Department of Mathematical Sciences Articles

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Browsing Department of Mathematical Sciences Articles by Subject "17B37"

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    Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions
    (Springer, 2014-08) Rimányi, R.; Tarasov, Vitaly; Varchenko, A.; Mathematical Sciences, School of Science
    We consider the conormal bundle of a Schubert variety SI in the cotangent bundle T∗Gr of the Grassmannian Gr of k-planes in Cn. This conormal bundle has a fundamental class κI in the equivariant cohomology H∗T(T∗Gr). Here T=(C∗)n×C∗. The torus (C∗)n acts on T∗Gr in the standard way and the last factor C∗ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum YI of the Yangian Y(gl2) weight functions (WJ)J. We describe a relation of YI with the double Schur polynomial [SI]. A modified version of the κI classes, named κ′I, satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold T∗Gr. This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on Gr. The classes (κ′I)I form a basis in the suitably localized equivariant cohomology H∗T(T∗Gr). This basis depends on the choice of the coordinate flag in Cn. We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.
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    Deformations of W algebras via quantum toroidal algebras
    (Springer, 2021-06) Feigin, B.; Jimbo, M.; Mukhin, E.; Vilkovisky, I.; Mathematical Sciences, School of Science
    We study the uniform description of deformed W algebras of type A including the supersymmetric case in terms of the quantum toroidal gl1 algebra E. In particular, we recover the deformed affine Cartan matrices and the deformed integrals of motion. We introduce a comodule algebra K over E which gives a uniform construction of basic deformed W currents and screening operators in types B,C,D including twisted and supersymmetric cases. We show that a completion of algebra K contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except D(2)ℓ+1. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.