A strong Dixmier-Moeglin equivalence for quantum Schubert cells

Brendan Nolan (Kent)

Frank Adams 1,

In the late 1970s and early 1980s, Dixmier and Moeglin gave algebraic and topological conditions for recognising the primitive ideals (namely the kernels of the irreducible representations) of the enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, such algebras are said to satisfy the "Dixmier-Moeglin equivalence". Many interesting families of algebras, including many families of quantum algebras, have since been shown to satisfy this condition.


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.

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