Moduli Techniques in Graded Ring Theory and Their Applications
Lead Research Organisation:
University of Edinburgh
Department Name: Sch of Mathematics
Abstract
A ring is a mathematical structure that models many types of symmetry. Most rings encountered "in nature" are noncommutative: the order of operations matters. This project will investigate deep relationships between noncommutative ring theory and geometry.
Rings are studied through their modules: objects that echo the symmetry encoded in the ring. The structure of a ring depends subtly and powerfully on the geometry of families of modules over that ring, and this connection has led to many advances. This project will explore this connection between the geometry of families of modules and the algebraic structure of rings in depth. I will extend current methods and develop new ones, and will apply my results to important unsolved algebraic problems.
An example of the power of this connection between geometry and algebra is given by the famous Virasoro algebra. The Virasoro algebra is renowned in mathematics and physics. It may be viewed as a mathematical model of statistical mechanics, and so is of deep importance to physics, particularly conformal field theory. The Virasoro algebra is a Lie algebra, rather than a ring; it can be turned into a ring by forming its so-called universal enveloping algebra.
Although the Virasoro algebra had been intensively studied for many years, important basic questions about its universal enveloping algebra remained unanswered. Specifically, for at least 25 years mathematicians had been asking if the enveloping algebra of the Virasoro algebra had the noetherian property. (Rings that are noetherian are relatively well-behaved; those that are not noetherian are more exotic.) In recent joint work with Walton, I applied geometry to solve this problem: the enveloping algebra of the Virasoro algebra is not noetherian. Our work shows the power of geometric techniques to address purely algebraic problems.
One key method of our proof that the enveloping algebra of the Virasoro algebra is not noetherian was to construct a simpler model, called the canonical birational commutative factor. Because it is simpler, the model is easier to study; on the other hand, passing to the model loses a great deal of information. In this project, I will develop a general method, which will apply to many more rings than the enveloping algebra of the Virasoro algebra, to construct other canonical factors that contain more information but are still amendable to study. A general construction of more complex canonical factors will be a significant advance.
Through the new techniques this project will develop, I will answer many important questions in ring theory. I will use geometry to get more information about the enveloping algebra of the Virasoro algebra. I will explore whether the noetherian property described above can be detected through geometry. I will apply geometric methods to a large class of rings, of which the enveloping algebra of the Virasoro is only one example: to universal enveloping algebras of graded infinite-dimensional Lie algebras. Through these methods, I will show these rings are not noetherian. These rings are famously intractable, and this problem is inaccessible without the new methods that I will bring to bear.
Rings are studied through their modules: objects that echo the symmetry encoded in the ring. The structure of a ring depends subtly and powerfully on the geometry of families of modules over that ring, and this connection has led to many advances. This project will explore this connection between the geometry of families of modules and the algebraic structure of rings in depth. I will extend current methods and develop new ones, and will apply my results to important unsolved algebraic problems.
An example of the power of this connection between geometry and algebra is given by the famous Virasoro algebra. The Virasoro algebra is renowned in mathematics and physics. It may be viewed as a mathematical model of statistical mechanics, and so is of deep importance to physics, particularly conformal field theory. The Virasoro algebra is a Lie algebra, rather than a ring; it can be turned into a ring by forming its so-called universal enveloping algebra.
Although the Virasoro algebra had been intensively studied for many years, important basic questions about its universal enveloping algebra remained unanswered. Specifically, for at least 25 years mathematicians had been asking if the enveloping algebra of the Virasoro algebra had the noetherian property. (Rings that are noetherian are relatively well-behaved; those that are not noetherian are more exotic.) In recent joint work with Walton, I applied geometry to solve this problem: the enveloping algebra of the Virasoro algebra is not noetherian. Our work shows the power of geometric techniques to address purely algebraic problems.
One key method of our proof that the enveloping algebra of the Virasoro algebra is not noetherian was to construct a simpler model, called the canonical birational commutative factor. Because it is simpler, the model is easier to study; on the other hand, passing to the model loses a great deal of information. In this project, I will develop a general method, which will apply to many more rings than the enveloping algebra of the Virasoro algebra, to construct other canonical factors that contain more information but are still amendable to study. A general construction of more complex canonical factors will be a significant advance.
Through the new techniques this project will develop, I will answer many important questions in ring theory. I will use geometry to get more information about the enveloping algebra of the Virasoro algebra. I will explore whether the noetherian property described above can be detected through geometry. I will apply geometric methods to a large class of rings, of which the enveloping algebra of the Virasoro is only one example: to universal enveloping algebras of graded infinite-dimensional Lie algebras. Through these methods, I will show these rings are not noetherian. These rings are famously intractable, and this problem is inaccessible without the new methods that I will bring to bear.
Planned Impact
The primary impact of this project will be on developing and maintaining expertise in the UK in the fields of noncommutative ring theory, noncommutative algebraic geometry, and representation theory. The project proposes to investigate problems of international relevance and interest. This will maintain and increase the world-leading role of UK researchers in these areas, which has been recognised in the past two international reviews of mathematics.
A key part of the proposed grant will be the training of a postdoctoral researcher (PDRA). Thus this project will have a direct impact on building the capacity of the UK in the mathematical sciences and in STEM fields more generally.
Mathematics is fundamental to the UK economy as explained in a recent study by EPSRC. The study showed that "[t]he quantified contribution of mathematical science research to the UK economy in 2010 is estimated to be approximately 2.8 million in employment terms (around 10 per cent of all jobs in the UK) and £208 billion in terms of GVA contribution (around 16 per cent of total UK GVA)." As well as these direct impacts, mathematical research activities by organisations and employees also has indirect and induced effects.
Research in pure mathematics is of benefit to society more broadly. As the 2010 International Review of Mathematical Sciences said, "the mathematical sciences provide a universal language for expressing abstractions in science, engineering, industry and medicine; mathematical ideas, even the most theoretical, can be useful or enlightening in unexpected ways, sometimes several decades after their appearance; the mathematical sciences play a central role in solving problems from every imaginable application domain; and, because of the unity of the mathematical sciences, advances in every sub-area enrich the entire field." Thus this research will have distributed impacts which are hard to quantify but are nonetheless profound.
A key part of the proposed grant will be the training of a postdoctoral researcher (PDRA). Thus this project will have a direct impact on building the capacity of the UK in the mathematical sciences and in STEM fields more generally.
Mathematics is fundamental to the UK economy as explained in a recent study by EPSRC. The study showed that "[t]he quantified contribution of mathematical science research to the UK economy in 2010 is estimated to be approximately 2.8 million in employment terms (around 10 per cent of all jobs in the UK) and £208 billion in terms of GVA contribution (around 16 per cent of total UK GVA)." As well as these direct impacts, mathematical research activities by organisations and employees also has indirect and induced effects.
Research in pure mathematics is of benefit to society more broadly. As the 2010 International Review of Mathematical Sciences said, "the mathematical sciences provide a universal language for expressing abstractions in science, engineering, industry and medicine; mathematical ideas, even the most theoretical, can be useful or enlightening in unexpected ways, sometimes several decades after their appearance; the mathematical sciences play a central role in solving problems from every imaginable application domain; and, because of the unity of the mathematical sciences, advances in every sub-area enrich the entire field." Thus this research will have distributed impacts which are hard to quantify but are nonetheless profound.
Organisations
People |
ORCID iD |
| Susan Sierra (Principal Investigator) |
Publications
Alexey Petukhov
(2021)
The Poisson spectrum of the symmetric algebra of the Virasoro algebra
Avesta A
(2023)
3D Capsule Networks for Brain Image Segmentation
in American Journal of Neuroradiology
Bell J
(2017)
On a dynamical Mordell-Lang conjecture for coherent sheaves ON A DYNAMICAL MORDELL-LANG CONJECTURE FOR COHERENT SHEAVES
in Journal of the London Mathematical Society
Bell Jason P.
(2016)
On a Dynamical Mordell-Lang Conjecture for Coherent Sheaves
in arXiv e-prints
Hennig J
(2016)
Path Algebras of Quivers and Representations of Locally Finite Lie Algebras
in International Mathematics Research Notices
Hennig J.
(2015)
Path algebras of quivers and representations of locally finite Lie algebras
in arXiv e-prints
Iyudu N
(2020)
Enveloping algebras with just infinite Gelfand-Kirillov dimension
in Arkiv för Matematik
IYUDU N
(2020)
POTENTIAL ALGEBRAS WITH FEW GENERATORS
in Glasgow Mathematical Journal
Iyudu N
(2021)
Pre-Calabi-Yau algebras as noncommutative Poisson structures
in Journal of Algebra
| Description | Sierra and Spenko studied the geometry of intermediate series representations of generalised Witt algebras and showed that these representations are naturally parameterised by the blowup of the affine plane at two points, unifying what had before been a case-by-case classification of these representations. This geometry also gives a new direct proof that the enveloping algebras of these Lie algebras are not noetherian. This addresses Problem 1.1 in Objective 1 of the original grant proposal. Spenko and Van den Bergh continued their work on noncommutative resolutions of singularities. They give the most general technique known so far for constructing noncommutative resolutions of toric singularities, and give new examples of such resolutions. They also use their powerful techniques to give semi-orthogonal decompositions of quotient stacks under extremely general conditions. Sierra and Lecoutre use Poisson geometry to construct new families of Poisson algebras and of Artin-Schelter regular algebras, including giving new examples of Calabi-Yau algebras in every dimension. Sierra, Bell, and Satriano used a techniques from algebraic geometry and p-adic analysis to prove a dynamical version of the Mordell-Lang conjecture: for surfaces or in the presence of an algebraic group, all but finitely many translates of two subvarieties by a power of an automorphism will be transverse, in the appropriate sense. . Sierra and Petukhov make two conjectures about ideals in the enveloping algebra of the positive Witt algebra: that these satisfy the ascending chain condition and that any proper factor of the enveloping algebra has finite Gelfand-Kirillov dimension, and make significant progress towards proving both conjectures. This addresses Problem 1.2 in the original grant proposal. Sierre, Spenko, Vancliff, Veerapan, and Wiesner study representations of the Lie superalgebra sl(1|1) and the colour Lie algebra sl_c(2). This research was originated at the workshop Women in Noncommutative Algebra and Representation Theory at the Banff International Research Centre in 2016. Rogalski, Stafford, and Sierra use their earlier proof of the noncommutative Castelnuovo contraction criterion to prove a striking property of Sklyanin algebras: the only connected graded noetherian algebras lying between a Sklyanin algebra S and it graded quotient ring Q are isomorphic to Veroneses of S, up to taking maximal orders. In more generality, they show that given appropriate conditions, all overrings of certain noncommutative rings are constructed by contracting finitely many (-1) lines. Iyudu and Shkarin classify cubic and quadratic potential algebras with few generators. They prove that if a two-generated potential algebra is finite-dimensional, then the potential contains terms of degree higher than two, asking an earlier question of Iyudu and Smoktunowicz. In a second paper, Iyudu and Shkarin classify quadratic algebras with the Hilbert series of a polynomial ring in three variables. Iyudu and Sierra used innovative combinatorics to prove one of the conjectures of Petukhov and Sierra, showing that any proper homomorphic image of the enveloping algebra of the Witt algebra has polynomial growth. This addresses Problem 1.2 in the original grant proposal. |
| Exploitation Route | Our findings are of interest to other mathematicians, and we have been invited to many conferences to speak about our work. |
| Sectors | Other |
| URL | https://www.maths.ed.ac.uk/~ssierra/preprints.html |
| Description | Enveloping Algebras of Infinite-Dimensional Lie Algebras |
| Amount | £555,385 (GBP) |
| Funding ID | EP/T018844/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 09/2020 |
| End | 08/2024 |