# University of Toronto's Symplectic Geometry Seminar

Sept. 22 2003, 2:10 - 3
SS5017A

## "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.