Representation homology of involutive algebras and hyperoctaherdral homology
Abstract
In this paper, we introduce and study the (involutive) representation homology of involutive algebras, following the development of representation homology of associative algebra in our earlier papers. Our main motivation is to understand the hyperoctahedral homology of involutive algebras. We show that over a field of characteristic $0$, the hyperoctahedral homology of an involutive algebra is isomorphic (as a graded commutative algebra) to its one dimensional involutive representation homology. This result enables the explicit computation of the hyperoctahedral homology of the universal enveloping algebras of involutive (DG) Lie algebras. For group algebras k[Γ] equipped with the involution g ↦ g−1, we relate the involutive representation homology of k[Γ] to the representation homology of the group Γ with coefficients in μ2, and we show that the latter is the homology of the loop space of the mod $2$ Dold-Thom space of BΓ; more generally, we compute the representation homology of Γ with coefficients in an arbitrary diagonalizable group scheme. As a consequence, the hyperoctahedral homology of k[Γ] is concentrated in degree $0$ in characteristic $0$, for every Γ.
Related
BibTeX
@misc{berest2026representation,
title = {{Representation homology of involutive algebras and hyperoctaherdral homology}},
author = {Yuri Berest and Ajay C. Ramadoss and John Wood},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14278},
eprint = {2610.14278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14278},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Representation homology of involutive algebras and hyperoctaherdral homology
AU - Yuri Berest
AU - Ajay C. Ramadoss
AU - John Wood
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14278
UR - https://arxiv.org/abs/2610.14278
AB - In this paper, we introduce and study the (involutive) representation homology of involutive algebras, following the development of representation homology of associative algebra in our earlier papers. Our main motivation is to understand the hyperoctahedral homology of involutive algebras. We show that over a field of characteristic $0$, the hyperoctahedral homology of an involutive algebra is isomorphic (as a graded commutative algebra) to its one dimensional involutive representation homology. This result enables the explicit computation of the hyperoctahedral homology of the universal enveloping algebras of involutive (DG) Lie algebras. For group algebras k[Γ] equipped with the involution g ↦ g^−1, we relate the involutive representation homology of k[Γ] to the representation homology of the group Γ with coefficients in μ_2, and we show that the latter is the homology of the loop space of the mod $2$ Dold-Thom space of BΓ; more generally, we compute the representation homology of Γ with coefficients in an arbitrary diagonalizable group scheme. As a consequence, the hyperoctahedral homology of k[Γ] is concentrated in degree $0$ in characteristic $0$, for every Γ.
ER -