Abstract
The loop representation of four dimensional quantum gravity is studied. We point out that there is a close connection between the T algebra in [C. Rovelli and L. Smolin, Phys. Rev. Lett. 61 (1988) 1155; Nucl. Phys. B 331 (1990) 80] and the W 1+∞ algebra familiar to 2D field theorists. Namely, the W 1+∞ algebra can be viewed as a sub-algebra of the T algebra. Under the reduction from T to W 1+∞, correspondingly, the space-time reduces to an (anti-) self-dual one.
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