IJPAM: Volume 43, No. 3 (2008)

RICCI OPERATOR ON $\text{\bf Diff}\,(S^1)/ S^1$

Helene Airault
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
LAMFA UMR 6140 CNRS
Université de Picardie Jules Verne
33, rue Saint Leu, Amiens, FRANCE
e-mail: hairault

insset.u-picardie.fr

Abstract.The bracket on the Lie algebra $\text{\rm diff}\,(S^1)$ is given by . Let and be two real numbers such that $\alpha(k)=bk^3\, -\, ak$ is strictly positive for $k\geq 1$ . The symplectic form is $\omega_{a,b}(u,v)=(1/2 \pi)\int_0^{\pi}(au'+bu''')v\, d\theta$ . For , $v\in \text{\rm diff}\,(S^1)$ , we put $(u\vert v)=\omega_{a,b}(u,Jv)$ , where be the Hilbert transform on $\text{\rm diff}\,(S^1)$ . Let be the subspace of $\text{\rm diff}\,(S^1)$ generated by $\{\cos(k\theta), \sin(k\theta)\}$ with $k\geq 1$ . For $v\in \text{\rm diff}\,(S^1)$ , the operator $\Gamma(v):V\to V$ is defined by $2 (\Gamma(v) u \vert w)=([w,v] \vert u) +( [w,u] \vert v)$ and $B(v):V\to V$ is defined by $\, 2\, B(v)u=\pi[v,u]$ , where $\pi$ is the projection on . We study $\Gamma_l(v)\, =\, \Gamma(v)+B(v)$ and $\, \, \Gamma_b(v)\, =\, \Gamma(v)\, -\, B(v)$ . Then $\sum_{p\geq 1} \alpha(p)^{-1} [ \Gamma_b(\cos p\theta)\Gamma_l(\cos p\theta) + \Gamma_b(\sin p\theta)\Gamma_l(\sin p\theta)]$ is a bounded operator on .