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 $(E) \subset L^2(E^*, μ) \subset (E)^*$ is of the form $R_κ= \int_{T\times T} κ(s,t) (a_s^* a_t - a_t^* a_s) ds dt$ where $κ\in E \otimes E^*$ is a skew-symmetric distribution. Hence $R_κ$ is twice the conservation operator associated with a skew-symmetric operator $S$. The Lie algebra containing $R_κ$, 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 $S$.

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