Hochschild and cyclic homology of the quantum multiparametric torus.
Résumé
The quantum multiparametric torus is the algebra generated over a field $k$ by the $2N$ variables $x_1,\ldots,x_N$ and $x_1^{-1},\ldots,x_N^{-1}$ and the relations $ x_ix_i^{-1}=1=x_i^{-1} x_i$ and $x_ix_j=q_{ij}x_jx_i$ for every $1\le i,j\le N$ and where $\{q_{ij}\}_{1\le i,j\le N}$ is a family of non-zero scalars of $k$ satisfying the relations $q_{ii}=1$ and $q_{ij}q_{ji}=1$ for every $1\le i,j,\le N$. We explicitly compute its Hochschild homology groups, using previously constructed ``quantum Koszul complexes''. We deduce the corresponding cyclic homology groups.
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