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CATEGORIES:Geometry and Mathematical Physics seminar
SUMMARY:Character manifolds and quantum cluster algebras -
Leonid Chekhov (Michigan State)
DTSTART:20190605T123000Z
DTEND:20190605T133000Z
UID:TALK3771AT
URL:/talk/index/3771
DESCRIPTION:We describe and quantise SL_N character manifold o
n a surface $\\Sigma_{g\,s\,n}$ of arbitrary genus
g\, $s>0$ holes and $n>0$ decorated boundary cusp
s (marked points on hole boundaries). All such man
ifolds can be constructed by amalgamation procedur
e from elementary blocks which are ideal triangles
$\\Sigma_{0\,1\,3}$ endowed with the Fock-Gonchar
ov cluster algebra structure. Elements of monodrom
y matrices correspond to sums over weighted paths\
, and we show that for any planar directed (acycli
c) network\, elements of this matrices satisfy qua
ntum R-matrix relations. From these elementary rel
ations\, under the satisfaction of the groupoid pr
operty\, we construct general quantum monodromy ma
trices\, which satisfy the Goldman bracket in the
semi-classical limit. Moreover\, a fresh view on t
he monodromy algebra in the above triangle allowed
us to solve an old problem of finding classical a
nd quantum Darboux coordinates for the groupoid of
upper-triangular matrices and presenting the brai
d-group action in this groupoid via mutations of q
uantum cluster variables in a special quiver obtai
ned from the triangle $\\Sigma_{0\,1\,3}$. (Forthc
oming joint paper with M.Shapiro).
LOCATION:Watson Building (Mathematics\, R15 on map) Lecture
Room C
CONTACT:Andrea Brini
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