Yangians and quantizations of slices in the affine Grassmannian

Kamnitzer, Joel; Webster, Ben; Weekes, Alex; Yacobi, Oded

doi:10.2140/ant.2014.8.857

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Yangians and quantizations of slices in the affine Grassmannian

Joel Kamnitzer, Ben Webster, Alex Weekes and Oded Yacobi

Vol. 8 (2014), No. 4, 857–893

DOI: 10.2140/ant.2014.8.857

Abstract

We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians — these are subalgebras of the Yangian we introduce which generalize the Brundan–Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov, Kharchev, Lebedev and Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for PGL $_{n}$ of Finkelberg and Rybnikov.

Keywords

quantization, affine Grassmannian, quantum groups, Yangian

Mathematical Subject Classification 2010

Primary: 20G42

Secondary: 53D55, 14M15, 14D24

Milestones

Received: 10 January 2013

Revised: 2 August 2013

Accepted: 31 August 2013

Published: 10 August 2014

Authors

Joel Kamnitzer
Department of Mathematics University of Toronto 40 St. George Street Toronto, ON M5S 2E4 Canada

Ben Webster
Department of Mathematics University of Virginia 141 Cabell Drive Charlottesville, VA 22904 United States

Alex Weekes
Department of Mathematics University of Toronto 40 St. George Street Toronto, ON M5S 2E4 Canada

Oded Yacobi
School of Mathematics and Statistics University of Sydney NSW 2006 Australia

Vol. 8, No. 4, 2014