We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl-Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. The result also extends to generic quantum grassmannians (by using noncommutative dehomogenisation) and to the quantum groups Oq(GLn) and Oq(SLn).

Let K be a field and q be a nonzero element of K that is not a root of unity. We give a criterion for 0 to be a primitive ideal of the algebra O q (M m,n ) of quantum matrices. Next, we describe all height one primes of O q (M m,n ); these two problems are actually interlinked since it turns out that 0 is a primitive ideal of O q (M m,n ) whenever O q (M m,n ) has only finitely many height one primes. Finally, we compute the automorphism group of O q (M m,n ) in the case where m = n. In order to do this, we first study the action of this group on the prime spectrum of O q (M m,n ). Then, by using the preferred basis of O q (M m,n ) and PBW bases, we prove that the automorphism group of O q (M m,n ) is isomorphic to the torus (K * ) m+n−1 when m = n and (m, n) = (1, 3), (3, 1).

Abstract. Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices.

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a natural torus action of H = (k * ) n on the quantum grassmannian Oq(Gm,n(k)) and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the nonnegative cells in recent studies concerning the totally nonnegative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity.

Let g be a complex simple Lie algebra of type B 2 and q be a nonzero complex number which is not a root of unity. In the classical case, a theorem of Dixmier asserts that the simple factor algebras of Gelfand-Kirillov dimension 2 of the positive part U + (g) of the enveloping algebra of g are isomorphic to the first Weyl algebra. In order to obtain some new quantized analogues of the first Weyl algebra, we explicitly describe the prime and primitive spectra of the positive part U + q (g) of the quantized enveloping algebra of g and then we study the simple factor algebras of Gelfand-Kirillov dimension 2 of U + q (g). In particular, we show that the centers of such simple factor algebras are reduced to the ground field C and we compute their group of invertible elements. These computations allow us to prove that the automorphism group of U + q (g) is isomorphic to the torus (C * ) 2 , as conjectured by Andruskiewitsch and Dumas.

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over Q.

For q ∈ C generic, we give an algorithmic construction of an ordered bijection between the set of H-primes of O q (M n (C)) and the sub-poset S of the (reverse) Bruhat order of the symmetric group S 2n consisting of those permutations that move any integer by no more than n positions. Further, we describe the permutations that correspond via this bijection to rank t H-primes. More precisely, we establish the following result. Imagine that there is a barrier between positions n and n + 1. Then a 2n-permutation σ ∈ S corresponds to a rank t H-invariant prime ideal of O q (M n (C)) if and only if the number of integers that are moved by σ from the right to the left of this barrier is exactly n − t.The existence of such a bijection was conjectured by Goodearl and Lenagan.2000 Mathematics subject classification: 16W35, 20G42, 06A07.Proposition 2.3 The map χ : S n × S n → H ′ R -Spec(O q (SL n (C))) defined by χ(w) = I w is an order-reversing bijection; its inverse is also an order-reversing bijection.Proof. In view of Proposition 2.2, it just remains to show that χ and χ −1 are decreasing. Concerning χ, it easily follows from the characterization of the (reverse) Bruhat ordering given in Proposition 1.2 and the definition of I w . So it just remains to deal with χ −1 . Let J ⊆ K be two H ′ R -invariant prime ideals of O q (SL n (C)). There exist w J and w K in S n × S n such that J = I w J and K = I w K . In order to prove that χ −1 is decreasing, we need to show thatAssume that this is not the case. Then we have for instance w + J w + K . It follows from Proposition 1.2 that there exists j ∈ [[1, n − 1]] such that w + J j w + K and so we have c + j,w + K ∈ I w J = J ⊆ K. Hence the quantum minor c + j,w + K must belong to K = I w K . But, on the other hand, Hodges and Levasseur have shown (see [11, Theorem 2.2.3]) that, for all i ∈ [[1, n − 1]], c + i,w + K ∈ I w K . This is a contradiction and so we have w + J ≥ w + K and w − J ≥ w − K , as required.