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| October 3, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Jacob Greenstein
Quotient categories and Directed Categories Abstract. Quotient categories were introduced to extend to the categorical setting the notion of a quotient module and to provide a formal way of "getting rid of" subquotients in module categories. In this talk we will discuss an application of quotient categories to structural properties of a special class of highest weight categories. |
| October 5, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Joel Kamnitzer (UC Berkeley/AIM)
Knot Homology via Derived Category of Coherent Sheaves Abstract. We will give a construction of a knot homology theory using the derived category of coherent sheaves of a certain variety arising in geometric representation theory. We conjecture that our knot homology is related to Khovanov homology (joint work with Sabin Cautis). |
| October 10, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Jacob Greenstein
Quotient categories and Directed Categories (cont.) |
| October 12, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Apoorva Khare
The BGG category O Abstract. The BGG (after Bernstein-Gelfand-Gelfand) category O is an important category of modules over a complex semisimple Lie algebra, that contains all finite-dimensional and all Verma modules. We show how category O decomposes into a direct sum of subcategories, and explore some of their properties. We further use category O to obtain certain well-known formulae, such as Weyl character formula and BGG reciprocity formula. |
| October 17, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Apoorva Khare
The BGG category O (cont.) |
| October 19, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Hans Wenzl (UC San-Diego)
Restriction coefficients and Brauer Algebras Abstract. We give a fairly simple proof for formulas for multiplicities for restricting representations of GL(N) to O(N) using Brauer algebras and fusion categories. The result is stated in tems of certain reflection groups. This formalism and results about tilting modules of quantum groups also suggest a description of the non-semisimple Brauer algebra in terms of certain parabolic Kazhdan-Lusztig polynomials. |
| October 24, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Andrew Linshaw (UC San-Diego)
Chiral equivariant cohomology Abstract. The equivariant cohomology ring HG(M) is an algebraic invariant one can attach to a smooth manifold M equipped with an action of a compact Lie group G. The chiral equivariant cohomology is a "chiralization" of HG(M), that is, a vertex algebra which contains HG(M) as the subspace of conformal weight zero. I will give a brief introduction to vertex algebras, and then discuss the construction of the new cohomology and some of the basic results and examples. This a joint work with Bong Lian and Bailin Song |
| October 26, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Prasad Senesi
Spectral Characters of Finite-Dimensional Representations of Twisted Affine Lie Algebras Abstract. A block decomposition of the category C of finite-dimensional representations of a twisted affine Lie algebra is examined. The result is an extension of one given by Chari and Moura for the untwisted affine Lie algebras, in which the blocks of C are shown to be in bijective correspondence with the spectral characters of the Lie algebra. |
| October 31, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Prasad Senesi
Spectral Characters of Finite-Dimensional Representations of Twisted Affine Lie Algebras (cont.) Abstract. A block decomposition of the category C of finite-dimensional representations of a twisted affine Lie algebra is examined. The result is an extension of one given by Chari and Moura for the untwisted affine Lie algebras, in which the blocks of C are shown to be in bijective correspondence with the spectral characters of the Lie algebra. |
| November 2, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Victor Ostrik
(University of Oregon, Eugene)
Tensor categories attached to cells in finite Weyl groups Abstract. For every two sided cell in a Weyl group Lusztig attached a tensor category (via suitably truncated convolution of perverse sheaves on the corresponding flag variety). Moreover, Lusztig proposed a precise conjecture which describes this category in elementary terms. In this talk we report on recent joint work with R.Bezrukavnikov and M.Finkelberg where this conjecture was proved for almost all two sided cells. On the other hand the conjecture fails in the remaining cases. |
| November 7, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Wee Liang Gan
Introduction to Dunkl operators Abstract. Dunkl operators are differential-difference operators introduced by Charles Dunkl in 1989. Due to the work of Opdam, Heckman and others, they are now a key tool in the theory of multivariable orthogonal polynomials. A major development was Cherednik's discovery of their intimate connection with degenerate affine Hecke algebras. I will give an introduction to Dunkl operators and some of their applications. |
| November 9, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Sebastian Zwicknagl (University of Oregon, Eugene/UC Riverside)
Equivariant Poisson structures and quantum symmetric algebras Abstract. M. Kontsevich showed that one can deform any Poisson algebra whose spectrum is a manifold or a variety. His results, however, leave the following natural questions: 1) Is it possible to find a presentation for the deformed algebra? 2) Can one deform Poisson algebras with zero-divisors? In this talk I will discuss these questions in the case of r-matrix brackets on the symmetric algebra of a Lie bialgebra module. I will classify all modules for which these brackets are Poisson and then explicitly describe their deformations as quantum symmetric algebras. I will then explain, why I conjecture that quantum symmetric algebras provide an affirmative answer to questions 1) and 2). |
| November 14, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Kobi Kremnizer (MIT)
Proof of the de Concini-Kac-Procesi conjecture Abstract. I will introduce the quantum flag variety and quantum D-modules on it. These are noncommutative algebro-geometric objects. In roots of unity they localize to the Springer resolution and allow for a computation of dimensions of modules over the quantum group. |
| December 7, 2006 | ||
| 1:00-2:00 p.m. | Surge 284 | Alexei Oblomkov (Princeton)
Quantum cohomology of Hilbert scheme of points of ADE resolution and loop algebras Abstract. Joint with D. Maulik. Let X be a resolution of the ADE singularity C2/Γ. We formulate the conjectural description of the structure of the ring of quantum equivariant cohomology of Hilbn(X). In particular the generators of the ring are given in terms of the loop algebra of the corresponding type. In the case of An singularity the conjecture is a theorem. In my talk I will mostly discuss the case of A1 singularity. All necessary geometric definitions unfamiliar to the audience will be reminded. |
| January 11, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Dimitar Grantcharov (CS San-Jose)
On the category of modules with bounded weight multiplicities Abstract. Let g be a finite dimensional simple Lie algebra. In this talk we will focus on the category B of all bounded weight g-modules, i.e. those that are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Fernando implies that bounded weight g-modules exist only for g = sl(n) and g = sp(2n). In the second case we show that the category B has enough projectives if and only if n>1 and is wild if and only if n>2. The case g = sl(n) is much more complicated as the description of each block Bχ of B depends on the type of the central character χ. |
| January 25, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Mark Colarusso (UCSD)
Gelfand-Zeitlin algebras and the polarization of generic adjoint orbits for gl(n) and so(n) Abstract. Let gn be either the n × n complex general linear Lie algebra gl(n) or the n × n complex orthogonal Lie algebra so(n) and let Gn be the corresponding adjoint group. Let P(gn) denote the algebra of polynomials on gn. The associative commutator on the universal enveloping algebra of gn induces a Poisson structure on P(gn). Let J(gn) be the commutative Poisson subalgebra of P(gn) generated by the invariants P(gm)Gm for m=1,…,n. J(gn) gives rise to a commutative Lie algebra of Hamiltonian vector fields on gn; V={ ξf | f ∈ J(gn)}. Choosing an appropriate set of generators for J(gn) gives rise to a subalgebra V′ ⊂ V. This subalgebra integrates to an action of a commutative, simply connected complex analytic group isomorphic to Cd/2 on gn, where d is the dimension of a generic adjoint orbit in gn. This fact should then allow one to polarize open submanifolds of generic adjoint orbits. We will discuss the orbit structure of the action of this group on gn. In the case of gn=gl(n), we will give a description of the work of Kostant-Wallach in the most generic case in such a form that can be used to establish a formalism for dealing with the less generic orbits studied by the speaker. In the case of gn=so(n), the speaker will discuss his work on analyzing the orbit structure of the group on certain sets of semi-simple elements. |
| February 1, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 |
Stephen Griffeth (University of Minnesota)
Finite dimensional representations of rational Cherednik algebras Abstract. The rational Cherednik algebras are an interesting family of algebras that can be attached to any complex reflection group. In this talk, I will show how to study finite dimensional modules for the rational Cherednik algebras attached to the infinite family of complex reflection groups G(r,p,n) via eigenspace decompositions. Our approach allows us to construct finite dimensional irreducible modules of dimension mn for each integer m coprime to the "Coxeter" number of G(r,p,n), to build "BGG" resolutions of these modules, and to construct a basis of the coinvariant ring for G(r,p,n) generalizing the Garsia-Stanton "descent monomial" basis for the coinvariant ring for the symmetric group G(1,1,n). |
| February 8, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Adriano de Moura (UNICAMP, Brazil)
Finite-Dimensional Representations of Hyper Loop Algebras Abstract. This talk is based on a joint work with D. Jakelic where we study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over a field of positive characteristic associated to non-twisted affine Kac-Moody algebras. In the talk we will go over the classification of the irreducible modules, a version of Steinberg's Tensor Product Theorem, and the construction of positive characteristic analogues of the Weyl modules as defined by Chari and Pressley in the characteristic zero case. We will also discuss reduction modulo p and a Conjecture regarding reduction modulo p of Weyl modules. |
| February 15, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Adriano de Moura (UNICAMP, Brazil)
On Applications Of Geometric Invariant Theory to Representation Theory Abstract. A. King has proposed a method for organizing the representation theory of wild algebras by using the concept of stability which originally arose in the context of Mumfors's geometric invariant theory. I will talk about a joint work with V. Futorny and M. Jardim where we explore some applications of these ideas to certain categories of modules for Lie algebras. |
| February 22, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Ghislain Fourier (Universität zu Köln, Germany)
Demazure modules for current algebras Abstract. This talk is based on joint works with P. Littelmann. We study finite dimensional modules for current algebras, starting with Demazure modules for (twisted) affine Kac-Moody algebras. A few useful properties of these Demazure modules are found and proved. With these properties one can prove that Weyl modules as well as Kirillov-Reshetikhin modules are (in the simply-laced case) in fact isomorphic to Demazure modules as modules for the current algebra. As some applications of this isomorphism one can prove the conjectured dimension formula for Weyl modules, some conjectures about fusion products, that the conjectural Kirillov-Reshetikhin crystals are unique etc |
| March 1, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Ghislain Fourier (Universität zu Köln, Germany)
Demazure modules for current algebras Abstract. This talk is based on joint works with P. Littelmann. We study finite dimensional modules for current algebras, starting with Demazure modules for (twisted) affine Kac-Moody algebras. A few useful properties of these Demazure modules are found and proved. With these properties one can prove that Weyl modules as well as Kirillov-Reshetikhin modules are (in the simply-laced case) in fact isomorphic to Demazure modules as modules for the current algebra. As some applications of this isomorphism one can prove the conjectured dimension formula for Weyl modules, some conjectures about fusion products, that the conjectural Kirillov-Reshetikhin crystals are unique etc |
| March 8, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Wee Liang Gan
Symplectic reflection algebras Abstract. I will give an introduction to symplectic reflection algebras. These algebras are closely related to quotient singularities of the form V/G, where V is a symplectic vector space, and G is a finite group of automorphisms of V. |
| March 15, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Apoorva Khare
A deformation-theoretic proof of the Poincare-Birkhoff-Witt Theorem for quadratic Koszul algebras Abstract. I will present a theorem by Braverman and Gaitsgory that characterizes what Koszul algebras generated by "quadratic" relations, have a PBW-type theorem. The usual PBW theorem for Lie algebras is an example, as are Weyl and Clifford algebras. |
| April 3, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Vyjayanthi Chari
Categorification and Representation theory |
| April 5, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 |
Rajeev Walia (Michigan State University)
Tensor factorization and Spin construction for Kac-Moody algebras Abstract. 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 introduce a category of “d-finite, Orthogonal Level zero” representations for which, surprisingly, the Spin gives a representation in the Bernstein-Gelfand- Gelfand category O. 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.e those representations whose Spin is irreducible. |
| April 10, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Vyjayanthi Chari
Categorification and Representation theory (cont.) |
| April 12, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Vyjayanthi Chari
Categorification and Representation theory (cont.) |
| April 24, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Vyjayanthi Chari
Categorification and Representation theory (cont.) |
| April 26, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Vyjayanthi Chari
Categorification and Representation theory (cont.) |
| May 1, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Vyjayanthi Chari
Categorification and Representation theory (cont.) |
| May 3, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Vyjayanthi Chari
Categorification and Representation theory (cont.) |
| May 15, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Wee Liang Gan
On quantization of Slodowy slices |
| May 17, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Wee Liang Gan
Khovanov homology. |
| May 22, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 |
Apoorva Khare
Categorification of the Khovanov algebra by projective-injective modules in the parabolic category O. Abstract. I will talk about recent work by Stroppel (0608234), which relates two algebras. The first is the endomorphism ring of a "minimal" projective-injective progenerator in the principal block of the parabolic BGG Category O, for gl(2n) and the maximal parabolic subalgebra for the partition (n,n). (This will occupy most of the talk.) The second is the Khovanov algebra obtained by considering the 2d TQFT associated to the Frobenius algebra of dual numbers. Stroppel establishes an isomorphism of both of these, as graded C-algebras. |
| October 2, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Vasiliy Dolgushev
The proof of the multiplicative part of Caldararu's conjecture Abstract. I am going to talk about recent preprint arXiv:0708.2725 "Hochschild cohomology and Atiyah classes" by D. 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. |
| October 9, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Vyjayanthi Chari
Current algebras, highest weight categories and quivers |
| October 11, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Vyjayanthi Chari
Current algebras, highest weight categories and quivers |
| October 16, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Vyjayanthi Chari
Current algebras, highest weight categories and quivers |
| October 18, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Jacob Greenstein
Kirillov-Reshetikhin modules and finite dimensional algebras |
| October 23, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Jacob Greenstein
Kirillov-Reshetikhin modules and finite dimensional algebras (cont.) |
| October 25, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Arkady Berenstein (University of Oregon, Eugene)
Lie algebras and Lie groups over noncommutative rings Abstract. 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. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F, I will define a Lie algebra (g, A)(F) functorially in F and A. In particular, if F is commutative, the Lie algebra (g, A)(F) is simply the loop Lie algebra of g with coefficients in F. In the case when g is semisimple or Kac-Moody and F is noncommutative, I will explicitly compute (g, A)(F) in terms of commutator ideals of F (surprisingly, these ideals have previously emerged as building blocks in M. Kapranov's approach to noncommutative geometry). Furthermore, to each Lie algebra (g, A)(F) one associates a "noncommutative algebraic" group which naturally acts on (g, A)(F) by conjugations. I will conclude my talk with examples of such groups and with the description of "noncommutative root systems" of rank 1. |
| October 30, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Apoorva Khare
The BGG Category O over tensor products and skew group rings Abstract. I study the Category O over the wreath product of sl(2,C) (15 copies of U(sl(2)) times S15). Complete reducibility and block decomposition hold here, because they hold for sl(2). Next, I tensor this algebra 77 times, and ask whether complete reducibility and block decomposition hold in its category O. Finally, I draw a "commuting cube" involving (sets of) simple modules in various categories O, where the "duality functor", "tensor product", and "wreath product induction" form the edges in the X,Y,Z-directions. (The numbers 2,15,77 above can be generalized to any n,m,k > 0 - and more.) |
| November 6, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Vasiliy Dolgushev
An algebraic index theorem for Poisson manifolds Abstract. 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. |
| November 8, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Vasiliy Dolgushev
An algebraic index theorem for Poisson manifolds (cont.) |
| November 15, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Sebastian Zwicknagl
Crystal Commutors and the unitarized R-matrix Abstract. 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. Then, we will lift them to the modules and show how they are related to Drinfeld's unitarized R-matrix, as shown recently by Kamnitzer and Tingley. |
| November 20, 2007 | ||
| 12:40-2:00 p.m. | Surge 284 | Sebastian Zwicknagl
Crystal Commutors and the unitarized R-matrix (cont.) |
| November 29, 2007 | ||
| 1:00-2:00 p.m. | Surge 284 | Farkhod Eshmatov (University of Michigan)
Deformed preprojective algebras and the Calogero-Moser correspondence Abstract. In this talk we discuss the relation between the following objects: rank 1 projective modules (ideals) over the first Weyl algebra A1(C), simple modules over deformed preprojective algebras Πλ(Q), and simple modules over the rational Cherednik algebras H0,c(Sn) associated to symmetric groups. The isomorphism classes of each type of these objects can be geometrically parametrized by the same space, the Calogero-Moser algebraic varieties. 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. Chalykh. |
| January 15, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Adriano de Moura (UNICAMP, Brazil)
Finite-Dimensional Representations of Hyper Loop Algebras over non algebraically closed fields Abstract. 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, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Benjamin Wilson (University of Sydney, Australia/Universidade de São Paulo, Brazil)
Representations of Polynomial Lie Algebras Abstract. Let g denote a Lie algebra over a field of characteristic zero, and let P(g) denote the tensor product of g with a ring of truncated polynomials. The Lie algebra P(g) is called a polynomial Lie algebra, a truncated current Lie algebra, or a generalized Takiff algebra. In this talk, we develop a highest-weight theory for P(g) when the underlying Lie algebra g possesses a triangular decomposition. We describe a reducibility criterion for the Verma modules of P(g) for a wide class of Lie algebras g, including the symmetrizable Kac-Moody Lie algebras, the Heisenberg algebra, and the Virasoro algebra. If time permits, we may discuss applications of this result to the study of "exponential-polynomial modules". |
| January 24, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Anthony Licata (Stanford)
Representations of affine lie algebras in type A and sheaves on CP2 Abstract. |
| January 31, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | David Hernandez (CNRS, Université de Versailles, France)
On the structure of minimal affinizations of representations of quantum groups Abstract. Minimal affinizations of representations of quantum groups introduced by Chari are relevant modules for quantum integrable systems. We present new results on their structure: we prove that all minimal affinizations in types A, B, G are "special" in the sense of monomials (an analog property is also proved for a large class in types C, D, F). As an application, the Frenkel-Mukhin algorithm works for these modules, and then we prove previously predicted explicit (q)-character formulas. |
| February 7, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Benjamin Jones (University of Georgia)
Singular Chern Classes of Schubert Varieties Abstract. Schubert varieties and their singularities are important in the study of representation theory and algebraic groups. 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. |
| February 12, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Rinat Kedem (University of Illinois at Urbana-Champaign)
The combinatorial Kirillov-Reshetikhin conjecture and fusion products Abstract. I will give an overview of the various statements which are called the Kirillov-Reshetikhin conjecture. These describe the structure of special modules of the Yangian of a Lie algebra g or the associated quantum affine algebra. I'll explain how to prove that all these conjectures are equivalent (and hence are now proven), and why it implies the Feigin-Loktev conjecture for the fusion product of the corresponding modules defined by Chari for the algebra of polynomials with coefficients in g, g[t]. |
| February 14, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Vyacheslav Futorny (IME - USP, Brazil)
Gelfand-Tsetlin modules over Yangians Abstract. We will discuss the classification problem of irreducible Gelfand-Tsetlin modules for Yangians and finite W-algebras associated with the Lie algebra gl(n). |
| February 21, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Yiqiang Li (Yale)
Geometric Realization of Irreducible Representations of Quantum Groups and their canonical basis Abstract. Let U be a quantum group. In this talk, I will discuss a geometric realization of certain simple U-modules and their canonical bases, via certain perverse sheaves on open subvarieties of the representation spaces of a quiver. |
| February 26, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Vasiliy Dolgushev
A curious L-infinity morphism for negative cyclic chains Abstract. For an associative algebra A negative cyclic chains CC-(A) form a module over the DG Lie algebra C(A) of Hochschild cochains. In recent preprint arXiv:0802.1706 A. Cattaneo and G. Felder consider this DG Lie algebra module for A being the algebra of functions on a smooth real manifold equipped with a volume form. Using an interesting modification of the Poisson sigma model the authors construct a curious L-infinity morphism (not a quasi-isomorphism!) from the DG Lie algebra module CC-(A) to a DG Lie algebra module modeled on polyvector fields using the volume form. 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. Cattaneo and G. Felder to the formality quasi-isomorphism is a mystery. |
| February 28, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Travis Schedler (University of Chicago)
Calabi-Yau Frobenius Algebras Abstract. n this talk, we will explore a generalization of symmetric Frobenius algebras (i.e., where the inner product is symmetric) to the case where the pairing is symmetric after some homological shift. 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". It turns out that the Hochschild (co)homology of such algebras has a very nice structure, and is best described by a Z-graded version of Hochschild (co)homology, which is a Hochschild analogue of Tate cohomology. 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, 2008 | ||
| 1:00-2:00 p.m. | Surge 284 | Reimundo Heluani (UC Berkeley)
Supersymmetry of the Chiral de Rham complex Abstract. The "chiral de Rham complex" of Malikov-Shechtman-Vaintrob is a sheaf of vertex superalgebras associated to any manifold X. We will show how, in the smooth context, extra geometric data on X (e.g. having special holonomy) translates into extra symmetries of the corresponding vertex superalgebras of global sections. |
