• DOI:10.1006/AIMA.1995.1011
• Corpus ID: 120128390

Hyperoctahedral Operations on Hochschild Homology

@article{Bergeron1995HyperoctahedralOO,
  title={Hyperoctahedral Operations on Hochschild Homology},
  author={Nantel Bergeron},
  journal={Advances in Mathematics},
  year={1995},
  volume={110},
  pages={255-276}
}

• N. Bergeron
• Published 1 February 1995
• Mathematics
• Advances in Mathematics

Abstract We construct Families {ρ l, k n } of orthogonal idempotents of the hyperoctahedral group algebras Q [ B n ], which commute with the Hochschild boundary operators b n =∑ n i=0 (−1) i d i . We show that those idempotents are projections onto some hyperoactahedral symmetric powers of the free Lie algebra Lie ( l, k ) n ( A ). The commutations above then decompose the Hochshild homology H n ( C ) obtained by any functor C : Δ op → K - module that factor through Fin ′ B , the hyperoctahedral category. Moreover, we show that this decomposition is the finest possible for any such functor. In particular, the Hochschild homology of a commutative algebra equipped with an involutive automorphism splits into components indexed by ( l, k ) and the corresponding Harrison homology splits into two components indexed by (0, 1) and (1, 0). Generalizing the Harrison complex, we show that H ( l,k ) n ( C )≅ H n (Sh ( l,k ) . /Sh ( l −1, k +1) . , where Sh ( r,s ) . are some shuffle complexes associated to C . We also give the characters of the representations related to Lie ( l, k ) n ( A ) as a direct sum of induced characters.