Operator Lie Algebras of Rotations and Transformations in White Noise

TL;DR

This paper explores the structure of operator Lie algebras related to rotations and transformations in white noise, focusing on their generators, conservation operators, and the associated orbit of skew-symmetric operators.

Contribution

It introduces a detailed analysis of the Lie algebra generated by operators in white noise analysis, linking it to the orbit of skew-symmetric operators and expanding understanding of their algebraic structure.

Findings

The generator of the rotation group is expressed as an integral involving skew-symmetric distributions.

The Lie algebra includes various operators such as the identity, annihilation, creation, and number operators.

The Lie algebra is shown to be associated with the orbit of a skew-symmetric operator S.

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^*, \mu) \subset (E)^* $ is of the form $$ R_\kappa = \int_{T\times T} \kappa(s,t) (a_s^* a_t - a_t^* a_s) ds dt $$ where $\kappa \in E \otimes E^*$ 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 $S$.