Separable Cowreaths Over Clifford Algebras

AbstractThe fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath $$ \left( A\otimes H_{4}^{op}, H_{4}, \psi \right) $$ A ⊗ H 4 op , H 4 , ψ , where $$H_{4}$$ H 4 is the Sweedler 4-dimensional Hopf algebra over a field k and $$A=Cl(\alpha , \beta , \gamma )$$ A = C l ( α , β , γ ) is the Clifford algebra generated by two elements G, X with relations $$G^{2}=\alpha $$ G 2 = α , $$X^{2}=\beta $$ X 2 = β and $$XG+GX=\gamma $$ X G + G X = γ , $$ (\alpha , \beta , \gamma \in k $$ ( α , β , γ ∈ k ) which becomes naturally an $$H_{4}$$ H 4 -comodule algebra. We show that, when $$\textrm{char}\left( k \right) \ne 2, $$ char k ≠ 2 , this cowreath is always separable and h-separable as well.