W-algebras and algebraic group actions

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project proposes research on two subjects: 1) Representation theory of W-algebras, 2) uniqueness properties for algebraic group actions. W-algebras (of finite type) are certain finitely generated associative algebras associated with nilpotent elements in semisimple Lie algebras. They originate from the work of B. Kostant of late 70's. In 90's they were studied by physicists. Starting from 2000 they attracted lot of attention of specialists in Representation Theory: Brundan, Ginzburg, Kleshchev, Premet, and others. In two recent years the investigator discovered a completely new approach to W-algebras based on Deformation quantization. This new approach allowed to him to prove many conjectures on W-algebras (mostly due to Premet) and, in particular, obtain the classification of their irreducible finite dimensional modules. The investigator plans to continue the study of representations of W-algebras and their q-deformations. In particular, he plans to prove a conjecture of Brundan-Goodwin- Kleshchev on the structure of the category O of W-algebras. Algebraic transformation group theory is a classical topic of algebraic geometry and group theory. One of major developments in algebraic transformation groups in recent 25 years is the theory of spherical varieties developed by Brion, Knop, Luna, Panyushev, Vinberg, Vust, and others. Spherical varieties are a particularly nice class of varieties equipped with a reductive group action. When the group is a torus, spherical is the same as toric. One of the nice features of spherical varieties is that their classification may be obtained in entirely combinatorial terms. In the recent few years the investigator obtained certain uniqueness properties of spherical varieties in terms of their combinatorial invariants proving conjectures due to Brion, Knop and Luna. The investigator plans to generalize these results to arbitrary varieties equipped with an action of a reductive group. In particular, he plans to prove that a smooth affine G-variety is uniquely determined by its algebra of U-invariants.This research projects deals with different kinds of symmetries arising both in pure mathematics and in physics. For instance, W-algebras are certain algebraic structures appeared in pure algebraic studies of Kostant in late 70's. Since then they found a number of applications in representation theory. On the other hand they are a manifestation of the notion of W-symmetry from Conformal field theory extensively studied by physicists. So the investigator's research project will contribute to pure mathematics and may have some applications to physics. The second part of this research project deals with a more classical notion of symmetries coming from geometry.