Symmetric homology and representation homology

Berest, Yuri; Ramadoss, Ajay

doi:10.1090/tran/8947

Symmetric homology and representation homology
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by Yuri Berest and Ajay C. Ramadoss;

Trans. Amer. Math. Soc. 376 (2023), 6475-6496

DOI: https://doi.org/10.1090/tran/8947

Published electronically: June 13, 2023

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Abstract:

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic $0$, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). 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, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras.

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