## A strong Dixmier-Moeglin equivalence for quantum Schubert cells

#### Brendan Nolan (Kent)

I will outline work of Jason Bell, Stephane Launois, and myself, showing that in several families of quantum algebras, an arbitrary prime ideal is equally close (in a manner which I will make precise) to being primitive, rational, and locally closed. The family on which I shall focus is that of the quantum Schubert cells $$U_q [w]$$; for an infinite field $$k$$, a simple complex Lie algebra $$\mathfrak{g}$$, a scalar $$q\in k$$ which is not a root of unity, and an element $$w$$ of the Weyl group of $$\mathfrak{g}$$, $$U_q [w]$$ is a subalgebra of $$U_q^+(\mathfrak{g})$$ constructed by De Concini, Kac, and Procesi; familiar examples include $$U_q^+(\mathfrak{g})$$ itself and the algebras of quantum matrices.