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 triplewhere κ ∈ E ⊗ E * is a skew-symmetric distribution. Hence Rκ is twice the conservation operator associated with a skew-symmetric operator S.The Lie algebra containing Rκ, 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.

On Solvable Lie Algebras of White Noise Operators

“…This has also been studied in [12,14] in terms of white noise operators. The following result can be easily shown, see [14] or [41] for proof of solvability and non-nilpotency.…”

“…Remark 2. As can be easily seen from Theorem 4, the number operator prevents nilpotency of h. Moreover, as a consequence of the canonical commutation relation of annihilation and creation operators, every nonzero ideal of h contains the identity (see [41]). Thus h cannot be semisimple; it cannot be written as direct sum of (simple) ideals.…”

“…In [41], we noted that h is finite dimensional if S is a finite rank operator. Here we provide a proof.…”