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Homology of the quantum algebra of pseudo-differential symbols on the circle. (Homologie de l’algèbre quantique des symboles pseudo-différentiels sur le cercle.) (French) Zbl 0870.18011
The homology of the algebra of usual differential operators and of its quantum analogon was explicitly calculated by C. Kassel [Commun. Math. Phys. 146, No. 2, 343-356 (1992; Zbl 0761.17020)], and a cyclic 1-cocycle for either was described, via quasi-isomorphisms between the standard Hochschild complex and the associated Koszul and de Rham complexes, as well. Corresponding results in the case of pseudo-differential symbols on the circle are obtained among others by J.-L. Brylinski and E. Getzler [$$K$$-Theory 1, 385-403 (1987; Zbl 0646.58026)]. This paper generalizes the foregoing by considering topological Hochschild and first cyclic homology groups relatively to the natural filtration of the algebra mentioned in the title.
##### MSC:
 18G60 Other (co)homology theories (MSC2010) 58J40 Pseudodifferential and Fourier integral operators on manifolds 81S05 Commutation relations and statistics as related to quantum mechanics (general) 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 16S32 Rings of differential operators (associative algebraic aspects) 17B55 Homological methods in Lie (super)algebras
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##### References:
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