Interactions between Algebraic Geometry and Noncommutative Algebra

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

In the 2012–13 academic year, the Mathematical Sciences Research Institute, Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more, the boundary between them has become increasingly blurred. This was apparent during the MSRI program, where there were a number of joint seminars on subjects of common interest: birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, and tilting theory, to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures, along with expository articles and research papers by participants of the programs. Volume 2 focuses on the most recent research.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

The workshop presented current developments exploring the boundaries and intersections between the fields of noncommutative algebra and representation theory on the one hand, and algebraic geometry on the other hand, with particular emphasis on topics where such interactions have led to substantial recent progress.

Geometric complexity theory and matrix powering

It is a remarkable fact that two prominent problems of algebraic complexity theory, the permanent versus determinant problem and the tensor rank problem (matrix multiplication), can be restated as explicit orbit closure problems. This offers the potential to prove lower complexity bounds by relying on methods from algebraic geometry and representation theory. While this basic idea for the tensor rank problem goes back to work by Volker Strassen from the mid eighties, the geometric complexity program has gained visibility and momentum in the past years. Some modest lower bounds for border rank have recently been proven by the construction of explicit obstructions. For further progress, a better understanding of irreducible representions of symmetric groups (tensor products and plethysms) is required. Interestingly, asymptotic versions of the the latter questions are of relevance in quantum information theory.

Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton–Okounkov bodies, PBW- filtrations and cluster algebras come to push the subject forward.

The new edition of this celebrated and long-unavailable book preserves the original book's content and structure and its unrivalled presentation of a universal method for the resolution of a class of singularities in algebraic geometry. At the same time, the book has been completely re-typeset, errors have been eliminated, proofs have been streamlined, the notation has been made consistent and uniform, an index has been added, and a guide to recent literature has been added. The book brings together ideas from algebraic geometry, differential geometry, representation theory and number theory, and will continue to prove of value for researchers and graduate students in these areas.

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Through basic research in mathematics, the NTT Institute for Fundamental Mathematics aims to enrich the "fountain of knowledge" that nourishes science and technology.In this article, we first provide an overview of the research being carried out at the Institute then introduce the Institute's core research areas: number theory, especially arithmetic dynamics; algebraic and arithmetic geometry; and representation theory and automorphic forms.

Written to honor the 80th birthday of William Fulton, the articles collected in this volume (the first of a pair) present substantial contributions to algebraic geometry and related fields, with an emphasis on combinatorial algebraic geometry and intersection theory. Featured topics include commutative algebra, moduli spaces, quantum cohomology, representation theory, Schubert calculus, and toric and tropical geometry. The range of these contributions is a testament to the breadth and depth of Fulton's mathematical influence. The authors are all internationally recognized experts, and include well-established researchers as well as rising stars of a new generation of mathematicians. The text aims to stimulate progress and provide inspiration to graduate students and researchers in the field.

Since its inception in the early seventies, the study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras. As indicated above, one of the main goals behind this meeting was to bring together a group of participants with a wide range of interests in and around the geometric and the combinatorial side of the representation theory of Lie groups and algebras. We strongly believe that such an approach to representation theory, in particular interaction between geometry and representation theory, will open up new avenues of thought and lead to progress in a number of areas. This diversity was well reflected in the expertize represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under the following three headings: (1) The study of the structure and representation theory through a) Slodowy slices and the Joseph ideal, b) Equivalence of categories of representations and certain geometric counterparts, c) Hall algebras, Quotient schemes and canonical bases, d) Richardson elements and birationality questions, e) free Lie algebras and current algebras, f) Dirac cohomology, h) Gelfand-Zeitlin theory (2) Combinatorial Aspects of Lie Systems particularly through a) Saturation problems and Buildings, b) Affine and double Affine Hecke algebras, Cherednik algebras, c) Solvable lattice models (3) Geometric Structures including a) Representation theoretic methods in enumerative geometry, b) Multiplicity free symplectic reduction, c) Monodromy actions of braid groups, d) Associated varieties for Lie super algebras, e) affine Grassmannians, f) Schubert varieties and Demazure modules The single most important development reported in this conference seems to us to be the generalization of the categorical Satake isomorphism to quantum groups at roots of unity, which allows to bring in a whole lot of new geometry into representation theory. It seems likely that it will lead to the proofs of some open conjectures of Lusztig. We were particularly happy to have quite a few very strong younger participants and some highly promising postdocs. We tried to accomodate young postdocs, as much as feasible, to present their work by devoting one afternoon to their (shorter) talks.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

This chapter is based on a series of lectures that I gave at the National University of Singapore in April 2013. The notes survey the representation theory of the cyclotomic Hecke algebras of type A with an emphasis on understanding the KLR grading and the connections between the classical ungraded representation theory and the rapidly emerging graded theory. They are fairly self-contained and they try to give a leisurely introduction to these algebras, with many examples and calculations that don't appear elsewhere. We make extensive use of the interactions between the ungraded and graded representation theory and try to explain what the grading gives us that we didn't have before. Combinatorics and cellular algebra techniques are used throughout, with a few results from geometry and 2-representation theory being quoted from the literature. Highlights include a complete description of the semisimple KLR algebras of type A using just the KLR relations, extensive discussion about graded Specht modules, a proof of the Ariki-Brundan-Kleshchev graded categorification theorem based on the graded branching rules, a cellular algebra approach to adjustment matrices and a (possibly optimistic) conjecture for the graded dimensions of the simple modules.