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SUMMARY:Okounkov's conjecture via BPS Lie algebras - Ben Davison (Universi
 ty of Edinburgh)
DTSTART:20240603T130000Z
DTEND:20240603T150000Z
UID:TALK217543@talks.cam.ac.uk
DESCRIPTION:Given a finite quiver Q\, using their theory of stable envelop
 es and resulting R matrices\, Maulik and Okounkov defined a new type of Ya
 ngian algebra.&nbsp\; This algebra is defined as a subalgebra of the endom
 orphism algebra of the (equivariant) cohomology of all the Nakajima quiver
  varieties associated to Q.\nIf Q is an orientation of a type ADE Dynkin d
 iagram\, the Maulik-Okounkov algebra recovers the usual Yangian algebra\, 
 a deformation of the universal enveloping algebra of the current algebra o
 f the associated ADE type Lie algebra.&nbsp\; If Q is the one-loop quiver\
 , their theory also recovers a well-known Yangian algebra\, and the Grojno
 wski-Nakajima action of an infinite-dimensional Heisenberg algebra on the 
 cohomology of Hilbert schemes of C^2 is recovered as a part of the theory.
 \nFor general quivers\, the picture is less clear.&nbsp\; Although\, as in
  the cases above\, there is a Lie algebra g_{MO} which generates the whole
  of the Maulik-Okounkov algebra\, even determining the dimensions of the g
 raded pieces remained an open question until quite recently.&nbsp\;\nOkoun
 kov's conjecture states that these dimensions are given by coefficients of
  Kac's polynomials\, which count isomorphism classes of Q-representations 
 over finite fields.&nbsp\; I'll present a proof of this conjecture\, which
  is joint work with Tomasso Botta.&nbsp\; The proof proceeds by identifyin
 g the Maulik-Okounkov Lie algebra g_Q with certain BPS Lie algebras (which
  I'll define during the talk).
LOCATION:Seminar Room 2\, Newton Institute
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