A Coxeter--Gram Classification of Positive Simply Laced Edge-Bipartite Graphs

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 is a report on advances in the representation theory of finite dimensional algebras in the years 1984 – 1990. During these years, the German research council (DFG) has sponsered a Forschungsschwerpunkt devoted to the representation theory of finite groups and finite dimensional algebras; it started in 1984 and will be finished by 1991.KeywordsEndomorphism RingIndecomposable ModuleFinite Dimensional AlgebraHall AlgebraHereditary AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The modular representation theory of finite groups has its origins in the work of Richard Brauer. In this survey article, we first discuss the work being done on some outstanding conjectures in the theory. We then describe work done in the eighties and nineties on modular representations in non-defining characteristic of finite groups of Lie type. In the second part of the article, we discuss some recent developments in the theory for symmetric groups and Hecke algebras, where remarkable connections with Lie theory and graded representation theory have been made.Modularrepresentations

Coxeter energy of graphs

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.

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,

This work presents a survey (with some proofs) of the recent developments in the study of Coxeter transformations and their applications to representation theory of associative algebras. Let Q be a quiver and C = C(Q) the associated Coxeter matrix. We consider results on the eigenvalues of C: multiplicities, distribution, bounds, relation with the symmetries of Q... We consider applications to the study of the indecomposable modules over the hereditary algebra k[Q] (k a field): position of modules, growth of the Auslander-Reiten translates... Other applications concern growth of components of the representation-quiver of finite dimensional algebras, towers of algebras...

Quasi-hereditary algebras were introduced by L. Scott [S] in order to study highest weight categories arising in the representation theory of semisimple complex Lie algebras and algebraic groups, and important results were proved by Cline, Parshall and Scott (see [CPS1,2]). These algebras can be defined entirely in ring-theoretic terms; and they were studied from this point of view by Dlab and Ringel (see [DR1,2], [R1,2]) and by others. In particular it turns out that quasi-hereditary algebras are quite common.

In this lecture we indicate how to complete the last step in the process outlined at the beginning of Part II: to take our knowledge of the classification and representation theory of complex algebras and groups and deduce the corresponding statements in the real case. We do this in the first section, giving a list of the simple classical real Lie algebras and saying a few words about the corresponding groups and their (complex) representations. The existence of a compact group whose Lie algebra has as complexification a given semisimple complex Lie algebra makes it possible to give another (indeed, the original) way to prove the Weyl character formula; we sketch this in §26.2. Finally, we can ask in regard to real Lie groupsGa question analogous to one asked for the representations of fmite groups in §3.5: which of the complex representationsVofGactually come from real ones. We answer this in the most commonly encountered cases in §26.3. In this final lecture, proofs, when we attempt them, are generally only sketched and may require more than the usual fortitude from the reader.KeywordsIrreducible RepresentationSpin RepresentationReal FormCartan SubalgebraExterior PowerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let A be a finite-dimensional algebra which is quasi-hereditary with re- spect to the poset (�,�), with standard modules �(�) for 2 �. Let F(�) be the category of A-modules which have filtrations where the quotients are standard modules. We determine some inductive results on the relative Auslander-Reiten quiver of F(�). Quasi-hereditary algebras were introduced by Cline, Parshall and Scott in the study of highest weight categories arising in the representation theory of semisimple Lie algebras and algebraic groups (2), and they were studied by Dlab and Ringel and others (4, 5). LetA be quasi-hereditary. Then the simple A-modules are labelled as L(�) forin some partially ordered set (�,≤). Of central importance are the standard modules, denoted by �(�), and the categoryF(�) of modules which have a filtration where the quotients are standard modules. In (12) it was proved that this category has relative Auslander-Reiten (AR) sequences. The aim of our paper is to present some inductive results on the relative Auslander-Reiten quiver ofF(�). For any subset of �, letF (�) be the category of modules inF(�) where all quotients are �(�) with �∈ . If � is an ideal in the poset (�,≤) then there is a subalgebra A of A which is quasi-hereditary with poset (, ≤). In this case, the categoryF (�) is equivalent toF(�A ). A main result in this paper relates, for a non-projective indecomposable mod- ule X inF (�), the two relative Auslander-Reiten sequences with cokernel X inF(�) and inF (�) (Theorem 2.5). By Ringel duality, there is an equivalent version for categories of modules filtered by costandard mod- ules. We describe this and some variations in Section 3. We continue with further results on relative Auslander-Reiten sequences. In particular, we determine certain relative Auslander-Reiten sequences where a projective- injective summand occurs; this generalizes the usual almost split sequence where a projective-injective module occurs.

Representations of Lie groups are abstractions of continuous symmetries of linear spaces.By considering their differentials (linear approximations), we obtain representations of Lie algebras.These are regarded as generalizations of the Fourier analysis and important tools not only in mathematics but also in physics.In this article, I present fundamental and important examples on representations of Lie groups and Lie algebras and present some of my latest results.

The representation of symmetry groups plays a crucial role in physics. In this chapter we discuss the elements of the representation theory of Lie groups and Lie algebras. In particular, we apply representations of the Lie group SU(3) and the Lie algebra su(3) to the quark model in strong interaction.

In this paper we show that the Deligne-Langlands-Lusztig classification of simple representations of an affine Hecke algebra remains valid if the parameter is not a root of the corresponding Poincaré polynomial. This verifies a conjecture of Lusztig proposed in 1989.

The Weyl algebra is one of the most important objects in mathematical physics and representation theory of Lie groups and Kac-Moody algebras.