Infinite dimensional geometry of $M_1=Diff_+(S^1)/PSL(2,R)$ and $q_R$--conformal symmetries

Abstract: A geometric interpretation of approximate (

H S

-projective or

T C

-projective) representations of the Witt algebra

w^{C}

by

q_{R}

-conformal symmetries in the Verma modules

V_{h}

over the Lie algebra

s l (2, C)

is established and some their characteristics are calculated. It is shown that the generators of representations coincide with the Nomizu operators of holomorphic

w^{C}

-invariant hermitean connections in the deformed holomorphic tangent bundles

T_{h} (M_{1})

over the infinite-dimensional K"ahler manifold

M_{1} = D i f f_{+} (S^{1}) / P S L (2, R)

, whereas the deviations

A_{X Y}

of the approximate representations coincide with the curvature operators for these connections, which supply the determinant bundle

d e t T_{h} (M_{1})

by a structure of the prequantization bundle over

M_{1}

. At

h = 2

the geometric picture reduces to one considered by A.A.Kirillov and the author [Funct.Anal.Appl. 21(4) (1987) 284-293] (without any relation to the approximate representations) for ordinary tangent bundles.

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