Constructions of Lie Algebras and their Modules

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Share This Paper. Tables from this paper. Citations Publications citing this paper. Quasi-particle bases of principal subspaces for the affine Lie algebras of types Bl 1 and Cl 1 Marijana Butorac. Penn , Christopher Sadowski. References Publications referenced by this paper. Foda , S. Ole Warnaar. This talk is intended as an elementary and informal introduction to the Beilinson-Drinfeld notion of a chiral algebra.

Representations of Lie Algebras

The Duflo theorem is a statement in Lie theory which allows us to compute the ring structure of the center of the universal enveloping algebra of a finite-dimensional Lie algebra. A categorical version of it was used by Maxim Kontsevich to give a spectacular proof of the so-called "Theorem on complex manifolds," which computes the multiplicative structure of Hochschild cohomology of a complex manifold in terms of the algebra of polyvector fields. In Lie theory there are also more general Duflo-type statements mostly conjectural , which study the case of a pair Lie algebra, Lie subalgebra.

I will explain how these translate into conjectures about the multiplicative structure of the Ext-algebra of the structure sheaf of a complex submanifold of a complex manifold, and how from this interaction we can hope to gain new insights into both algebraic geometry and Lie theory. Based on discussions with Damien Callaque. December 1, p. Connections to classical Lie theory are provided via the theory of double affine Hecke algebras and their degenerations. Winter January 13, p. This will be a series of introductory talks on linear algebraic groups.

The goal is to acquire a working knowledge of the subject rather than a systematic development of the theory. Prerequisites will be minimal and I will recall anything from algebraic geometry which we need. January 15, p. Surge Wee Liang Gan Introduction to linear algebraic groups cont. January 20, p. January 22, p.

Submission history

January 27, p. We discuss total positivity for matrices and use this as an motivation for the definition of cluster algebras which we will recall. January 29, p. We introduce preprojective algebras and basic properties of their representation theory. February 3, p. This is fundamental for the construction of our cluster character. February 5, p. We explain how certain subcategories of the representations of a preprojective algebra categorify the cluster algebra structure for the unipotent cells of the corresponding Lie group - where we come back to total positivity. February 12, p. Surge Konstantina Christodoulopoulou The boson-fermion correspondence Abstract.

I will describe some classical results for the boson-fermion correspondence in the context of affine Lie algebras. February 17, p. I will introduce new families of quantum algebras of double affine type which can be seen as Lie algebra analogs of certain algebras of Hecke type which have become of interest in the past ten years, namely Cherednik algebras, symplectic reflection algebras and deformed preprojective algebras. They are related to Yangians.

February 19, p. February 24, p. February 26, p. March 3, p. March 5, p. The talk will be based on Kostant's paper in Inventiones. October 2, p. Surge Jacob Greenstein An introduction to crystals October 7, p. I will give an interpretation of Littelmann's generalized Gelfand-Tsetlin patterns as saying that computing weight multiplicities for semisimple complex Lie algebras is equivalent to counting points with integral coordinates in certain families of polytopes.

The notion of a "chopped and sliced cone" formalizes this kind of families. Using the Blakley-Sturmfels theorem on vector partition functions I obtain properties of functions described by chopped and sliced cones, notably a version of the Duistermaat-Heckman theorem in this context. Surge Jacob Greenstein An introduction to crystals cont.

October 14, p.

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  • Achieving Project Management Success Using Virtual Teams;
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  • Constructions of Lie Algebras and their Modules : George B. Seligman : !

I will review some results by Milicic-Soergel on modules induced from Whittaker modules in the sense of Kostant in the setting of complex semisimple Lie algebras. Then I will describe some extensions of these results in the context of affine Lie algebras. October 16, p.

I will describe the irreducible Whittaker modules for the Lie algebra formed by adjoining a degree derivation to an infinite-dimensional Heisenberg Lie algebra. I will use these modules to construct a new class of modules for non-twisted affine Lie algebras and I will describe an irreducibility criterion for them. October 21, p. In this talk I will introduce co-Poisson module algebras and their quantizations as a framework for studying quantizations of module algebras. We study a family of infinite-dimensional algebras that are similar to semisimple Lie algebras as well as symplectic reflection algebras.

Finally, we discuss some questions about the higher rank cases. Joint with A. Tikaradze, and also with W. We study a class of structures in groups which provide a powerful tool in classifying the finite simple groups of Lie type. In this talk, we shall discuss the definitions and some basic properties of these systems, provide some examples, and provide further motivation, including a proof that any group with a Tits system admits a Bruhat decomposition. October 30, p. Using the results established by the other talks, I will work up to a theorem which establishes a simplicity condition on certain subgroups of a group with a Tits system, and exhibit an example of its use.

Surge Tim Ridenour Finite order automorphisms of simple Lie algebras. Arakawa and T. Calaque, B. Enriquez and P. Etingof, R. Freund and X. Then we will talk about what kinds of dAHA modules we get from above Lie-theoretic constructions. November 18, p. Surge Apoorva Khare Quivers, with a view toward Gabriel's theorem cont. November 25, p.

1.1 What is a Lie Algebra?

We will also see that this setting unifies many known types of Hecke algebras - usual finite , affine, double affine Cherednik , Hecke algebras of complex reflection groups Broue-Malle-Rouquier , and many others. In particular, there are orbifold Hecke algebras which provide quantization of Del Pezzo surfaces and their Hilbert schemes. Spring April 3, p. Conformal superalgebras describe symmetries of superconformal field theories and come equipped with an infinite family of products.

They also arise as singular parts of the vertex operator superalgebras associated with some well-known Lie structures e. In joint work with Arturo Pianzola and Victor Kac, we classify forms of conformal superalgebras using a non-abelian Cech-like cohomology set. As the products in scalar extensions are not given by linear extension of the products in the base ring, the usual descent formalism cannot be applied blindly.

April 8, p. I will give a quick introduction to the theory of cluster algebras introduced by Fomin and Zelevinsky. I will illustrate it by examples like coordinate rings of unipotent groups and flag varieties. April 10, p. I will introduce the notion of monoidal categorification of a cluster algebra, and will give examples coming from the representation theory of quantum affine algebras. Amiot Abstract.

Fomin and Zelevinsky invented cluster algebras in Soon, it became clear that these new algebras were intimately related to quiver representations. Cluster categories, introduced in , have provided a beautiful framework for making this relation precise. However, cluster categories are only defined for quivers without oriented cycles.

Building on Derksen-Weyman-Zelevinsky's fundamental work on quivers with potentials Claire Amiot has recently been able to extend the construction of the cluster category to a large class of quivers admitting oriented cycles and endowed with a potential, namely the so-called Jacobi-finite quivers with potential. April 22, p. It is a well known result due to D. Furthermore, I will give the details of a simple proof of Peterson's theorem and give a method for explicitly defining all such ideals. April 29, p. We shall show that they play a crucial role for instance in the theory of quantum invariants for 3-manifolds, in the theory of quantum Schur algebras at roots of unity.

May 6, p. For each partition we construct a natural representation of the Lie algebra of matrix-valued polynomials. We discuss universality properties of these repreresntations as well as combinatorics of their characters. We present explicit answers for currents in up to three variables. This construction can be viewed as a geometric analogue of the classical result that all irreducible representations of a finite nilpotent group are induced from 1-dimensional representations of suitable subgroups.

May 13, p. The Bethe Ansatz is a method to find eigenvectors of a certain family of commutative matrices. This method is often more complicated than the standard methods of linear algebra, moreover, sometimes it fails to produce the complete set of the eigenvectors. However, the attempts to understand it lead to a number of interesting connections with surprisingly many areas of mathematics - and to new results in those areas. In this talk I will try to give an introduction to the Bethe Ansatz method. The focus will be on more recent results of Cohen-Manin-Zagier, Duval, Lecomte, Ovsienko, Roger, and others on modules of differential operators between tensor density modules.

We will conclude with a brief look at similar problems over higher dimensional manifolds which are still open. The talk is based on a joint work with Punita Batra June 3, p. Inhomogeneous Lie groups and algebras play an important role in physics, and so do some inhomogeneous quantum groups.

In this talk I will introduce the notions of a inhomogeneous Lie algebra and Lie bialgebra and show how one can obtain classification results for the inhomogeneous quantum groups by studying inhomogeneous Lie bialgebras. If time permits I shall explain how one obtains quantum symmetric algebras from inhomogeneous quantum groups.

June 5, p. Surge R. Title: The talk will focus on finite-dimensional representations of hyper loop algebras over arbitrary fields. Hyperalgebras are certain Hopf algebras related to algebraic groups. When the field is of characteristic zero, a given hyper loop algebra coincide with the universal enveloping algebra of a certain "classical" loop algebra.

The main results we will discuss are: the classification of the irreducible representations, construction of the Weyl modules, a study of base change forms , and tensor products of irreducible modules. Some of these results are more interesting when the field is not algebraically closed and are beautifully related to the study of irreducible representations of polynomial algebras and field theory. January 17, p. If time permits, we may discuss applications of this result to the study of "exponential-polynomial modules".

January 24, p. January 31, p. Minimal affinizations of representations of quantum groups introduced by Chari are relevant modules for quantum integrable systems. Schubert varieties and their singularities are important in the study of representation theory and algebraic groups.


A new family of algebras underlying the Rogers-Ramanujan identities and generalizations.

In this talk I will describe one aspect of this story which involves singular Chern classes, characteristic cycles, and small resolutions of singularities. For concreteness, I'll focus on the case of Schubert varieties in the Grassmannian. In this context there is an open "positivity conjecture" which is interesting from both the geometric and combinatorial points of view.

I will give an overview of the various statements which are called the Kirillov-Reshetikhin conjecture. February 14, p. In recent preprint arXiv Cattaneo and G. Using an interesting modification of the Poisson sigma model the authors construct a curious L-infinity morphism not a quasi-isomorphism!

The authors also apply this result to a construction of a specific trace on the deformation quantization algebra of a unimodular Poisson manifold. Although this trace can be constructed using the formality quasi- isomorphism for Hochschild chains, the relation of the L-infinity morphism of A.

Felder to the formality quasi-isomorphism is a mystery. February 28, p. In this talk, we will explore a generalization of symmetric Frobenius algebras i. We will explain how this property closely resembles the Calabi-Yau property for infinite-dimensional algebras, and will call such algebras "Calabi-Yau Frobenius algebras". The Hochschild cohomology is then a Frobenius algebra. In the case of periodic algebras algebras which have a periodic bimodule resolution , we obtain a Batalin-Vilkovisky structure on Hochschild cohomology, which is conjecturally selfadjoint with respect to the Frobenius structure.

We will explain these results in detail in the case of preprojective algebras of Dynkin quivers, giving a full computation of their Hochschild co homology over the integers. March 6, p. Fall October 2, p. I am going to talk about recent preprint arXiv Calaque and M. Van den Bergh. In this paper they proved a multiplicative version of Caldararu's conjecture which describes the Hochschild cohomology of a smooth algebraic variety as a graded ring.

I will formulate the result of Calaque and Van den Bergh and explain how they proved it using Kontsevich's formality quasi-isomorphism. Surge Vyjayanthi Chari Current algebras, highest weight categories and quivers October 11, p. Surge Vyjayanthi Chari Current algebras, highest weight categories and quivers October 16, p.

Surge Vyjayanthi Chari Current algebras, highest weight categories and quivers October 18, p. Surge Jacob Greenstein Kirillov-Reshetikhin modules and finite dimensional algebras October 23, p. Surge Jacob Greenstein Kirillov-Reshetikhin modules and finite dimensional algebras cont. In my talk based on the joint paper with Vladimir Retakh I will introduce a version of Lie algebras and Lie groups over noncommutative rings. Kapranov's approach to noncommutative geometry. I will conclude my talk with examples of such groups and with the description of "noncommutative root systems" of rank 1.

The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. I will talk about my recent paper with V.

Rubtsov in which we propose a version of the algebraic index theorem for a Poisson manifold based on this trace density map. Surge Vasiliy Dolgushev An algebraic index theorem for Poisson manifolds cont. November 15, p. In my talk I will report on commutors for crystals which were introduced and studied by Kamnitzer and Henriques. We will define the commutors associated to tensor products of crystal bases of modules over quantized enveloping algebras. November 29, p. The isomorphism classes of each type of these objects can be geometrically parametrized by the same space, the Calogero-Moser algebraic varieties.

A new family of algebras underlying the Rogers-Ramanujan identities and generalizations.

We will give a conceptual explanation of this bijection by constructing a natu- ral functor between the corresponding module categories. This is joint work with Y. Berest and O. We will discuss the "Factorization Phenomenon" which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into smaller representations of the subalgebra. The original Lie algebra may be any symmetrizable Kac-Moody algebra including finite-dimensional, semi-simple Lie algebras.

We will provide an algebraic explanation for such a phenomenon using "Spin construction". We will present a few Factorization results for any embedding of a symmetrizable Kac-Moody algebra into another, using Spin construction and give some combinatorial consequences of it. We will extend the notion of Spin from finite-dimensional to symmetrizable Kac-Moody algebras which requires a very delicate treatment. We will give the formula for the character of Spin for the above category and refine the factorization results in the case of affine Lie algebras. Finally, we will discuss classification of "Coprimary representations" i.

Surge Vyjayanthi Chari Categorification and Representation theory cont. April 12, p. May 1, p. Surge Wee Liang Gan Khovanov homology. May 22, p. I will talk about recent work by Stroppel , which relates two algebras. This will occupy most of the talk. Winter January 11, p. January 25, p. This fact should then allow one to polarize open submanifolds of generic adjoint orbits. February 1, p. The rational Cherednik algebras are an interesting family of algebras that can be attached to any complex reflection group.

February 8, p. This talk is based on a joint work with D. Exact results Alexander Berkovich. Functional models for representations of current algebras and semi-infinite Schubert cells A. Stoyanovsky , Boris L.

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