## Journal of Mathematics of Kyoto University

### One-parameter Subgroups and a Lie Subgroup of an Infinite Dimensional Rotation Group.

Hiroshi Sato

#### Abstract

Let $\mathscr{S}_{r}$, be the real topological vector space of real-valued rapidly decreasing functions and let $\mathcal{O}(\mathscr{S}_{r})$ be the group of rotations of $\mathscr{S}_{r}$. Then every one-parameter subgroup of $\mathcal{O}(\mathscr{S}_{r})$ induces a flow in $\mathscr{S}_{r}^{*}$ the conjugate space of $\mathscr{S}_{r}$ with the Gaussian White Noise as an invariant measure.

The author constructed a group of functions which is isomorphic to a subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and some of its one-parameter subgroups.

But the problem whether it contains sufficiently many one-parameter subgroups has been a problem. In Part I of the present paper, we answer this problem affirmatively by constructing two classes of one-parameter subgroups in a concrete way.

In Part II, we construct an infinite dimensional Lie subgroup of $\mathcal{O}(\mathscr{S}_{r})$ and the corresponding Lie algebra. Namely, we construct a topological subgroup $\mathfrak{G}$ of $\mathcal{O}(\mathscr{S}_{r})$ which is coordinated by the nuclear space $\mathscr{S}_{r}$ and the algebra $\mathfrak{a}$ of generators of one-parameter subgroups of $\mathfrak{G}$ which is closed under the commutation. Furthermore, we establish the exponential map from $\mathfrak{a}$ into $\mathfrak{G}$ and prove continuity.

#### Article information

Source
J. Math. Kyoto Univ., Volume 11, Number 2 (1971), 253-300.

Dates
First available in Project Euclid: 17 August 2009

https://projecteuclid.org/euclid.kjm/1250523648

Digital Object Identifier
doi:10.1215/kjm/1250523648

Mathematical Reviews number (MathSciNet)
MR285037

Zentralblatt MATH identifier
0218.22019

#### Citation

Sato, Hiroshi. One-parameter Subgroups and a Lie Subgroup of an Infinite Dimensional Rotation Group. J. Math. Kyoto Univ. 11 (1971), no. 2, 253--300. doi:10.1215/kjm/1250523648. https://projecteuclid.org/euclid.kjm/1250523648