Dual Hodge decompositions and derived Poisson brackets (GPS)
Student No.:50
Time:Tue 10:30-12:00, 2017-6-27
Instructor:Yining Zhang  []
Place:Conference room 4, Floor 2, Jin Chun Yuan West Building
Starting Date:2017-6-27
Ending Date:2017-6-27





After briefly recall the definition of cyclic homology and classical Hodge decomposition, I will state a theorem by B. Feigin and B. Tsygan. Using this theorem, I shall construct a Hodge type decomposition (dual Hodge decomposition) of (reduced) cyclic homology of universal enveloping algebras of Lie algebras. In fact this decomposition can be viewed as being Koszul dual to the Hodge decomposition of cyclic homology of cocommutative coalgebras. Next, I will introduce the notion of noncommutative Poisson bracket in the sense of Crawley-Boevey and its homological generalization derived Poisson bracket. As a consequence, if an algebra admits a derived Poisson structure, then its cyclic homology would be a graded Lie algebra. Lastly, I want to talk about the interactions between this Lie bracket on cyclic homology and dual Hodge decomposition. If time permits, I will mentions some applications to string topology. This is based on a joint work with Yuri Berest and Ajay C. Ramadoss.


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