Operator Lie Algebras of Rotations and Transformations in White Noise

The infinitesimal generator of a one-parameter subgroup of the infinite dimensional rotation group associated with the complex Gelfand triple $ (E) \subset L^2(E^, \mu) \subset (E)^ $isoftheform$$ R_\kappa = \int_{T\times T} \kappa(s,t) (a_s^* a_t - a_t^* a_s) ds dt $$where$\kappa \in E \otimes E^*$isaskew-symmetricdistribution\mathrm{.}Hence$R_\kappa$ is twice the conservation operator associated with a skew-symmetric operator $S$. The Lie algebra containing $R_{\kappa }$, identity operator, annihilation operator, creation operator, number operator, (generalized) Gross Laplacian is discussed. We show that this Lie algebra is associated with the orbit of the skew-symmetric operator $S$.