Hyperoctahedral Operations on Hochschild Homology

✍ Scribed by N. Bergeron

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 680 KB
• Volume: 110
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1995.1011

✦ Synopsis

We construct families (\left{\rho_{n}^{(t, k)}\right}) of orthogonal idempotents of the hyperoctahedral group algebras (Q\left[B_{n}\right]), which commute with the Hochschild boundary operators (h_{n}=\sum_{i=0}^{n}(-1)^{i} d). We show that those idempotents are projections onto some hyperoactahedral symmetric powers of the free Lie algebra (\operatorname{Lie}{n}^{(1 / k)}\left(. \alpha^{\prime}\right)). The commutations above then decompose the Hochshild homology (H{n}(\mathrm{C})) obtained by any functor (C: A^{\wedge \Gamma} \rightarrow K)-module that factor through (\mathbf{F i n}{B}^{\prime}). 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{n}^{(t, k)}(C) \cong H_{n}\left(\mathrm{Sh}^{(!\cdot k)} / \mathrm{Sh}!{ }^{1, k+1)}\right)), where (\mathrm{Sh}^{(k, s)}) are some shufle complexes associated to (C). We also give the characters of the representations related to (\mathrm{Lie}_{n}^{(1, k)(}(\mathrm{Cl})) as a direct sum of induced characters. ' 1995 Academic Press, Inc.