Laurent Rigal

QUANTUM UNIQUE FACTORISATION DOMAINS

We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchons 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).

RING THEORETIC PROPERTIES OF QUANTUM GRASSMANNIANS

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

Quantum analogues of Schubert varieties in the Grassmannian

We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains which are maximal orders and are AS-CohenMacaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders. 2000 Mathematics subject classification: 16W35, 16P40, 16S38, 17B37, 20G42.

PRIME SPECTRUM AND AUTOMORPHISMS FOR 2×2 JORDANIAN MATRICES

ABSTRACT This paper is devoted to some ring theoretic properties of the jordanian deformation of the algebra of regular functions on the matrices with coefficients in an algebraically closed field of characteristic zero, and of the associated factor algebra . We prove in particular that the prime spectrum of is the disjoint union of five components, each of which being homeomorphic to the spectrum of a commutative (possibly localised) polynomial ring. So we can give an explicit description of the prime spectrum of , and check that any prime factor of satisfies the Gelfand-Kirillov property. Then we study the automorphism groups of the algebras and and prove that they are generated by linear automorphisms and exponentials of locally nilpotent derivations.

THE MAXIMAL ORDER PROPERTY FOR QUANTUM DETERMINANTAL RINGS

We develop a method of reducing the size of quantum minors in the algebra of quantum matrices

Dimensions of quantum determinantal rings

\mathcal{O}_q(M_n)

Prime ideals in the quantum grassmannian

. We use the method to show that the quantum determinantal factor rings of

Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum grassmannians☆

\mathcal{O}_q(M_n)c

The first fundamental theorem of coinvariant theory for the quantum general linear group

are maximal orders, for

Twisted Semigroup Algebras

q