CONFIGURATION SPACES AND ALGEBROIDS

VITALY KUSHNIREVITCH and ROMAN KADOBIANSKI

Abstract

Let M be complete, connected, oriented, C^{∞} (noncompact) Riemannian manifold of dimension d. The configuration space Γ_{M} over M is the set of locally finite subsets in M:

Γ_{M} := {γ ⊂ M : card (γ ∩ K) < ∞ for each compact K ⊂ M}.

Any γ ∈ Γ_{M} is identified with positive integer-valued Radon measure. The tangent space to Γ_{M} at a point γ is defined as Hilbert space T_{γ}Γ_{M} := L^{2}(M → T M; γ) (or, equivalently, T_{γ}Γ_{M} := ⨁_{x∈γ} T_{x}M). (See S.Albeverio, Yu.G.Kondratiev,

M.Röckner, JFA 157 (1998).) High order differential forms and de Rham cohomology on configuration spaces can also be considered (see S.Albeverio, A.Daletskii, E.Lytvynov, JGeomPhys, to appear). The main topic of discussion is to consider these objects from the Lie algebriods theory point of view.

Roman KADOBIANSKI, National Technical University of Ukraine ”Kiev Polytechnic Institute”, UKRAINE, e-mail: tskafits@adam.kiev.ua

Vitaly KUSHNIREVITCH, National Technical University of Ukraine ”Kiev Polytechnic Institute”, UKRAINE, e-mail: tskafits@adam.kiev.ua