Braided Clifford Algebras as Quantum Deformations

Abstract

We present a general algebraic framework for the study of quantum/braided Clifford algebras. We allow that the quadratic form g on the base vector space \({\mathbb{V}}\) takes values from a noncommutative algebra Σ. Clifford algebra is understood as a Chevalley—Kahler deformation of the braided exterior algebra built from V, Σ, and the initial braid operator σ: \({\mathbb{V}}\) ⊗ \({\mathbb{V}}\) → \({\mathbb{V}}\) ⊗ \({\mathbb{V}}\). The new product is canonically associated to g, σ, and Σ, and it is constructed by applying Rota's and Stein Cliffordization.