This book is based on a DMV seminar held at Oberwolfach in 1995: the main topics come from recent intensive interactions between theoretical physics and mathematics. It contains six survey papers systematically explaining many topics connected with infinite-dimensional Lie groups and Lie algebras, their representations and infinite-dimensional homogeneous spaces.

The first contribution (by A. Huckleberry) is a useful summary of basic facts needed from the finite-dimensional situation (differentiable manifolds, fibre bundles, symplectic and complex geometry and their equivariant versions, the Borel-Weil realisations of irreducible representations of compact Lie groups). The first contribution by K.-H. Neeb describes a generalisation of infinite-dimensional groups and their representations to the setting of groups modelled over sequentially complete locally convex spaces. This is then used in his second paper devoted to the Borel-Weil theory for positive energy representations of loop groups. The coadjoint representations of the Virasoro algebra and its generalisations is treated by V. Yu. Ovsienko: it describes a geometrical realisation of these coadjoint representations in terms of representations of suitable Lie superalgebras on spaces of linear differential operators and their connection to the Adler-Gelfand-Dickey bracket. C. Paycha describes renormalisation techniques needed to define determinant bundles in the infinite-dimensional situation and their use in quantisation of gauge field theories. T. Wurzbacher describes a relation between the fermionic second quantisation and the geometry of restricted Grassmannians, including the C*-algebra approach to the Grassmannian and its determinant bundle.

This book can be recommended to a wide range of interested readers, students and researchers, from mathematics and from theoretical and mathematical physics.

Reviewer:

vs