Volume 8 (2008), Number 1. Abstracts
A. Beilinson. Remarks on Topological Algebras [PDF]
The article discusses some subjects of topological polylinear algebra closely related to the theory of chiral/vertex algebras.
Keywords. Topological chiral algebras, topological chiral differential operators.
2000 Mathematics Subject Classification. 14A22.
Yu. Berest, O. Chalykh, and F. Eshmatov. Recollement of Deformed Preprojective Algebras and the Calogero–Moser Correspondence [PDF]
The aim of this paper is to clarify the relation between the following objects: (a) rank 1 projective modules (ideals) over the first Weyl algebra A1(C); (b) simple modules over deformed preprojective algebras Πλ(Q) introduced by Crawley-Boevey and Holland; and (c) 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 parametrized naturally by the same space (namely, the Calogero–Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on A∞-modules over A1 to a more familiar setting of representation theory.
Keywords. Weyl algebra, Calogero–Moser space, preprojective algebra, recollement, Cherednik algebra, Kleinian singularity.
2000 Mathematics Subject Classification. Primary 16S32, 16S38; Secondary 16G20, 17B10.
R. Bezrukavnikov and M. Finkelberg. Equivariant Satake Category and Kostant–Whittaker Reduction [PDF]
We explain (following V. Drinfeld) how the G(C[[t]]) equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some old results of V. Ginzburg. The global cohomology functor corresponds under this identification to restriction to the Kostant slice. We extend this description to loop rotation equivariant derived category, linking it to Harish-Chandra bimodules for the Langlands dual Lie algebra, so that the global cohomology functor corresponds to the quantum Kostant–Whittaker reduction of a Harish-Chandra bimodule. We derive a conjecture by the authors and I. MirkoviŠ, which identifies the loop-rotation equivariant homology of the affine Grassmannian with quantized Toda lattice.
Keywords. Affine Grassmannian, Langlands dual group, Toda lattice.
2000 Mathematics Subject Classification. Primary 19E08; Secondary 22E65, 37K10.
M. Garay. Vanishing Cycles in Complex Symplectic Geometry [PDF]
We study the vanishing cycles on the Milnor fibre for some non-isolated singularities which appear naturally in symplectic geometry. Under assumptions given in the text, we show that the vanishing cycles associated to a distinguished basis freely generate the corresponding homology groups of the Milnor fibre. We derive some consequences of this fact, in particular for the study of integrable systems and of adjoint orbits in Lie algebras.
Keywords. Monodromy, vanishing cycles, integrable systems, symplectic geometry, lagrangian varieties, involutive varieties, simple Lie algebras.
2000 Mathematics Subject Classification. 32S50.
M. Gorelik and V. Serganova. On Representations of the Affine Superalgebra q(n)(2) [PDF]
In this paper we study highest weight representations of the affine Lie superalgebra q(n)(2). We prove that any Verma module over this algebra is reducible and calculate the character of an irreducible q(n)(2)-module with a generic highest weight. This formula is analogous to the Kac–Kazhdan formula for generic irreducible modules over affine Lie algebras at the critical level.
Keywords. Affine Lie superalgebra, highest weight representation, Shapovalov form.
2000 Mathematics Subject Classification. 32S50.
V. Ostrik. Pre-Modular Categories of Rank 3 [PDF]
We classify ribbon semisimple monoidal categories with three isomorphism classes of simple objects over the field of complex numbers.
Keywords. Braided tensor categories.
2000 Mathematics Subject Classification. 18D10.
R. Rouquier. q-Schur Algebras and Complex Reflection Groups [PDF]
We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a q-Schur algebra (parameter ∉½ + Z), providing thus character formulas for simple modules. We give some generalization to Bn(d). We prove an “abstract” translation principle. These results follow from the unicity of certain highest weight categories covering Hecke algebras. We also provide a semi-simplicity criterion for Hecke algebras of complex reflection groups and show the isomorphism type of Hecke algebras is invariant under field automorphisms acting on parameters.
Keywords. Hecke algebra, reflection group, Schur algebra, Cherednik algebra, highest weight category.
2000 Mathematics Subject Classification. 20C08, 20C30, 20F55.
A. Rybko and S. Shlosman. Phase Transitions in the Queuing Networks and the Violation of the Poisson Hypothesis [PDF]
We present examples of queuing networks that never come to equilibrium. That is achieved by constructing certain non-linear Markov evolutions, which are non-ergodic and possess eternal transience property.
Keywords. Self-averaging, mean field, eternal transience.
2000 Mathematics Subject Classification. Primary 82C20; Secondary 60J25.
Moscow Mathematical Journal