Seminar

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Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. For associative algebras, the symmetric homology theory  was introduced by Z. Fiedorowicz (1991) based on his earlier work with J.-L. Loday. In this talk, after giving a survey of known results on symmetric homology, we show that, for algebras over a field of characteristic 0, this homology theory is naturally equivalent to the (one-dimensional) representation homology. Then, using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz (2007), including their main conjecture on topological realization of symmetric homology of polynomial algebras.

Symmetric homology and derived character maps of representations