[Paper Review] Traces on finite W-algebras

Pavel Etingof, Travis Schedler|arXiv (Cornell University)|Apr 26, 2010

Algebraic structures and combinatorial models12 references3 citations

TL;DR

This paper computes the zeroth Poisson homology of classical finite W-algebras and the zeroth Hochschild homology of their quantum counterparts, showing both are isomorphic to the top cohomology of the corresponding Springer fiber modulo any central character. This yields an upper bound on the number of irreducible finite-dimensional representations of quantum W-algebras, consistent with earlier results by Dodd.

ABSTRACT

We compute the space of Poisson traces on a classical W-algebra modulo an arbitrary central character, i.e., linear functionals on such an algebra invariant under Hamiltonian derivations. This space identifies with the top cohomology of the corresponding Springer fiber. As a consequence, we deduce that the zeroth Hochschild homology of the corresponding quantum W-algebra modulo a central character identifies with the top cohomology of the corresponding Springer fiber. This implies that the number of irreducible finite-dimensional representations of this algebra is bounded by the dimension of this top cohomology, which was established earlier by C. Dodd using reduction to positive characteristic. Finally, we prove that the entire cohomology of the Springer fiber identifies with the so-called Poisson-de Rham homology (defined previously by the authors) of the classical W-algebra modulo a central character.

Motivation & Objective

• To compute the zeroth Poisson homology of classical finite W-algebras and the zeroth Hochschild homology of their quantum deformations.
• To establish that both homology spaces are isomorphic to the top cohomology of the Springer fiber when reduced modulo any central character.
• To provide a geometric, characteristic-zero proof of an upper bound on the number of irreducible finite-dimensional representations of quantum W-algebras, previously shown via positive characteristic methods.
• To show that the Poisson-de Rham homology of the centrally reduced classical W-algebra is isomorphic to the cohomology of the Springer fiber in complementary degree.
• To unify representation-theoretic and geometric invariants via D-module theory and Hamiltonian reduction.

Proposed method

• Utilizes the Hotta-Kashiwara presentation of the Springer D-module on the nilpotent cone via generators and relations.
• Applies earlier results of the authors on characterizing zeroth Poisson homology via D-modules and Hamiltonian derivations.
• Employs the Grothendieck-Springer resolution and the Springer correspondence to relate cohomology of Springer fibers to W-algebra invariants.
• Uses deformation quantization and the Rees algebra construction to relate quantum W-algebras to their classical limits.
• Applies results from Nest-Tsygan and Brylinski on Hochschild homology of quantized algebras and their relation to Poisson homology.
• Leverages topological triviality of the Springer resolution family to equate cohomology dimensions across generic and special fibers.

Experimental results

Research questions

• RQ1 What is the structure of the zeroth Poisson homology of a classical finite W-algebra?
• RQ2 How does the zeroth Hochschild homology of a quantum W-algebra relate to the geometry of the Springer fiber?
• RQ3 Can the number of irreducible finite-dimensional representations of a quantum W-algebra be bounded using geometric invariants?
• RQ4 What is the relationship between the Poisson-de Rham homology of a centrally reduced classical W-algebra and the cohomology of the Springer fiber?
• RQ5 How do the homological invariants of W-algebras behave under central reduction and deformation quantization?

Key findings

• The zeroth Poisson homology of a classical finite W-algebra modulo any central character is isomorphic to the top cohomology group of the corresponding Springer fiber.
• The zeroth Hochschild homology of a quantum W-algebra modulo a central character is isomorphic to the same top cohomology group of the Springer fiber.
• The number of irreducible finite-dimensional representations of a quantum W-algebra with a fixed central character is bounded above by the dimension of the top cohomology of the Springer fiber.
• The Poisson-de Rham homology of the centrally reduced classical W-algebra is isomorphic to the cohomology of the Springer fiber in complementary degree.
• For generic central characters, the Hochschild homology of the quantum W-algebra is isomorphic to the Poisson homology of the classical W-algebra tensored with the power series ring in the deformation parameter.
• The cohomology of the Springer fiber has constant dimension across the family of central reductions, implying invariance of homological invariants under generic deformation.