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Hidden symmetries in equivariant cohomology related to quantum groups

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Vassily Gorbounov
Die, 2016-03-22 15:00 - 16:00
MPIM Lecture Hall

The standard torus action on a vector spaces $C^n$ induces the action on the partial flag
varieties in $C^n$. The scalar matrices act trivially on the partial flags, hence in fact we
have an action of the torus of one dimension less. We will show how this "missed in
action" one dimensional torus shows up in the "hidden" action of the Yang Baxter algebras
of a certain quantum Integrable systems from statistical physics on the equivariant
cohomology of partial flag varieties and related varieties. This missed one dimensional
torus is responsible for appearance of the spectral parameter in the language of quantum
groups. The first example of this phenomenon as far as we know was discovered in the
work of Maulik and Okounkov. In their case, in particular, the Yangian acts on the
equivariant cohomology of cotangent spaces to Grassmanians. We also present other
examples of these type hidden symmetries from the recent work of C. Korff, C. Stroppel,
P. Zinn-Justen.

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