The q-analog of higher order Hochschild homology and the Lie derivative

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Let A be a commutative algebra over C. Given a pointed simplicial finite set Y and q∈ C a primitive N-th root of unity, we define the q-Hochschild homology groups of A of order Y. When D is a derivation on A, we construct the corresponding Lie derivative on these groups. We also define the Lie derivative for a higher derivation {Dₙ}ₙ≥ ₀ on A. Finally, we describe the morphisms induced on the bivariant q-Hochschild cohomology groups of order Y by a derivation D on A.