Operator Lie Algebras of Rotations and Transformations in White Noise

Abstract

The infinitesimal generator of a one-parameter subgroup of the infinite dimensional rotation group associated with the complex Gelfand triple $(E)\subset L^2(E^{\ast },\mu )\subset (E{)}^{\ast }$ is of the form $$R_{\kappa }={\int }_{T\times T}\kappa (s,t)(a_s^{\ast }{a}_t-a_t^{\ast }{a}_s)dsdt$$ where $\kappa \in E\otimes E^{\ast }$ is a skew-symmetric distribution. 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 $S$.