## Haar measure on infinite dimensional Lie groups?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like this:

Def. element on the group by "Euler coordinates",

$g \doteq \prod\limits_{i=-\infty}^{\infty} e^{\omega^i X_i}$, with $\left[ X_i ,X_j \right] = (j-i)X_{i+j}$.

Now I could define a (left invariant) Maurer-Cartan form as $\Omega_L \doteq g^{-1} dg = X_i \otimes \theta^i$, where $\theta^i = \mathcal L^i_j d\omega^j$. Then the Haar measure is

$d\mu (g) \doteq ||\mathcal L || \bigwedge\limits_i d\omega^i$.

Elements of $\mathcal L$ can be written as

$\mathcal L^i_j = \left( \prod\limits_{n=\infty}^{j+1} \exp(-\omega^n adX_n) \right)^i_j$

Clearly the determinant $||\mathcal L ||$ will be horrible... is there any hope for a manageable explicit expression? I couldn't find any literature on the subject (yet), so I'd appreciate any hints to the right direction.

EDIT: umm and of course the whole question of existence of such a measure should probably addressed...

EDIT 2: I realized that the question setup is a bit misleading: I'm actually looking for a measure on the Virasoro group (with zero central charge), i.e. the Lie group corresponding to the algebra above... maybe the Shavgulidze measure has something to do with it, I don't know...

-

1
I think usually people restrict to fairly "tame" subspaces of infinite-dimensional groups in order to construct an actual measure that's useful. I don't know the literature as well as I'd like but my impression was that the kind of measures you want simply don't exist until you specialize to more modest spaces. – Ryan Budney Dec 6 2011 at 21:56

2
I think you might need to give up some of the properties of a measure (so there might be some kind of left-invariant form, in a weak sense, but not a measure per se). The reason is that a separated measurable group - roughly speaking, a measure space equipped with compatible group topology - carries a natural topology with respect to which it is a topological group, and that topological group has a "completion" to a locally compact group. (This is in Section 62 of Halmos's book on Measure Theory.) – Yemon Choi Dec 6 2011 at 22:58

2
IIRC if there was such a thing quantum field theory would be a lot easier. – Steve Huntsman Dec 7 2011 at 3:27
Small correction to my comment: "compatible group topology" should have been "compatible group structure". [The point is that the Weil topology can be defined in terms of the measure algebra structure] – Yemon Choi Dec 7 2011 at 4:26

1
Doug Pickrell did some work in this direction. You might want to check out arxiv.org/pdf/funct-an/9612009v1. If I remember correctly, this paper appeared as part of a book/monograph. – Antun Milas Dec 15 2011 at 20:57
show 1 more comment

## 1 Answer

There is something called the Shavgulidze, or the Malliavin-Shavgulidze measure on Diff of smooth manifolds. You can find a discussion in Differentiable measures and the Malliavin calculus (p. 397, available on google books). It is not quite invariant, but quasi-invariant.

-

Thanks for pointng that out! I can't get my hands on the book, and google won't show me some important pages... but at least I have something to google and I've found some references. – H. Arponen Dec 7 2011 at 9:03
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 9, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9422798156738281, "perplexity_flag": "middle"}