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
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 $[u,v]=uv' - u'v$. Let $a$ and $b$ 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 $u$, $v\in \text{\rm diff}\,(S^1)$, we put $(u\vert v)=\omega_{a,b}(u,Jv)$, where $J$ be the Hilbert transform on $\text{\rm diff}\,(S^1)$. Let $V$ 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 $V$. 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 $V$.

Received: October 8, 2007

AMS Subject Classification: 17B66, 30B50

Key Words and Phrases: Ricci curvature, infinite dimensional groups

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 43
Issue: 3