The associated graded variety of a two-sided ideal

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra, with PBW filtration $F_{n}U(\mathfrak{g})$ and associated graded algebra $\mathrm{gr}U(\mathfrak{g})$ (The PBW filtration by tensor degree on the enveloping algebra), and let $I⊴U(\mathfrak{g})$ be a two-sided ideal (Left, right and two-sided ideals). For each $n\ge 0$ put $F_{n}I:=I\cap F_{n}U(\mathfrak{g})$ with $F_{-1}I=0$, and define the associated graded ideal

$\mathrm{gr}I:={⨁}_{n\ge 0}{F}_{n}I/F_{n-1}I\subseteq \mathrm{gr}U(\mathfrak{g})\mathrm{.}$

Fix an ordered basis $x_1,\dots ,x_n$ of $\mathfrak{g}$. By PBW gives an ordered monomial basis for the enveloping algebra, its ordered monomials form a basis of $U(\mathfrak{g})$ and multiplication identifies $\mathrm{gr}U(\mathfrak{g})$ with the symmetric algebra $S(\mathfrak{g})$, which under this basis is the polynomial algebra $\mathbb{C}[x_1,\dots ,x_n]$ on the symbols; by The associated graded algebra of the PBW filtration is commutative this algebra is commutative, so $\mathrm{gr}I$ is an ideal of it. The dual basis of $x_1,\dots ,x_n$ identifies ${\mathfrak{g}}^{\ast }$ with ${\mathbb{C}}^n$, so elements of $S(\mathfrak{g})$ are polynomial functions on ${\mathfrak{g}}^{\ast }$. The associated variety of $I$ is the classical affine algebraic set (Classical affine algebraic sets, including the empty boundaries)

$\mathcal{V}(I):=\{f\in {\mathfrak{g}}^{\ast }:p(f)=0\text{ for every }p\in \mathrm{gr}I\}\mathrm{.}$

It is the zero locus of the family $\mathrm{gr}I$ in the polynomial ring on ${\mathfrak{g}}^{\ast }$; equivalently $\mathcal{V}(I)=V(\mathrm{gr}I)$ in the notation of the classical zero loci, a Zariski closed subset of ${\mathfrak{g}}^{\ast }$ (Classical affine zero loci form the Zariski closed sets).

• $\mathrm{gr}I$ is a graded ideal, not merely a graded subspace. If $u\in F_{m}U(\mathfrak{g})$ and $v\in F_{n}I$, then $uv\in I\cap F_{m+n}U(\mathfrak{g})$, and the symbol of $uv$ is the product of the symbols of $u$ and $v$; hence $\mathrm{gr}I$ is closed under multiplication by the whole of $\mathrm{gr}U(\mathfrak{g})$.
• The definition does not depend on the ordered basis. The subspaces $F_{n}U(\mathfrak{g})$ are defined by tensor degree with no reference to a basis, so $\mathrm{gr}I$ is intrinsic; changing the ordered basis changes the identification of $S(\mathfrak{g})$ with a polynomial ring by an invertible linear change of variables, whose induced map on ${\mathbb{C}}^n$ is a linear isomorphism carrying one zero locus onto the other. The associated variety $\mathcal{V}(I)$ as a subset of ${\mathfrak{g}}^{\ast }$ is therefore well defined (The classical vanishing ideal, The coordinate ring of a classical affine algebraic set).
• Proper ideals and the empty case. If $I=U(\mathfrak{g})$ then $F_{n}I=F_{n}U(\mathfrak{g})$ for all $n$, so $\mathrm{gr}I=\mathrm{gr}U(\mathfrak{g})$ and $\mathcal{V}(I)=\mathrm{\varnothing }$; if $I=0$ then $\mathrm{gr}I=0$ and $\mathcal{V}(I)={\mathfrak{g}}^{\ast }$. Both boundary cases are allowed by the definition. Primitive ideals are proper, but may be zero; for example, when $\mathfrak{g}=0$, the simple $U(0)=\mathbb{C}$-module $\mathbb{C}$ has annihilator $I=0$.