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\title{A Lie--algebraic reduction scheme for the BKP, CKP, DKP
hierarchies}
\author{Paolo Casati
\\ 
I University of Milan Bicocca, Dept of Mathematics, \\ 
via Bicocca degli Arcimboldi 8, I-20126 Milano, Italy \\
E-mail:  casati@matapp.unimib.it  
} \date{} \maketitle


\bigskip \noindent
The hierarchies BKP, CKP, DKP are complete integrable hierarchies 
corresponding to the infinite dimensional Lie algebras $B_\infty$ 
$C_\infty$ $D_\infty$ respectively. In this talk it will be shown how they 
can be recovered by performing a reduction process from the same 
dynamical system in an  infinite number of evolution equations, which 
lives on the infinite dimensional Grassmanian of Sato.\par\noindent
Moreover it will shown that the    integrable systems corresponding to 
the simple Lie algebras $B_n$, $C_n$, $D_n$ discovered 
by Drinfeld  and Sokolov can be obtained by constraining the hierarchies
BKP, CKP, DKP respectively, in a different way to that usually used in the 
 approach with the Lax matrix. This constrains have a direct Lie--algebraic 
interpretation, which will be explained in the talk. 
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