arXiv subjects

Susan J. Sierra

Publications and source records attributed to Susan J. Sierra.

At least 19 recordsLinked to original sources

arXiv2025

Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities

Let $B = \Bbbk_q[u,v]^{C_{n+1}}$ be a Type $\mathbb{A}_n$ quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution $Λ$, which is an "algebraic" resolution of $B$. We construct a category $\mathcal{X}$ which serves as a "geometric" resolution of $B$ by adapting techniques from quiver GIT and show that $\mathcal{X}$ and $\text{mod-}Λ$ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of $\mathcal{X}$ corresponds to a Type $\mathbb{A}_n$ Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities.

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arXiv2023

A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras

We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending chain condition (ACC) on Poisson ideals. We define a combinatorial condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $\mathbb N^k$. Our main result is: Theorem. If $\mathfrak g$ is a Dicksonian graded Lie algebra over a field of characteristic zero, then the symmetric algebra $S(\mathfrak g)$ satisfies the ACC on radical Poisson ideals. As an application, we establish this ACC for the symmetric algebra of any graded simple Lie algebra of polynomial growth over an algebraically closed field of characteristic zero, and for the symmetric algebra of the Virasoro algebra. We also derive some consequences connected to the Poisson primitive spectrum of finitely Poisson-generated algebras.

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arXiv2022

The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Let $W = \mathbb{C}[t,t^{-1}]\partial_t$ be the Witt algebra of algebraic vector fields on $\mathbb{C}^\times$ and let $Vir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In this paper, we study the Poisson ideal structure of the symmetric algebras of $Vir$ and $W$, as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of $S(Vir)$ and $S(W)$. In particular, we show that the only functions in $W^*$ which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of $S(W)$ with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions local. Given a local function $χ\in W^*$, we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of $χ$, which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of $Vir$ of finite codimension and show in particular that any such subalgebra of $Vir$ contains the central element $z$, substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension 1. As a consequence, we deduce that $S(Vir)/(z-λ)$ is Poisson simple if and only if $λ\neq 0$.

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arXiv2022

Ideals in enveloping algebras of affine Kac-Moody algebras

Let $L$ be an affine Kac-Moody algebra, with central element $c$, and let $λ\in \mathbb C$. We study two-sided ideals in the central quotient $U_λ(L):= U(L)/(c-λ)$ of the universal enveloping algebra of $L$, and prove: Theorem 1. If $λ\neq 0$ then $U_λ(L)$ is simple. Theorem 2. The algebra $U_0(L)$ has just-infinite growth, in the sense that any proper quotient has polynomial growth. As an immediate corollary, we show that the annihilator of any nontrivial integrable highest weight representation of $L$ is centrally generated, extending a result of Chari for Verma modules. We also show that universal enveloping algebras of loop algebras and current algebras of finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to Theorems 1 and 2 for quotients of symmetric algebras of these Lie algebras by Poisson ideals.

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arXiv2020

Enveloping algebras with just infinite Gelfand-Kirillov dimension

Let $\mf g$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(\mf g )$ has intermediate growth and thus infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of $U(\mf g)$ is {\em just infinite} in the sense that any proper quotient of $U(\mf g)$ has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra $S(\mf g)$ by proper Poisson ideals. In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.

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arXiv2020

Associating Geometry to the Lie Superalgebra $\mathfrak{sl}(1|1)$ and to the Color Lie Algebra $\mathfrak{sl}^c_2(\Bbbk)$

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, there is a correspondence between Verma modules and certain line modules that associates a pair $(\mathfrak{h},\,ϕ)$, where $\mathfrak{h}$ is a two-dimensional Lie subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ and $ϕ\in \mathfrak{h}^*$ satisfies $ϕ([\mathfrak{h}, \, \mathfrak{h}]) = 0$, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra $\mathfrak{sl}(1|1)$ and for a color Lie algebra associated to the Lie algebra $\mathfrak{sl}_2$.

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arXiv2019

Ideals in the enveloping algebra of the positive Witt algebra

Let $W_+$ be the positive Witt algebra, which has a $C$-basis $\{e_n: n \in Z_{\geq 1}\}$, with Lie bracket $[ e_i, e_j] = (j-i) e_{i+j}$. We study the two-sided ideal structure of the universal enveloping algebra $U(W_+)$ of $W_+$. We show that if $I$ is a (two-sided) ideal of $U(W_+)$ generated by quadratic expressions in the $e_i$, then $U(W_+)/I$ has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of $U(W_+)$, and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra $S(W_+)$.

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arXiv2017

A new family of Poisson algebras and their deformations

Let $\Bbbk$ be a field of characteristic zero. For any positive integer $n$ and any scalar $a\in\Bbbk$, we construct a family of Artin-Schelter regular algebras $R(n,a)$, which are quantisations of Poisson structures on $\Bbbk[x_0,\dots,x_n]$. This generalises an example given by Pym when $n=3$. For a particular choice of the parameter $a$ we obtain new examples of Calabi-Yau algebras when $n\geq 4$. We also study the ring theoretic properties of the algebras $R(n,a)$. We show that the point modules of $R(n,a)$ are parameterised by a bouquet of rational normal curves in $\mathbb{P}^{n}$, and that the prime spectrum of $R(n,a)$ is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe ${\rm Spec}\ R(n,a)$ as a union of commutative strata.

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arXiv2017

On a Dynamical Mordell-Lang Conjecture for Coherent Sheaves

We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassman's theorem that we prove in the appendix.

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arXiv2017

Generalised Witt algebras and idealizers

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $Γ$ be an additive subgroup of $\Bbbk$. Results of Kaplansky-Santharoubane and Su classify intermediate series representations of the generalised Witt algebra $W_Γ$ in terms of three families, one parameterised by ${\mathbb A}^2$ and two by ${\mathbb P}^1$. In this note, we use the first family to construct a homomorphism $Φ$ from the enveloping algebra $U(W_Γ)$ to a skew extension of ${\Bbbk}[a,b]$. We show that the image of $Φ$ is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of $U(W_Γ)$ under $Φ$ is not left or right noetherian, giving a new proof that $U(W_Γ)$ is not noetherian. We construct $Φ$ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let $G$ be an arbitrary group and let $A$ be a $G$-graded ring. A graded $A$-module $M$ is an intermediate series module if $M_g$ is one-dimensional for all $g \in G$. Given a shift-invariant family of intermediate series $A$-modules parametrised by a scheme $X$, we construct a homomorphism $Φ$ from $A$ to a skew-extension of ${\Bbbk}[X]$. The kernel of $Φ$ consists of those elements which annihilate all modules in $X$.

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arXiv2016

Maps from the enveloping algebra of the positive Witt algebra to regular algebras

We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither left nor right noetherian.

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arXiv2014

The universal enveloping algebra of the Witt algebra is not noetherian

This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces. As a consequence of our main result, we also show that the enveloping algebras of many other infinite dimensional Lie algebras are not noetherian. These Lie algebras include the Virasoro algebra and all infinite dimensional Z-graded simple Lie algebras of polynomial growth.

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arXiv2012

Moduli spaces for point modules on naive blowups

The naive blow-up algebras developed by Keeler-Rogalski-Stafford, after examples of Rogalski, are the first known class of connected graded algebras that are noetherian but not strongly noetherian. This failure of the strong noetherian property is intimately related to the failure of the point modules over such algebras to behave well in families: puzzlingly, there is no fine moduli scheme for such modules, although point modules correspond bijectively with the points of a projective variety X. We give a geometric structure to this bijection and prove that the variety X is a coarse moduli space for point modules. We also describe the natural moduli stack \tilde{X} for embedded point modules---an analog of a "Hilbert scheme of one point"---as an infinite blow-up of X and establish good properties of \tilde{X}. The natural map \tilde{X} -> X is thus a kind of "Hilbert-Chow morphism of one point" for the naive blow-up algebra.

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arXiv2011

Prime spectra of derived quiver representations

We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered this way.

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arXiv2010

Geometric algebras on projective surfaces

Let X be a projective surface, let σbe an automorphism of X, and let L be a σ-ample invertible sheaf on X. We study the properties of a family of subrings, parameterized by geometric data, of the twisted homogeneous coordinate ring B(X, L, σ). In particular, we find necessary and sufficient conditions for these subrings to be noetherian. We also study their homological properties, their associated noncommutative projective schemes, and when they are maximal orders. In the process, we produce new examples of maximal orders; these are graded and have the property that no Veronese subring is generated in degree 1. Our results are used in a companion paper to give defining data for a large class of noncommutative projective surfaces.

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arXiv2009

Classifying birationally commutative projective surfaces

Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3 over an uncountable algebraically closed field. Suppose that the graded quotient ring of R is a skew-Laurent ring over a field; we say that R is a birationally commutative projective surface. We classify birationally commutative projective surfaces and show that they fall into four families, parameterized by geometric data. This generalizes work of Rogalski and Stafford on birationally commutative projective surfaces generated in degree 1; our proof techniques are quite different.

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arXiv2009

A general homological Kleiman-Bertini theorem

Let G be a smooth algebraic group acting on a variety X. Let F and E be coherent sheaves on X. We show that if all the higher Tor sheaves of F against G-orbits vanish, then for generic g in G, the sheaf Tor^X_j(gF, E) vanishes for all j >0. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman-Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.

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arXiv2008

Rings graded equivalent to the Weyl algebra

We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rings B that are graded equivalent to A: that is, the categories gr-A and gr-B are equivalent. This includes some surprising examples: in particular, we show that A is graded equivalent to an idealizer in a localization of A. We obtain this classification as an application of a general Morita-type characterization of equivalences of graded module categories.

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