Localization algebras and deformations of Koszul algebras |
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Authors: | Tom Braden Anthony Licata Christopher Phan Nicholas Proudfoot Ben Webster |
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Institution: | (1) Nagoya University, Nagoya, Japan |
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Abstract: | We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular,
the center of A acts by characters on the deformed standard modules, providing a “localization map”. We construct a universal graded deformation
of A and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the
arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings
of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal
deformation for the ring governing a block of parabolic category O{\mathcal{O}} for
\mathfrakgln{\mathfrak{gl}_n} is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version
of the “category O{\mathcal{O}}” of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric
variety. |
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