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Mathematics, Representation Theory, Rings and Algebras
Abstract:
Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra
over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and
$\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its
symmetric algebra respectively. Consider the Jacobson topology on the primitive
spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the
primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a
homeomorphism between the corresponding topological spaces (on the level of
points, it gives a bijection between the primitive ideals of
$\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$). We also show that
all primitive ideals of $\mathrm{S}(\mathfrak{n})$ from an open set in a
properly chosen topology are generated by their intersections with the Poisson
center. Under the assumption that $\mathfrak{n}$ is a nil-Dynkin Lie algebra,
we give two criteria for primitive ideals
$I(\lambda)\subset\mathrm{S}(\mathfrak{n})$ and
$J(\lambda)\subset\mathrm{U}(\mathfrak{n})$, $\lambda\in\mathfrak{n}^*$, to be
nonzero. Most of these results generalize the known facts about primitive and
Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that
for a finite-dimensional nilpotent Lie algebra all primitive ideals
$I(\lambda)$, $J(\lambda)$ are nonzero).
