{1}, …, i_{k+1}), i_{s} ≥ 0, i_{s} ∈ ℤ, k + i_{1} + ⋯ + i_{k+1} = n of semiprime ideals with zero intersections with C. There exists the denominator subset S_{μ} ⊂ R/P_{μ} generated by k q-commuting elements. The localization (R/P_{μ})S_{μ}^{−1} is isomorphic to a factor algebra of an algebra of twisted Laurent polynomials.

2) Any prime ideal I of R with zero intesection with C (recall that indeed all such ideals are completely prime) contains a unique ideal P_{μ} such that S_{μ} ⋂(ImodP_{μ}) = ∅.

The stratification of Spec(R) takes place. We shall recall the stratification of prime ideals of R, refering to ([P2], Theorem 3.4) for complete proof.

Consider the subset L_{1} = {x_{n}, x_{n−1}, …, x_{n−i_{1}+1}} of last i_{1} generators of R. Let J_{1} be minimal semiprime ideal containing L_{1}. Denote by X_{1} the set of minimal prime ideals over J. We decompose X_{1} = X′_{1} ⋃ X′′_{1} ⋃ X′′′_{1} where

X′_{1} = {Q ∈ X_{1} : Q ⋂ C = 0, x_{n−i_{1}} ∉ Q, },
X′′_{1} = {Q ∈ X_{1} : Q ⋂ C ≠ 0, x_{n−i_{1}} ∉ Q},
X′′′_{1} = {Q ∈ X_{1} : x_{n−i_{1}} ∈ Q}.

We denote

P_{i_{1}} = ⋂_{Q∈X′_{1}} Q.  (2.2)

The ideal P_{i_{1}} contains L_{1}, does not contain x_{n−i_{1}} and P_{i_{1}} ⋂ C = 0. Denote by S_{1∗} the denominator subset {x_{n−i_{1}}^{m}}_{m∈ℕ}. Consider the localizaton R^{(1)} = (R/P_{i_{1}})S_{1∗}^{−1}. The algebra R^{(1)} is generated by x_{1}, …, x_{n−i_{1}−1}, x_{n−i_{1}}^{±1} and admits the filtration

R^{(1)} = R_{1}^{(1)} ⊃ ⋯ ⊃ R_{n−i_{1}}^{(1)} ⊃ R_{n−i_{1}+1}^{(1)} = ⋯ = R_{n}^{(1)} = 0

with R_{j}^{(1)} = (R/P_{i_{1}} ⋂ R_{j})S_{1∗}^{−1}. The subalgebra R_{j}^{(1)} is generated by x_{j} and R_{j+1}^{(1)}.

There exists a localization of C over some finitely generated denominator subset N_{1} ⊂ C such that the algebra R^{(1)}N_{1}^{−1} obtains the new system of generators x′_{1}, ⋯, x′_{n−i_{1}−1}, x_{n−i_{1}}^{±1} related x′_{j}x_{n−i_{1}} = q_{j,n−i_{1}}x_{n−i_{1}}x′_{j} ([P2]. Lemma 3.3). The algebra R_{j}^{(1)}N_{1}^{−1} is generated by x′_{j} and R_{j+1}^{(1)}N_{1}^{−1}. Notice that the above q_{j,n−i_{1}} is the entries of matrix ℚ (see Definition 2.1). Denote by S_{1} the denominator set generated by N_{1} and S_{1∗} and save the notaton R^{(1)} for localization R^{(1)}N_{1}^{−1}.

We continue the stratification process. Consider the subset L_{2} = {x′_{n−i_{1}−1}, …, x′_{n−i_{1}−i_{2}}} of i_{2} generators of R^{(1)}. Let J_{2} be the minimal semiprime ideal containing L_{2}. We denote by X_{2} the set of minimal prime ideals over J_{2}. As above we decompose X_{2} = X′_{2} ⋃ X′′_{2} ⋃ X′′′_{2} where

X′_{2} = {Q′ ∈ X_{2} : Q′ ⋂ C_{1} = 0, x′_{n−i_{1}−i_{2}−1} ∉ Q, },
X′′_{2} = {Q′ ∈ X_{2} : Q′ ⋂ C_{1} ≠ 0, x′_{n−i_{1}−i_{2}−1} ∉ Q},
X′′′_{2} = {Q′ ∈ X_{2} : x′_{n−i_{1}−i_{2}−1} ∈ Q}.

We denote

P′_{i_{1}i_{2}} = ⋂_{Q′∈X′_{2}} Q′.  (2.3)

There exists a semiprime ideal P_{i_{1}i_{2}} is R such that

(P_{i_{1}i_{2}}/P_{i_{1}})S_{1}^{−1} = P′_{i_{1}i_{2}}.

Consider the denominator set S′_{2∗} in R^{(1)} generated by x′_{n−i_{1}−i_{2}−1}. Let S_{2∗} = S_{1} ⋂(R/P_{i_{1}}). Denote R^{(2)} = (R/P_{i_{1}i_{2}})S_{2∗}^{−1}. The algebra R^{(2)} is generated by

x′_{1}, ⋯, x′_{n−i_{1}−i_{2}−2}, (x′_{n−i_{1}−i_{2}−1})^{±1}, x_{n−i_{1}}^{±1}

and admites the filtration

R^{(2)} = R_{1}^{(2)} ⊃ ⋯ ⊃ R_{n−i_{1}−i_{2}}^{(2)} = ⋯ = R_{n−i_{1}}^{(2)} ⊃ R_{n−i_{1}+1}^{(2)} = ⋯ = R_{n}^{(2)} = 0.

There exists localization of C over some finitely generated denominator subset N_{2} ⊂ C such that the algebra R^{(2)}N_{1}^{−1} obtains the new system of generators

x′′_{1}, ⋯, x′′_{n−i_{1}−i_{2}−2}, (x′_{n−i_{1}−i_{2}−1})^{±1}, x_{n−i_{1}}^{±1}

where x′′_{j} q-commutes with x′_{n−i_{1}−i_{2}−1} and x_{n−i_{1}}. As above we denote by S_{2} denominator set generated by S_{2∗} and N_{2}. We save the notation R^{(2)} for localization R^{(2)}N_{2}^{−1}.

We continue the stratification process. Finally, we get a semiprime ideal P_{μ} = P_{i_{1}i_{2}...i_{k+1}} and a denominator subset S_{μ} in R/P_{μ}. Denote B_{μ} = (R/P_{μ})S_{μ}^{−1}. The algebra B_{μ} is generated by q-commuting elements u_{1}^{±1}, …, u_{k}^{±1} with

u_{1} = x_{n−i_{1}}, u_{2} = x′_{n−i_{1}−i_{2}−1}, ⋯, u_{k} = x_{n−i_{1}−···−i_{k}−(k−1)}^{(k−1)}.

For any s = 1, 2, ⋯, k we denote ψ(s) = n − i_{1} − ⋯ − i_{s} − (s − 1) and

u_{s} = x_{ψ(s)}^{(s−1)}.

The generators u_{1}, ⋯, u_{k} are related

u_{i}u_{j} = q′_{ij}u_{j}u_{i}  (2.4)

and the matrix

ℚ_{μ} = (q′_{ij})_{i,j=1}^{k}  (2.5)

is a submatrix of ℚ. We denote by 𝕊_{μ} the corresponding submatrix of 𝕊. The algebra B_{μ} admits the filtration

B_{μ} = B_{μ,1} ⊃ B_{μ,2} ⊃ ⋯ ⊃ B_{μ,n}  (2.6)

with

B_{μ,j} = (R/(P_{μ} ⋂ R_{i}))S_{μ}^{−1}.

Notice that some subalgebras B_{μ,j} and B_{μ,j+1} can coinside. If B_{μ,j} ≠ B_{μ,j+1}, then j = ψ(s) for some s ∈ {1, 2, …, k} and B_{μ,j} is generated by u_{s} and B_{μ,j+1}. Recall that R_{j} is a skew extension of R_{j+1} and x_{j} (see Definition 2.1). We save the notation x_{j} for image of x_{j} in B_{μ,j}. We get that B_{μ,j} is generated by x_{j} and B_{μ,j+1}. The elements u_{s} and x_{j} are related as follows

u_{s} = x_{j} + r_{j+1}  (2.7)

with r_{j+1} ∈ B_{μ,j+1}.

Our next goal is to study stratification of prime ideals of specializations of R. For any λ ∈ K we consider specislization ρ_{λ} : R ↦ R_{λ} = Rmod(q − λ). For any ideal I in R_{λ} we denote I = ρ_{λ}^{−1}(I). The above ideal I is prime in R and I ⋂ C = C(q − λ). Denote as usual P_{μλ} = P_{μ}mod(q − λ) = P_{μ} + R(q − λ)/R(q − λ) and S_{μλ} the image of S_{μ} in R_{λ}/P_{μλ} = (R/P_{μ})mod(q − λ). Since I ⋂ C ≠ 0, we can’t apply Theorem 2.5 directly. The next Example shows that above stratification is not valid for all prime ideals. Example 2.6. Let R_{q} = 𝔸_{q,1} be Quantum Weyl algebra generated by x, y related xy − qyx = 1. Any prime ideal of R_{q} with zero intersection with K[q, q^{−1}] admits localization over y. The localization of R_{q} over y is generated by y^{±1} and u = (q − 1)yx + 1 related yu = quy. Consider specialization q ↦ ε = −1. The algebra R_{−1} is generated by x, y related xy +yx = 1. Let I be the kernel of irreducible representation

x ↦ ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) y ↦ ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )

The ideal I is prime, but does not admit localization over y. □ Definition 2.7. Let λ ∈ K be not a root of unity. We say that λ is is a point of a good reduction of stratification 𝕄 = {P_{μ}}, if the following properties hold. Property 1. Every ideal P_{μλ} is a semiprime ideal in R_{λ}, S_{μλ} is a denominator set of R_{λ}/P_{μλ}. Property 2. For any prime ideal I in R_{λ} there exists a standart ideal P_{μλ} of R_{λ} such that I ⊃ P_{μλ} and S_{μλ} ⋂(ImodP_{μλ}) = ∅. (i.e I ⊃ P_{μ} and S_{μ} ⋂(ImodP_{μ}) = ∅). For a point of good reduction we denote

B_{μλ} = (R_{λ}/P_{μλ})S_{μλ}^{−1} = B_{μ}mod(q − λ)

and we can study any prime ideal I in R_{λ} as an ideal of B_{μλ}. That is we can reduce the study of I to the study of an ideal in the algebra of twisted Laurent polynomials.

Let R be a quantum solvable algebra obeying Conditions 2.2-2.4. As above ε if a primitive l-root of unity. Suppose that Condition 2.4 is true for choosen l. Denote by D_{0} the set of derivations D_{x_{1}^{l}}, …, D_{x_{n}^{l}} defined in Secton 1. We say that an ideal I of R_{ε} is D_{0}-stable if I is stable with respect to all derivations of set D_{0}. Definition 2.8. Let λ = ε be a primitive l-root of unity. We say that ε is is a point of a good reduction of stratification 𝕄 = {P_{μ}}, if Property 1 of above Definition 2.7 holds and Property 2 holds for any prime D_{0}-stable ideal I.

We say that λ ∈ K is a point of bad reduction is the statements of Definitions 2.7 and 2.8 is not true for λ. We say further that a property E(λ) is true for almost all λ ∈ K if there exists a finite set F such that this property is true for all elements λ ∈ K − F. Theorem 2.9 The number of points of bad reduction is finite. Proof. Step 1. Suppose that λ is not a root of unity. First consider the semiprime H-stable (see Conditon 2.3) ideal P_{i_{1}} defined in (2.2) as an intersection of prime ideals Q ∈ X′_{1}. These prime ideals are H-stable ([P2], Prop.2.1). For almost all λ estableach ideal Q_{λ} is semiprime ([P2], Prop.2.7). Recall that x_{n−i_{1}} ∉ Q for Q ∈ X′_{1}. For almost all λ the element x_{n−i_{1},λ} is regular in R_{λ}/Q_{λ} ([P2], Prop.2.7). This proves P_{i_{1},λ} is semiprime and the element x_{n−i_{1},λ} is regular in R_{λ}/P_{i_{1},λ}. The set S_{1∗} = x_{n−i_{1}}^{m}m ∈ ℕ obeys Ore condition in R/P_{i_{1}} ([P2], Theorem 3.4). Then S_{1∗,λ} obeys Ore condition in R_{λ}/P_{i_{1},λ}. We conclude that S_{1∗,λ} is a denominator set and consider localization R^{(}1)_{λ} of R_{λ}/P_{i_{1},λ} over S_{1∗,λ} .

For almost λ the elements of denonimator set N_{1} ⊂ C (see stratification process) is nonsero modulo q−λ. We can consider localization over N_{1} modulo q−λ. Following the stratification process, we see that all ideals P_{μλ} are semiprime and S_{μλ} is a denominator set in factor algebra of R_{λ} over P_{μλ}. This proves Property 1 of Definition 2.7.

Notice that, since λ is not a root of unity, then all prime ideals in R are completely prime ([GL],Theorem 2.3).

Consider the nonempty Zariski open set 𝕆_{st} ⊂ C defined as follows. A point λ lies in K − 𝕆_{st} if λ annihilate some of ideal in X′′_{m} ⋂ C (see stratification process) or some element in S ⋂ C. The Property 2 of Definition 2.7 is true for all λ in 𝕆_{st} ([P2], Theorem 3.10). Step 2. Suppose that λ = ε is a primitive l-root of unity. The proof of Property 1) of Definition 2.8 is simillar to Step 1. We are going to prove Property 2) of the above definition. Let I be a prime D_{0}-stable ideal in R_{ε}. As usual I = ρ_{ε}^{−1}(I) with I ⋂ C = (q − ε)C.

Recall that Theorem 1.13 asserts that for all j the ideal I ⋂ R_{ε,j} is prime in R_{ε,j}. It follows that I_{j} = I ⋂ R_{j} is a prime ideal in R_{j}.

In particular, I ⋂ R_{n} is a prime ideal in R_{n} = C[x_{n}]. Hence, I ⋂ R_{n} is completely prime in R_{n}. If x_{n} ∉ I, then I has zero intersection with {x_{n}^{m}}_{m∈ℕ}. Therefore, the ideal I admits localization over x_{n}.

Otherwise, x_{n} ∈ I. Suppose that some subset L_{1} = {x_{n}, …, x_{n−i_{1}−1}} ⊂ I and x_{n−i_{1}} ∉ I. The ideal I contains a minimal prime ideal, say Q, in the set X_{1} (see stratification process). Since x_{n−i_{1}} ∉ I, then Q ∈ X′_{1} ⋃ X′′_{1}. If Q ∈ X′′_{1}, then ε annihilates Q ⋂ C. This contradicts the claim ε ∈ 𝕆_{st}. We get Q ∈ X′_{1} and I ⊃ P_{i_{1}}. We see above that I_{n−i_{1}} = I ⋂ R_{n−i_{1}} is a prime ideal in R_{n−i_{1}}. Since R_{n−i_{1}}/I_{n−i_{1}} is a factor ring of K[x_{n−i_{1}}], then the ideal I_{n−i_{1}} is completely prime. It follows that I_{n−i_{1}} (and I) have empty intersection with denominator set {x_{n−i_{1}}^{m}}_{m∈ℕ} ⊂ R/P_{i_{1}}. Ideal I admits localization over x_{n−i_{1}}. The ideal I^{(1)} = I_{0}/P_{i_{1}} admits localization over N_{1} (see stratification process).

Suppose that I^{(1)} contains some subset L_{2} = {x′_{n−i_{1}−1}, …, x′_{n−i_{1}−i_{2}}} and does not contain x′_{n−i_{1}−i_{2}−1}. As above the claim ε ∈ 𝕆_{st} implies I_{0} ⊃ P_{i_{1}i_{2}} (see also [P2], Theorem 3.10). Since I_{n−i_{1}−i_{2}−1} is prime ideal in R_{n−i_{1}−i_{2}−1} and R_{n−i_{1}}^{(1)}/I_{n−i_{1}}^{(1)} is a factor ring of the algebra of twisted polynomials K_{q}[x_{n−i_{1}}^{±1}, x′_{n−i_{1}−i_{2}−1}], then the ideal I_{n−i_{1}−i_{2}−1}^{(1)} (and I_{n−i_{1}−i_{2}−1}) is completely prime. It follows that I_{n−i_{1}−i_{2}−1}^{(1)} have empty intersection with denominator set {(x′_{n−i_{1}−i_{2}−1})^{m}}_{m∈ℕ}. Ideal I^{(1)} admits localization over x′_{n−i_{1}−i_{2}−1}. The ideal I^{(2)} = I/P_{i_{1}}S_{2∗}^{−1} admits localization over N_{2}. And so on.

Finally, we prove that ideal I contains a unique ideal P_{μ} such that S_{μ} ⋂(ImodP_{μ}) = ∅.□ Notations 2.10.

1) Let R be a domain and a C algebra. Let I be an ideal in R_{ε} = Rmod(q − ε). We denote by I_{D} the greatest D-stable (i.e. stable with respect to quantum adjoint action) ideal in I.

2) Let R be quantum solvable algebra, obeying Condition 2.4. We denote by I_{0} = I(D_{0}) the greatest D_{0} ideal in I. Remark 2.11. If I is prime ideal in R_{ε}, then I_{0} and I_{D} are also prime ideals in R_{ε} ([MC-R],14.2.3; [G],6.4). Definition 2.12. We say, that two R_{ε}-ideals I_{1}, I_{2} are D-equivalent, if I_{1D} = I_{2D}. We shall refer the class of equivalent to I ideals, as an orbit of I with respect to quantum adjoint action. We need further a few Lemmas. Lemma 2.13. Let R be a domain and a C-algebra. Let P be a semiprime ideal in R with P ⋂ C = 0 and P_{ε} = (P + (q − ε)R)mod(q − ε). We claim the following.

1) P_{ε} is D-stable;

2) Let u_{ε} = umod(q − ε) ∈ Z_{ε} and D_{u} the quantum adjoint action of u. We denote by ū is the image of u in B = R/P. As usual B_{ε} = Bmod(q − ε). We denote by \overline{D}ū the quantum adjoint action of ū via reduction B_{ε} = Bmod(q − ε). Then D_{u} : B_{ε} ↦ B_{ε} coinsides with \overline{D}ū;

3) Denote by Z′ the image of the center Z_{ε} of R_{ε} under the map R_{ε} ↦ R_{ε}/P_{ε} = B_{ε}. The subalgebra Z′ lies in the center Z(B_{ε}). We equip Z′ with Poisson structure inherited by the Poisson structure in Z_{ε}. We consider the Poisson structure ob Z(B_{ε}) defined via reduction B_{ε} = Bmod(q − ε) (Section 1). We assert that Z′ is a Poisson subalgebra of Z(B_{ε}), i.e. the Poisson bracket of two elements in Z′ coinsides with their Poisson bracket as elements Z(B_{ε}). Proof. 1) For any a ∈ P_{ε} = (P + R(q − ε))mod(q − ε) one can choose \underline{a} ∈ P such that a = \underline{a}mod(q − ε). Let u ∈ R such that u_{ε} = umod(q − ε) ∈ Z_{ε}, then b = u\underline{a} − \underline{a}u ∈ R(q − ε) ⋂ P = (q − ε)P. Then b ∈ (q − ε)P) and D_{u}(a) ∈ P_{ε}.

The claim 2) is obtained by reduction ua − au ∈ (q − ε)R modulo P. The claim 3) is a corollary of claim 2). □

We shall remind some well knows statements on algeras of twisted polynomials. Let ℚ and 𝕊 be as above matrices with q_{ij} = q^{s_{ij}}. Let B = B_{ℚ} be an algebra generated over C by the generators u_{1}^{±1}, …, u_{k}^{±1} related u_{i}u_{j} = q_{ij}u_{j}u_{i}. That is B is a factor algebra of an algebra of twisted Laurent plynomials. One can decompose B = A⊗Z(B) where A is an algebra of twisted Laurent polynomials and Z(B) is the center of B. Generators of A can be presented as y_{1}^{±1}, …, y_{2r}^{±1} related y_{1}y_{2} = q^{d_{1}}y_{2}y_{1}, …, y_{2r−1}y_{2r} = q^{d_{r}}y_{2r}y_{2r−1}. All other pair of generators commutes. Here 2r is the rank 𝕊. We shall refer 2r as a rank of B. The center Z(B) is generated by monomials. Any ideal of B is generated by its intersection with the center Z(B). Any prime ideal of B is completely prime.

One can consider specialization B_{ε} of B at primitive l-root of unity ε. We obtain B_{ε} = A_{ε} ⊗ Z(B)_{ε}. The elements u_{1}^{l}, …, u_{k}^{l} lie in the center of Z(B_{ε}) of B_{ε}. Further we denote by \overline{D}_{u} the quantum adjoint action of an element u ∈ B with u_{ε} ∈ Z(B_{ε}). For u_{i}^{l} we calculate

\overline{D}_{u_{i}^{l}}(u_{j}) = s_{ij}lε^{−1}u_{j}u_{i}^{l}.

Any \overline{D}-ideal in generated by its intersection with Z(B_{ε}). Lemma 2.14. Suppose that l is relatively prime with elementary divisors of matrix 𝕊. Then 1) the center Z(B_{ε}) is generated as K-algebra by u_{1}^{±l}, …, u_{k}^{±l} and the reduction Z(B)_{ε} of Z(B) modulo q − ε;

2) if the field K is algebraically closed, then the dimension of every irreducible representation of B_{ε} equals to l^{r} with r = \frac{1}{2}rank(𝕊). Proof. One can choose another system y_{1}^{±1}, …, y_{2r}^{±1}, z_{1}^{±1}, …, z_{t}^{±1}, 2r + t = k of generators of B related y_{1}y_{2} = q^{d_{1}}y_{2}y_{1}, …, y_{2r−1}y_{2r} = q^{d_{r}}y_{2r}y_{2r−1}. All other pair of generators commutes. Since (l, d_{i}) = 1 for all i, then the l-powes of generators and z_{1}, …, z_{t} generate the center of B_{ε}.□

Let B = B_{ℚ} be as above. Consider the filtration of B by subalgebras B_{j} generated by u_{j}^{±1}, …, u_{k}^{±1}. We put B_{k+1} = C. The center Z(B) have a filtration

Z(B) = Z(B)_{1} ⊃ ⋯ ⊃ Z(B)_{j} ⊃ ⋯ ⊃ Z(B)_{k} ⊃ Z(B_{k+1}) = C

where Z(B) ⋂ B_{j} = Z(B)_{j}. Similarly, B_{ε} = Bmod(q − ε) have the filtration B_{jε} with Z(B_{ε}) ⋂ B_{jε} = Z(B_{ε})_{j}. Lemma 2.15. Let 𝕊_{j} be a S-submatrix of (s_{iβ}), i = \overline{1, k}, β = \overline{j, k}. Suppose that l is relatively prime with elementary divisors of 𝕊_{j} . We assert that if a a = u_{j}^{n_{1}} ⋯ u_{k}^{n_{k}}, n_{i} ∈ ℤ lies in the center Z(B_{ℚ,ε}) then there exist the m_{β} ∈ ℤ, β = \overline{j, k}, m_{β} = n_{β}mod(l) such that a = u_{j}^{m_{1}} ⋯ u_{k}^{m_{k}} lies in the center Z(B). Proof. A monomial a lies in Z(B_{ε}) if (n_{j}, …, n_{k}) are a solution of system of equations

∑_{β=1}^{k} s_{iβ}n_{β} = 0mod(l), i = \overline{1, k}.

Under the requirements on l any solution modulo l is a reduction modulo l of a solution over ℤ.□ Corollary 2.16. Suppose that B_{j} = B_{j+1}[u_{j}^{±1}; τ_{j}] with τ_{j} = Ad_{u_{j}}. Suppose that Z(B)_{j} = Z(B)_{j+1}. If a polynomial u_{j}^{m}b with b ∈ B_{j+1}, 0 ≤ m < l lies the center Z(B_{ε})_{j}, then m = 0 and b ∈ Z(B_{ε})_{j+1}. The algebra Z(B_{ε})_{j} is generated by u_{j}^{l} and Z(B_{ε})_{j+1}. □

Now we are returning to ideals. Let R be a quantum solvable algebra obeying Conditions 2.2-2.4. Definition 2.17 Let ℚ and 𝕊 are matrices of Definition 2.1. We say that primitive l-root of inity ε is admissible if ε is a point of good reduction and l is relatively prime with all nonzero minors of 𝕊. By Proposition 2.9, the set of admissible roots of unity is infinite.

Suppose that ε is admissible. Consider the reduction of ideals P_{μ} and denominator sets S_{μ} ⊂ R/P_{μ} modulo q − ε. We denote B = B_{μ} = (R/P_{μ})S_{μ}^{−1} and B_{ε} = Bmod(q − ε). We get D-stable semiprime ideal P_{με} in R_{ε}. Recall that B_{ε} is generated by qcommuting elements u_{1}^{±1}, …, u_{k}^{±1} (see (2.4),(2.5)). Since ε is admissible, l is relatively prime with elementary divisors of matrix 𝕊_{μ}.

One can extend D-action from R_{ε} to B_{ε}. The algebra B_{ε} admites also the quantum adjoint action \overline{D} of its center Z(B_{ε}). Our next goal is to compair this two actions. We consider Z(B_{ε})\overline{D} the Z(B_{ε})-module spanned by z_{ε}\overline{D}_{b} with b ∈ B, b_{ε} = bmod(q − ε) ∈ Z(B_{ε}) and z_{ε} ∈ Z(B_{ε}). We consider the Z(B_{ε})-submodule ad_{B_{ε}} of inner derivations in B_{ε}. We denote by Z(B_{ε})D_{0} the Z(B_{ε})-submodule

Z(B_{ε})D_{x_{1}^{l}} + ⋯ + Z(B_{ε})D_{x_{n}^{l}}

of derivations of B_{ε}. Proposition 2.18. As above R is quantum solvable algebra obeying Conditions 2.2-2.4 and ε is admissible. Then Z(B_{ε})D_{0} = Z(B_{ε})\overline{D}mod(ad_{B_{ε}}). Proof. First Z(B_{ε})D_{0} ⊂ Z(B_{ε})\overline{D} ( the derivation D_{x_{i}^{l}} : B_{ε} ↦ B_{ε} coinsides with \overline{D} derivation of the image of x_{i}^{l} in B). To conclude the proof we are going to prove Z(B_{ε})D_{0} containes \overline{D} modulo inner derivations.

Consider the filtration of B = B_{μ} (see (2.4)). Recall that by Z(B_{ε})_{j} we denote the intersection of Z(B_{ε}) with B_{jε}. We consider

\overline{D}_{j} = \overline{D}_{Z(B_{ε})_{j}} = \{\overline{D}_{u} : u ∈ B, u_{ε} = umod(q − ε) ∈ Z(B_{ε})_{j}\}.

Considering \overline{D}_{u} upto inner derivations, we may choose above u ∈ B_{j}. We denote Z(B_{ε})_{j}D_{0j} = Z(B_{ε})_{j}D_{x_{j}^{l}} + ⋯ + Z(B_{ε})_{j}D_{x_{n}^{l}}. We shall prove by induction on j that \overline{D}_{j} equals to Z(B_{ε})_{j}D_{0j} modulo inner derivations.

The statement is obviously true for j = k + 1. Assume that the statement is true for j + 1. Our goal is to prove it for j.

Let B_{j} ≠ B_{j+1}. The algebra B_{j} is generated by B_{j+1}, x_{j} and also by B_{j+1}, u_{s}. The elements u_{s} and x_{j} are related as (2.7). To simplify the notations we put u_{s} = u. We save the notations for images of x_{j} and u in B_{ε}. Recall that u^{l} ∈ Z(B_{ε}) and x_{j}^{l} ∈ Z(R_{ε}).

1) Suppose that there exists a monomial z = u^{m}b ∈ Z(B)_{j} with m ≠ 0 and b ∈ B_{j+1} (remark that here z may be zero). We denote by m_{0} the least positive integer with the following property. The element v = u^{m_{0}}b_{0} modulo q − ε lies in Z(B_{ε})_{j} for some b ∈ B_{j+1}. Here one may consider b_{0} to be a monomial.

The algebra Z(B_{ε})_{j} is generated by v_{ε} = vmod(q − ε) and Z(B_{ε})_{j+1} Then m_{0} divides m. Put m = pm_{0}. We obtain z = v^{p}a with a monomial a ∈ B_{j+1}. Notice that amod(q − ε) lies in Z(B_{ε})_{j+1}.

Since z ∈ Z(B)_{j}, we have \overline{D}_{z} = 0. We obtain

0 = \overline{D}_{z} = pv^{p−1}a\overline{D}_{v} + v^{p}\overline{D}_{a}.

Then \overline{D}_{v} = −p^{−1}va^{−1}\overline{D}_{a}. By assumption of induction, the derivation \overline{D}_{a} lies in Z(B_{ε})_{j+1}D_{0,j+1}. This proves the statement.

2) Suppose that the condition z = u^{m}b ∈ Z(B)_{j} implies m = 0. This proves that Z(B)_{j} = Z(B)_{j+1}. Consider the homomorphism Φ of the skew extension B_{j+1}[u, τ_{j}] onto B_{j}. Every ideal in B_{j+1}[u, τ_{j}] is generated by its intersection with the center. The kernel of Φ is zero (otherwise, there exists a central , in particular zero, element of the point 1). We identify B_{j+1}[u, τ_{j}] and B_{j}. By Lemma 2.16, Z(B_{ε})_{j} is generated by u^{l} and Z(B_{ε})_{j+1} By (2.7), u = x_{j} + r_{j+1} with r_{j+1} ∈ B_{j+1}. Then u^{l} = x_{j}^{l} + F (u). Where F (u) a polynomial of u of degree ≤ l − 1 and F (u)_{ε} lies in Z(B_{ε})_{j}. The algebra Z(B_{ε})_{j} is generated by monomials. Applying Lemma 2.16, we prove that F (u)mod(q − ε) ∈ Z(B_{ε})_{j+1}. We obtain

\overline{D}_{u^{l}} = \overline{D}_{x_{j}^{l}} + \overline{D}_{F (u)} ∈ D_{x_{j}^{l}} + \overline{D}_{Z(B_{ε})_{j+1}}.

Using the assumption of induction, we prove the claim. □. Corollary 2.19. Let L be either a D_{0}-stable ideal in B_{ε} or D_{0}-stable ideal in Z(B_{ε}), then L is \overline{D}-stable. Theorem 2.20. As above R is a quantum solvable algebra obeying Conditions 2.2-2.4, and ε is admissible.

1) For any prime ideal I in R_{ε} the ideals I_{D_{0}} and I_{D} are completely prime. Any prime D_{0}-stable (in particular, D-stable) ideal P is completely prime.

2) For above P skew field of fractions Fract(R_{ε}/P) is isomorphic to a skew field of an algebra of twisted polynomials. Proof. For any prime ideal I the ideals I_{D_{0}} and I_{D} are also prime ([MC-R],14.2.3). These ideals are correspond to prime \overline{D}-ideals in B_{με} which is a factor algebra of an algebra of twisted Laurent polynomials. Hence, these ideals are completely prime.□.

4 Representations at roots of unity

Let K = ℂ be the field of complex numbers. Let R be a quantum solvable algebra obeying Conditions 2.2-2.4. Suppose that ε is admissible. As above, Z_{ε} is the center of R_{ε}. Let π be an irreducible complex representation of a quantum solvable algebra R_{ε}. Since R_{ε} is finite over its center, then π has finite dimentsion. Denote by I(π) the kernal of representation π in R_{ε}. For I(π) we consider I(π)_{D}-the greatest D-stable ideal in I(π). We denote

R_{π} = R_{ε}/I(π)_{D}.

The ideal I(π)_{D} is completely prime (Theorem 2.20) (i.e. R_{π} is a domain). There exists a unique standart semiprime D-stable ideal P_{με} and a denominator set S_{με} in R_{ε}/P_{με} such that

1) (R_{ε}/P_{με})S_{με}^{−1} is a factor algebra of an algebra of an algebra twisted Laurent polynomials;

2) I(π)_{D} ⊃ P_{με} and S_{με} ⋂ I(mod(P_{με})) = ∅. (see Theorem 2.9, Definitions 2.8, 2.17). The algebra R_{π} contains image of S_{με} (we save the notaton S_{με} for image of S_{με} in R_{π}). The localization B_{π} = R_{π}S_{με}^{−1} is the factor algebra of B_{με} = (R_{ε}/P_{με})S_{με}^{−1} over prime D-stable ideal

Î(π)_{D} = (I(π)_{D}/P_{με})S_{με}^{−1}.

Recall that B_{μ} = A_{μ} ⊗ Z(B_{μ}) where A_{μ} is generated by y_{1}^{±1}, …, y_{2r}^{±1} with y_{1}y_{2} = q^{d_{1}}y_{2}y_{1}, …, y_{2r−1}y_{2r} = q^{d_{r}}y_{2r}u_{2r−1} and Z(B_{μ}) is the center of B_{μ} (see Section 2). Since ε is admissible, l is relatively prime with d_{1}, …, d_{r}. We have B_{με} = A_{με} ⊗ Z(B_{μ})_{ε} (see Lemma 2.14). The center Z(B_{με}) of B_{με} is generated by y_{1}^{±l}, …, y_{2r}^{±l}, Z(B_{μ})_{ε}. Poisson bracket in Z(B_{με}) is determined by reduction B_{με} = B_{μ}mod(q − ε). All symplectic leaves of Z(B_{με}) have dimension 2r, algebraic and determined by characters of Z(B_{μ})_{ε}(see Section 2).

By Corollary 2.20, any D-stable ideal (in particular Î(π)_{D}) is \overline{D}-stable. Any \overline{D}- stable ideal in B_{με} is generated by its intersection with Z(B_{μ})_{ε}. We obtain Î(π)_{D} = B_{με}(Î(π)_{D} ⋂ Z(B_{μ})_{ε}). We denote

E_{π} = \frac{Z(B_{μ})_{ε}}{Î(π)_{D} ⋂ Z(B_{μ})_{ε}} .

We have B_{π} = A_{με} ⊗E_{π}. Here E_{π} is a commutative domain (Later we shall prove that E_{π} = ℂ). The center if B_{π} is generated by images of y_{1}^{±l}, …, y_{2r}^{±l} and E_{π}. Quantum adjoint action \overline{D} provides the Poisson structure in Z(B_{π}). All symplectic leaves of Z(B_{π}) have dimension 2r, algebraic and determined by characters of E. Definition 3.1. We say that two irredicible representation π_{1} and π_{2} are D-equivalent if I(π_{1})_{D} = I(π_{2})_{D} (see Definition 2.12).

Denote by χ = χ(π) : Z_{ε} ↦ K the central character of π. The center Z_{ε} determines the algebraic Poisson manifold M. Further we refer Z_{ε} as an algebra ℂ[M] of regular functions on M and χ is a point of M. The intersection I(π) ⋂ Z_{ε} is the maximal ideal m(χ) for χ ∈ Z_{ε}. Let R_{χ} be R_{ε}/m(χ).

We denote by Ω_{χ} the symplectic leaf of χ with respect to Poisson bracket on Z_{ε}. We consider the greatest D-stable ideal m(χ)_{D} in m(χ). The ideal m(χ)_{D} is prime. We denote by M_{χ} the annihilator set Ann(m(χ)_{D}). Remark that M_{χ} is the Zariski closure symplectic leaf Ω_{χ}. We shall prove further (Theorem 3.3) that Ω_{χ} is open in its closure M_{χ}. The algebra of regular functions Z_{χ} = ℂ[M_{χ}] coinsides with Z_{ε}/m(χ)_{D}. The Poisson structure on Z_{ε} provides the Poisson structure on Z_{χ}.

Notice that an element a lies in I(π)_{D} iff for any system of elements u_{1}, …, u_{s} ∈ R, obeying u_{1ε}, …, u_{sε} ∈ Z_{ε}, the element D_{u_{1}} ⋯ D_{u_{s}}(a) lies in I(π). This proves that I(π)_{D} ⋂ Z_{ε} = m(χ)_{D} and Z_{χ} = Z_{ε}/m(χ)_{D} = Z_{ε}/(I(π)_{D} ⋂ Z_{ε}). The algebra Z_{χ} lies in the center of R_{π} = R_{ε}/I(π)_{D} and R_{π} is finite over Z_{χ}.

According to Lemma 2.13, Z_{χ} is a Poisson subalgebra of Z(R_{π}). Denote by Y_{π} the Poisson manifold of Z(B_{π}). Lemma 3.2. Let f ∈ Z_{χ} = ℂ[M_{χ}], p ∈ M_{χ} and f ≠ 0. There exists p′ ∈ M_{χ} such that p′ lie in the symplectic leaf of p, f(p′) ≠ 0 and R_{p} is isomorphic to R_{p′}. Proof. Since f is nonzero element of Z_{χ} = Z_{ε}/m(χ)_{D}, then there exist elements u_{1}, …, u_{s} such that u_{1ε}, …, u_{nε} ∈ Z_{ε} and the element D_{u_{1}} ⋯ D_{u_{s}}(f)(p) ≠ 0. Let G_{u_{1}}(t_{1}), …, G_{u_{s}}(t_{s}) be local flows of the above derivations defined for small |t_{i}|. Put p′ = G_{u_{s}}(t_{s}) ⋯ G_{u_{1}}(t_{1})(p). We have f(p′) ≠ 0. The algebras R_{p} and R_{p′} are isomorphic ([DC-P], 11.8). □ Theorem 3.3 As above R is quantum solvable algebra obeying Conditions 2.2-2.4 and ε is admissible. We assert that:

1) Dimension of any irreducible representation equals to l^{\frac{1}{2}dim(Ω_{χ})};

2) Any symplectic leaf Ω_{χ} is Zariski-open in its Zariski closure;

3) If two points χ_{1} and χ_{2} belong to a common symplectic leaf, then the algebras R_{χ_{1}} and R_{χ_{2}} are isomorphic;

4) Let π, π′ are irreducible representations of R_{ε} and χ, χ′ their central characters. If π and π are quantun equivalent (see Definition 3.1), then χ and χ′ lie in the common symplectic leaf. Proof. Let χ be a central character of Z_{ε}. Let π be an irreducible representation over χ.

Recall that R_{π} is finite over Z_{χ} = ℂ[M_{χ}], M_{χ} = \overline{Ω_{χ}}. Let v_{1}, …, v_{k} be q-commuting generators of S_{μ}. As above v_{1ε} … v_{kε} their images in R_{π}. Since R_{π} is finite over Z_{χ}, the R_{π}Z_{χ}^{−1} is a skew field. The element v_{iε}^{−1} can be presented in the form

v_{iε}^{−1} = a_{i}w_{i}^{−1},

where a_{i} ∈ R_{π}, w_{i} ∈ Z_{χ}. Let N denote the denominator subset generated by w_{1}, …, w_{k}. The localization R_{π}N^{−1} contains R_{π}S^{−1} = B_{π} and and R_{π}N^{−1} = B_{π}N^{−1}. Denote

f_{0} = w_{1} ⋯ w_{k} ∈ Z_{χ}.

The algebra B_{π}N^{−1} (as B_{π}) finite over its center Z(B_{π}N^{−1}). Denote by Y the Poisson manifold of Z(B_{π}N^{−1}). Notice that

Y = {p ∈ Y_{π} : f_{0}(p) ≠ 0}.

As Y_{π}, symplectic leaves of Y are algebraic and have dimension 2r. Any symplectic leaf V ⊂ Y is Ann(J_{V}), where J_{V} = Z(B_{π}N^{−1})(J_{V} ⋂ E_{π}) is a prime ideal generated by maximal ideal J_{V} ⋂ E_{π} in E_{π}.

The algebra Z_{χ}N^{−1} is a commutative Poisson subalgebra of Z(B_{π}N^{−1}). Denote

O = {p ∈ M_{χ} : f_{0}(p) ≠ 0},
ℂ[O] = Z_{χ}N^{−1}.

Step 1. At this step we shall prove that O ⊂ Ω_{χ}. Since Ω_{χ} is Zariski dense in M_{χ} and O is Zariski open in M_{χ}, then the set O_{χ} = Ω_{χ} ⋂ O is Zariski dense in O.

The inclusion of Poisson algebras ℂ[O] = Z_{χ}N^{−1} ⊂ Z(B_{π}N^{−1}) = ℂ[Y] yields the Poisson map ϕ : Y ↦ O of Poisson manifolds. It follows that an image ϕ(V) of any symplectic leaf V is a Poisson subset (i.e a union of symplectic leaves).

Cosider the algebra C^{∞}(O) (respect. C^{∞}(Y)) of C^{∞}-smooth functions on O (respect. Y). Let J^{∞}(O_{χ}) be an ideal in C^{∞}(O), cosisting of functions vanishing on O_{χ}. The ideal J^{∞}(O_{χ}) is D-stable . We denote by J_{Y}^{∞} the ideal C^{∞}(Y)ϕ^{∗}(J^{∞}(O_{χ})) of functions vanishing on π^{−1}(O_{χ}). By Proposition 2.18, the ideal J_{Y}^{∞} is \overline{D}-stable. This proves that ϕ^{−1}(O_{χ}) is a Poisson subset. Choose a symplectic leaf V ⊂ ϕ^{−1}(O_{χ}). We obtain

ϕ(V) ⊂ ϕϕ^{−1}(O_{χ}) = O_{χ}.

Since ϕ(V) is Poisson subset, we obtain ϕ(V) = O_{χ}. The symplectic leaf V is algebraic subset V = Ann(J_{V}). This implies that O_{χ} is annihilated by J_{V} ⋂ ℂ[O]. But O_{χ} is Zariski dense in O. We get J_{V} ⋂ ℂ[O] = 0. The ideal J_{V} corresponds to an ideal in localization F = ℂ[Y](ℂ[O]^{∗})^{−1}. Since ℂ[Y] is finite over ℂ[O], the algebra F is a field. This proves that J_{V} = 0. Whence Y = V and O_{χ} = ϕ(V) = ϕ(Y) = O. This concludes Step 1. In addition we notice that J_{V} = 0 yields that 0 = J_{V} ⋂ E_{π} is a maximal ideal in E_{π}. We obtain E_{π} = ℂ. Step 2. In this step we conclude the proof. Recall that χ ∈ M and M_{χ} is an annihilator of m(χ)_{D}.

1) Let π be an irreducible representation with Z_{ε} character χ. We apply Lemma 3.2 for f = f_{0}. There exists p ∈ Ω_{χ} such that f_{0}(p) ≠ 0 (i.e p ∈ O) such that R_{χ} is isomorphic to R_{p}. Representations of R_{p} are passed through R_{ε} ↦ B_{π}N^{−1}. The algebra B_{π} is an algebra of twisted Laurent polynomials. Statement 1) is well known for algebras of twisted Laurent polynomials. This proves that

dim(π) = l^{\frac{1}{2}dim(O)} = l^{\frac{1}{2}dim(Ω_{χ})}.

2) We have proved in Step 1 that Ω_{χ} contains a Zariski-open subset O. For any element p ∈ \overline{Ω_{χ}} − Ω_{χ} the dimension of corresponding symplectic leaf Ω_{p} is less than r = dim(Ω_{χ}). The set \{p ∈ M : dim(Ω_{p}) < r\} is Zariski-closed in M. The set \overline{Ω_{χ}} −Ω_{χ} is Zariski closed. This proves statement 2.

3) For any two points χ_{1} and χ_{2} of Ω_{χ}. Simillar to 1) there exist p_{1}, p_{2} ∈ O such that R_{χ_{1}} is isomorphic to R_{p_{1}} and R_{χ_{2}} is isomorphic to R_{p_{2}}. The algebras R_{p_{1}} and R_{p_{2}} are isomorphic as fibers of an algebra of twisted Laurent polynomials. This proves 3.

4) Let π_{1} and π_{2} be two D-equivalent representations of R_{ε}. Then I(π_{1})_{D} = I(π_{2})_{D}. Then m(χ_{1})_{D} = m(χ_{2}). By 2), the points χ_{1} and χ_{2} lie in a common symplectic leaf.□

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