Nelson Laplacian - Imported Knowlpedia Content

Definition

Let $\pi$ be a strongly continuous unitary representation of a finite-dimensional Lie group $G$ on $H$ , and choose a basis $X_1,\ldots,X_n$ of its real Lie algebra. On the common invariant domain $H^\infty$ of smooth vectors, put

\Delta_{\pi}=-\sum_{j=1}^{n}d\pi(X_j)^2.

This nonnegative symmetric operator is the Nelson Laplacian of the representation. Some authors instead call $1+\Delta_\pi$ the Nelson operator, or use the opposite sign for the Laplacian. The sign convention above makes $\Delta_\pi$ positive because each infinitesimal generator $d\pi(X_j)$ is skew-symmetric on $H^\infty$ .

Closure and smooth vectors

The operator $\Delta_\pi$ on $H^\infty$ is essentially self-adjoint. If $\overline{\Delta_\pi}$ denotes its nonnegative self-adjoint closure, then

H^\infty=\bigcap_{k\geq 1} \operatorname{Dom}\!\left(\overline{\Delta_\pi}^{\,k}\right).

Thus powers of one elliptic operator encode simultaneous differentiability under every derived operator. More generally, the topology on $H^\infty$ defined by the derived action of the universal [[lie-groups/universal-enveloping-algebra|enveloping algebra]] is equivalent to the graph topology defined by the powers of $1+\overline{\Delta_\pi}$ .

Dependence on choices

The displayed operator depends on the chosen basis, or equivalently on a choice of inner product on the Lie algebra. Different choices yield equivalent regularity scales and the same space $H^\infty$ , but they need not yield the same operator. Unless the inner product is $\operatorname{Ad}(G)$ -invariant, the quadratic element $-\sum_jX_j^2$ is not central in the universal enveloping algebra and $\Delta_\pi$ need not commute with the representation.

Analytic vectors and Nelson's theorem

Vectors analytic for $\overline{\Delta_\pi}$ are analytic vectors for the group representation: their orbit maps $g\mapsto\pi(g)v$ are real analytic near the identity. Nelson's heat-kernel argument produces a dense supply of such vectors. This converts an infinite family of infinitesimal generators into a single elliptic regularity problem and is a key ingredient in criteria for integrating representations of a Lie algebra to unitary representations of a Lie group.

Terminological warning

The Nelson Laplacian is attached to a representation and a Lie-algebra basis. It should not automatically be identified with a geometric Laplace–Beltrami operator or with a Casimir operator. In special representations these operators can be related, but the identifications require additional invariant geometric structure.

References

Edward Nelson, “Analytic Vectors,” Annals of Mathematics 70 (1959), 572–615. DOI record. Relevant: the Laplacian criterion, essential self-adjointness, and analytic vectors.
Roe Goodman, “Analytic and Entire Vectors for Representations of Lie Groups,” Transactions of the American Mathematical Society 143 (1969), 55–76. DOI record. Relevant: analytic-vector regularity for Lie-group representations.