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Let V be a ℂ-space, $\sigma \in End\left(V^{\otimes 2}\right)$ be a pre-braid operator and let $F\in lin(V^{\otimes 2},ℂ).$ This paper offers a sufficient condition on (σ,F) that there exists a Clifford algebra Cl(V,σ,F) as the Chevalley F-dependent deformation of an exterior algebra $Cl(V,\sigma ,0)\equiv V^{\wedge }\left(\sigma \right)$. If $\sigma \ne {\sigma }^{-1}$ and F is non-degenerate then F is not a σ-morphism in σ-braided monoidal category. A spinor representation as a left Cl(V,σ,F)-module is identified with an exterior algebra over F-isotropic ℂ-subspace of V. We give a sufficient condition on braid σ that the spinor representation...

The purpose of this note is to show how calculi on unital associative algebra with universal right bimodule generalize previously studied constructions by Pusz and Woronowicz [1989] and by Wess and Zumino [1990] and that in this language results are in a natural context, are easier to describe and handle. As a by-product we obtain intrinsic, coordinate-free and basis-independent generalization of the first order noncommutative differential calculi with partial derivatives.