A Kähler metric on a based loop group and a covariant differentiation

Ichiro Shigekawa and Setsuo Taniguchi
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Ichiro Shigekawa: Kyoto University, Department of Mathematics, Graduate School of Science
Setsuo Taniguchi: Kyushu University, Graduate School of Mathematics

Chapter 21 in Itô’s Stochastic Calculus and Probability Theory, 1996, pp 327-346 from Springer

Abstract: Abstract Loop groups have been attracting many authors recently. In this paper, we are discussing a Kähler metric on a loop group. Let G be a d-dimensional compact Lie group and g be its Lie algebra (- the space of left invariant vector fields). Then, g admits an Ad(G)-invariant inner product (η, η)g and we fix it through the paper. We denote the G-valued path space on [0, 1] by 1.1 $$PG:=\left\{ \gamma :\left[ 0,1 \right]\to G;continuous and \gamma \left( 0 \right)=e \right\}$$ e being the unit element of G.

Date: 1996
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DOI: 10.1007/978-4-431-68532-6_21

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