No abstract

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).

The m × n quantum grassmannian, G q (m, n), with m ≤ n, is the subalgebra of the algebra O q (M mn ) of quantum m × n matrices that is generated by the maximal m×m quantum minors. Several properties of G q (m, n) are established. In particular, a k-basis of G q (m, n) is obtained, and it is shown that G q (m, n) is a noetherian domain of Gelfand-Kirillov dimension m(n − m) + 1. The algebra G q (m, n) is identified as the subalgebra of coinvariants of a natural left coaction of O q (SL m ) on O q (M mn ) and it is shown that G q (m, n) is a maximal order.

We study quantum analogues of quotient varieties, namely quantum grassmannians and quantum determinantal rings, from the point of view of regularity conditions. More precisely, we show that these rings are AS-Cohen-Macaulay and determine which of them are AS-Gorenstein. Our method is inspired by the one developed by De Concini, Eisenbud and Procesi in the commutative case. Thus, we introduce and study the notion of a quantum graded algebra with a straightening law on a partially ordered set, showing in particular that, among such algebras, those whose underlying poset is wonderful are AS-Cohen-Macaulay. Then, we prove that both quantum grassmannians and quantum determinantal rings are quantum graded algebras with a straightening law on a wonderful poset, hence showing that they are AS-Cohen-Macaulay. In this last step, we are led to introduce and study (to some extent) natural quantum analogues of Schubert varieties.

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).

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.

A quantum deformation of the classical conjugation action of GL(N, C) on the space of N × N matrices M(N, C) is defined via a coaction of the quantum general linear group O(GL q (N, C)) on the algebra of quantum matrices O(M q (N, C)). The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of O(M q (N, C)) generated by (weighted) sums of principal quantum minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction.

The famous 1960’s construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not locally nilpotent, algebra which has polynomially bounded growth.