Spinors in braided geometry

Let V be a ℂ-space, ${\displaystyle \sigma \in End\left(V^{\otimes 2}\right)}$ be a pre-braid operator and let ${\displaystyle F\in l\in (V^{\otimes 2},ℂ).}$ 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 ${\displaystyle Cl(V,\sigma ,0)\equiv V^{\wedge }(\sigma )}$. If ${\displaystyle \sigma \ne {\sigma }^{-1}}$ 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 is faithful and irreducible.