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...

We establish several properties of a quadratic algebra over a field k, which is a deformation of the symmetric algebra Sk³. In particular, we prove that A is an integral domain, noetherian and Koszul; we compute its first Hochschild cohomology group; we determine the corresponding Poisson structure on k³ and its symplectic leaves; we define an involution on A and describe the corresponding irreducible involutive representations.

We focus on connected graded algebras and their PBW-deformations endowed with additional symmetric structures. Many well-known algebras such as negative parts of Drinfeld-Jimbo’s quantum groups, cubic Artin-Schelter algebras and three-dimensional Sklyanin algebras appear in our research framework. As an application, we investigate a ${𝒦}_2$ algebra $𝒜$ which was introduced to compute the cohomology ring of the Fomin-Kirillov algebra ${\mathrm{ℱ𝒦}}_3$, and explicitly construct all the (self-)symmetric and sign-(self-)symmetric...