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# Timetable (ALTW02)

## Algebraic Lie Structures with Origins in Physics

Monday 23rd March 2009 to Friday 27th March 2009

 08:30 to 10:00 Registration 10:00 to 11:00 Crystal structure for representations of Cherednik algebras I will report on recent work of Peng Shan (Paris VII). Using the parabolic induction and restriction functors of Bezrukavnikov and Etingof she showed the existence of a crystal structure on category O of the rational Cherednik algebras of type G(m,1,n) as n varies. Independently, Maurizio Martino (Bonn) and I proved the same result, with a slightly different construction which I shall outline. INI 1 11:00 to 11:30 Coffee and Posters 11:30 to 12:30 Yangians and Dorey's rule We discuss finite--dimensional representations of Yangians and quantum affine algebras. Dorey's rule can be viewed as an analog of the famous PRV conjecture proved by Kumar and Mathieu and predicts the multiplicity of the trivial representation in a tensor product of three fundamental representations of the Yangian. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 WL Gan ([UC at Riverside])Dunkl operators for symplectic reflection algebras of wreath-product types I will speak on an analogue of Dunkl operators attached to wreath-product of a symmetric group and a finite subgroup of SL(2,C). These operators are a main ingredient in the realization of the spherical subalgebra of symplectic reflection algebras of wreath-product types by quantum hamiltonian reduction. This is a joint work with P. Etingof, V. Ginzburg and A. Oblomkov which appeared in Publ. Math IHES no. 105. INI 1 15:00 to 15:30 Tea and Posters 15:30 to 16:30 Affine W-algebras and Zhu's Poisson varieties associated with Kac-Moody vertex algebras The C2 cofiniteness condition is an important finiteness condition on a vertex algebra which guarantees the finite-dimensionality of the corresponding conformal blocks. In this talk I will talk about the relationship between the C2 cofiniteness condition of affine W- algebras and the certain invariants of Kac-Moody vertex algebras. INI 1 16:30 to 17:30 E Ragoucy ([LAPTH (CNRS)])Nested Bethe ansatz for spin chains We present in a unified way the nested Bethe ansatz for spin chains based on gl(n), gl(m|n) and their deformations. We perform the ansatz for closed and open spin chains. In the case of open spin chains, we use diagonal boundary conditions. In all cases, we deduce a recursion formula, and a trace formula for Bethe vectors. INI 1 17:30 to 18:30 Wine Reception and Poster Session 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Yangians, finite W-algebras and Hecke algebras I'll talk about representations of finite W-algebras with special reference to the type A case, where there are interesting connections to Yangians and to cyclotomic Hecke algebras. INI 1 10:00 to 11:00 D Rumynin ([Warwick])Modular representations of Lie algebras We will discuss geometric issues that appear in modular representation theory of simple Lie algebras. INI 1 11:00 to 11:30 Coffee and Posters 11:30 to 12:30 Equidimensionality of characteristic varieties over Cherednik algebras This talk will report on joint work with Victor Ginzburg and Iain Gordon. Type A Cheredink algebras H_c, which are particular deformations of the twisted group ring of the n-th Weyl algebra by the symmetric group S_n, form an intriguing class of algebras with many interactions with other areas of mathematics. In earlier work with Iain Gordon we used ideas from noncommutative geometry to prove a sort of Beilinson-Bernstein equivalence of categories, thereby showing that H_c (or more formally its spherical subalgebra U_c) is a noncommutative deformation of the Hilbert scheme Hilb(n) of n points in the plane. There is however a second way of relating U_c to Hilbert schemes, which uses the quantum Hamiltonian reduction of Gan and Ginzburg. In the first part of the talk we will show that these two methods are actually equivalent. In the second part of the talk we will use this to prove that the characteristic varieties of irreducible U_c-modules are equidimensional subshemes of Hilb(n), thereby answering a question from the original work with Gordon. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Algebraic group analogues of the Slodowy slices and deformed Poisson W-algebras We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group with Lie algebra g. These slices are associated to the pairs (p,s), where p is a parabolic subalgebra in g and s is an element of the Weyl group W of g. In the algebraic group framework simple Kleinian singularities are realized as the singularities of the fibers of the restriction of the conjugation quotient map to the slices associated to pairs (b,s), where b is a Borel subalgebra in g and s is an element of W whose representative in G is subregular. We also define some Poisson structures on the slices associated to the pairs (p,s). These structures are analogous to the Poisson structures introduced by DeBoer, Tjin and Premet on the Slodowy slices in complex simple Lie algebras. The quantum deformations of these Poisson structures are known as W-algebras of finite type. One of applications of our construction gives rise to new Poisson structures on the coordinate rings of simple Kleinian singularities. INI 1 15:00 to 15:30 Tea and Posters 15:30 to 16:30 T Suzuki ([Okayama])Conformal field theory and Cherednik algebras I would like to discuss about a physical background of the connection between representaions of Lie, Hecke and Cherednik algebras. For example, I will explain how conformal field theory gives a natural construction of the functors between the category of highest weight representations of the affine Lie algebras and the category of highest weight representations of the rational/trigonometric Cherednik algebras of type A. INI 1 16:30 to 17:30 Langlands duality for representations of quantum groups This is a joint work with E. Frenkel. We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an "interpolating quantum group" depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Irreducible finite dimensional representations of W-algebras W-algebras (of finite type) are certain associative algebras constructed from nilpotent orbits in semisimple Lie algebras. In this talk I am going to describe a relation between irreducible finite dimensional representations of a W-algebra and primitive ideals of the universal enveloping algebra whose associated variety coincides with the closure of the given orbit. INI 1 10:00 to 11:00 Mickelsson algebras and irreducible representations of Yangians Yangians and their twisted analogues first appeared as basic examples of deformations of affine Lie algebras, and found numerous applications in Quantum Integrable Systems. Classification of the irreducible finite-dimensional representations of Yangians has been known for a long time, but explicit realizations of irreducibles were not known except in special cases. In this talk, we present such a realization. It has basic features of representation theory of reductive groups and uses the theory of Mickelsson algebras as an important technical tool. This a joint work with M.Nazarov. INI 1 11:00 to 11:30 Coffee and Posters 11:30 to 12:30 Cherednik algebra, Calogero-Moser space and Bethe ansatz We show that the center of the rational Cherednik algebra is naturally identified with a Bethe algebra of the Gaudin model. At the same time, using the algebraic Bethe ansatz we prove that the Bethe algebra is isomorphic to the space of regular functions on the Calogero-Moser space. We discuss the implications of these constructions. INI 1 12:30 to 13:30 Lunch at Wolfson Court 19:30 to 23:00 Conference Dinner at St John's College
 09:00 to 10:00 V Futorny ([Sao Paulo])Parabolic induction for Affine Kac-Moody algebras We will discuss the category of weight modules with nonzero central charge over affine Lie algebras. Though the classification of irreducible modules with finite-dimensional weight spaces in this category is known, it remains open in general. The talk will focus on recent progress in the study of parabolic induction for affine Kac-Moody algebras which provides a recipe to construct new irreducible modules is the above category. The talk is based on joint results with I.Kashuba. INI 1 10:00 to 11:00 M Finkelberg ([State, Moscow])Affine Gelfand-Tsetlin bases and affine Laumon spaces Affine Laumon space P is the moduli space of parabolic sheaves of rank n on the product of 2 projective lines. The natural correspondences give rise to the action of affine Yangian of sl(n) on the equivariant cohomology of P. The resulting module M is isomorphic to the universal Verma module over the affine gl(n). The classes of torus fixed points form a basis of M which is an affine analogue of the classical Gelfand-Tsetlin basis. The Chern classes of tautological vector bundles on P can be computed in terms of the affine Yangian action on M. This is a joint work with B.Feigin, A.Negut, and L.Rybnikov. INI 1 11:00 to 11:30 Coffee and Posters 11:30 to 12:30 The impact of Goldie’s theorem on primitive ideal theory In 1958 Alfred Goldie published a seemingly rather abstract theorem stating that many important rings admit a calculation of fractions. This was soon realized to be a deep and fundamental result, particularly leading to a numerical invariant known as Goldie rank. Through Duflo’s theorem one may parameterize the primitive spectrum of an enveloping algebra in the semisimple case by the dual of the Cartan. Then astonishingly, Goldie rank is given through a family of polynomials. Moreover these polynomials have some remarkable properties. For example they form a basis of a multiplicity free representation of the Weyl group. One thereby obtains a quite unattended connection with the Springer theory relating Weyl group representations to the geometry of nilpotent orbits. The polynomials that define Goldie rank are determined up to a scale factor by an explicit formula involving the Kazhdan-Lusztig polynomials and were even a motivation for the precise definition of the latter. These scale factors can be largely determined by finding the locus of Goldie rank one, a problem which has remained open for some thirty years. Another related question is to describe the Goldie rank one sheets (of which there are just finitely many, by virtue of a positivity property of Goldie rank polynomials coming from geometry) and in particular to determine their topology. Combined with the Gelfand-Kirillov conjecture, primitive quotients of enveloping algebras are described (in principle) as matrix rings over differential operators linked to symplectic structure, exactly like Dirac’s relativistic quantum mechanical equation. Thus primitive ideal theory is intimately related to Quantization. In this lecture we review the main results and open problems of the theory. INI 1 12:30 to 13:30 Lunch at Wolfson Court 15:00 to 15:30 Tea and Posters 15:30 to 16:30 V Kac ([MIT])An explicit construction of the complex of variational calculus and Lie conformal algebra cohomology Lie conformal algebras encode the singular part of the operator product expansion of chiral fields in conformal field theory, and, at the same time, the local Poisson brackets in the theory of soliton equations. That is why they form an essential part of the vertex algebra and Poisson vertex algebra theories. The structure and cohomology theory of Lie conformal algebras was developed about 10 years ago. In a recent joint work with Alberto De Sole we show that the Lie conformal algebra cohomology can be used to explicitly construct the complex of calculus of variations, which is the resolution of the variational derivative map of Euler and Lagrange. INI 1 16:30 to 17:30 Poisson vertex algebras in the theory of Hamiltonian equations We describe the interplay between the theory of Poisson vertex algebras (and, more in general, of conformal algebras) and the theory of Hamiltonian equations and their integrability. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 A Molev ([Sydney])Higher Sugawara operators and the classical W-algebra for gl n We give an explicit description of the center of the affine vertex algebra for gl n at the critical level. The results are applied to construct generators of the classical W-algebra for gl n and calculate the eigenvalues of the central elements in the Wakimoto modules at the critical level. INI 1 10:00 to 11:00 Non-reductive lie algebras and their representations with good invariant-theoretic properties One of the main topics of Invariant Theory is to describe finite-dimensional representations of complex reductive groups with good properties of algebras of invariants. For instance, it is known that, for simple algebraic groups, most of irreducible representations with polynomial algebras of invariants occur in connection with periodic automorphisms of semisimple Lie algebras. (These are the so-called "$\Theta$-groups" of Vinberg.) In my talk, I will discuss several constructions of non-reductive algebras having polynomial algebras of invariants for the adjoint or coadjoint representations (e.g. contractions of semisimple algebras and iterated semi-direct products). Some of these coadjoint representations can also be understood as $\Theta$-group associated with periodic automorphisms of non-reductive Lie algebras. There are also possibilities for constructing more general representations with polynomial algebras of invariants using bi-periodic gradings and contractions. INI 1 11:00 to 11:30 Coffee and Posters 12:30 to 13:30 Lunch at Wolfson Court 15:00 to 15:30 Tea and Posters 15:30 to 16:30 V Serganova ([UC at Berkeley])Geometric induction for algebraic supergroups Let G be a classical algebraic supergroup, and H be its subgroup. The geometric induction functor is the derived functor from the category of H-modules to the category of G-modules. It is defined as the cohomology of vector bundles on G/H. We study this functor in detail in case when H is a parabolic subgroup and G=SL(m,n) or OSP(m,2n) and use this result to find the characters of all irreducible representations of G. INI 1 16:30 to 17:30 P Etingof ([MIT])Representation theory in complex rank The subject of representation theory in complex rank was started by P.Deligne, who defined complex rank analogs for the classical complex groups GL(N),O(N),and Sp(N), and (later) for the symnmetric group S_N. These are certain symmetric tensor categories of superexponential growth (i.e. non-Tannakian), in which the dimension N of the generating object is a generic complex number. Later F.Knop generalized the latter construction to a large class of finite groups (such as general linear groups over a finite field). For generic N, these categories are semisimple, so one may think of these results as "compact" representation theory in complex rank. I will speak about "noncompact" representation theory in complex rank. The discussion will include classical real groups (i.e., classical symmetric pairs), degenerate affine Hecke algebras, rational Cherednik algebras, affine Lie algebras, Yangians, and so on. By definition, a representation of a "noncompact algebra" of complex rank is a representation of its "maximal compact subalgebra" (i.e. an (ind)-object of the corresponding tensor category) together with some additional structure (morphisms satisfying some relations). I will discuss the explicit form of such morphisms and relations in several special cases (such as rational Cherednik algebras). This approach leads to a multitude of new interesting representation categories, which, in a sense, capture the phenomenon of "stabilization with respect to rank" in representation theory of classical groups and algebras. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)