Pure Rickart Modules

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.

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.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Totally acyclic complexes

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.

The aim of this workshop was to communicate the most current developments in the field of noncommutative algebra and its interactions with algebraic geometry and representation theory.

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.

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.

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.

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.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

The basic structure and representation theory of finite-dimensional simple Lie algebras over \(\mathbb{C} \), as developed by W. Killing, E. Cartan and H. Weyl, is a well-known and frequently used branch of mathematics. It is a striking fact that very significant new results in this classical area of mathematics have come to light during the past decade. These new results have originated from the development of the theory of quantum groups. The purpose of this article is to survey briefly the classical results on the representation theory of simple Lie algebras and then to describe the recent developments primarily due to G. Lusztig, M. Kashiwara and P. Littelmann.

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

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.