De Rham–Hodge–Kodaira Operator on Loop Groups

✍ Scribed by Shizan Fang; Jacques Franchi

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 376 KB
• Volume: 148
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3077

✦ Synopsis

We consider a based loop group L e (G ) over a compact Lie group G, endowed with its pinned Wiener measure & (the law of the Brownian bridge on G) and we shall calculate the Ricci curvature for differential n-forms over L e (G ). A type of Bochner Weitzenbo ck formula for general differential n-forms (or Shigekawa identity) will be established. 1997 Academic Press 1. INTRODUCTION Let G be a compact Lie group, with unit e. Consider the following path group over G : P e (G )=[#: [0, 1] Ä G continuous; #(0)=e]. Let + be the Wiener measure on P e (G ) induced by a G-valued Brownian motion over [0, 1], starting from e. Consider the based loop group L e (G )=[l # P e (G), l(1)=e]. The associated Wiener measure & over L e (G ) is the Brownian bridge law. The quasi-invariance of the measure & has been discussed by M. P. Malliavin and P. Malliavin [MM1]. The differential and geometric calculus on L e (G ) were extensively investigated these last years by many authors.