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author:
- 'Mikhail V. Ignatyev'
title:  
---

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The theory of primitive ideals in enveloping algebras of Lie algebras has its roots in the representation theory of Lie algebras. However, classifying irreducible representations of Lie algebras is not feasible except in few very special cases, while a classification of annihilators of irreducible representations, i.e., of primitive ideals, can be achieved in much greater generality. This idea goes back to J. Dixmier and his seminar, and for semisimple or solvable finite-dimensional Lie algebras there is an extensive theory of primitive ideals.

In the case when ${\mathfrak{n}}$ is a finite-dimensional nilpotent Lie algebra, the primitive ideals in the universal enveloping algebra $U({\mathfrak{n}})$ can be described in terms of the Dixmier map assigning to any linear form $f\in{\mathfrak{n}}^*$ a primitive ideal $J(f)$ of $U({\mathfrak{n}})$. If ${\mathfrak{n}}$ is abelian, $J(f)$ is simply the annihilator of $f$. For a general finite-dimensional nilpotent Lie algebra ${\mathfrak{n}}$, the theory of primitive ideals retains many properties from the abelian case: in particular, $J(f)$ is always a maximal ideal and every primitive ideal in $U({\mathfrak{n}})$ is of the form $J(f)$ for some $f\in{\mathfrak{n}}^*$. Moreover, $J(f)=J(f')$ if and only if $f$ and $f'$ belong to the same 