# University of Toronto's Symplectic Geometry Seminar

Sept. 22 2003, 2:10 - 3

SS5017A

## Megumi Harada

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University of Toronto

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"The Gel'fand-Cetlin-Molev integrable system"

** Abstract: ** Gel'fand and Cetlin constructed in the 1950s a
canonical basis for a finite-dimensional representation $V(\lambda)$
of $U(n,\C)$ by successive decompositions of the representation by a
chain of subgroups. Guillemin and Sternberg constructed in the 1980s
the Gel'fand-Cetlin integrable system on the coadjoint orbits of
$U(n,\C)$, which is the symplectic geometric version, via geometric
quantization, of the Gel'fand-Cetlin basis. (Much the same
construction works for representations of $O(n) = U(n,\R)$.) A. Molev
recently found a Gel'fand-Cetlin type basis for representations of the
symplectic group, using essentially new ideas. An important new role
is played by the Yangian $Y(2)$, an infinite-dimensional Hopf algebra,
and a subalgebra of $Y(2)$ called the twisted Yangian $Y^{-}(2)$. In
this talk we show how deformation theory give rise to the analogous
symplectic-geometric results for the case of $U(n,\H)$, i.e. we
construct a completely integrable system on the coadjoint orbits of
$U(n,\H)$. We call this the Gel'fand-Cetlin-Molev integrable system.