Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathrm{H}\mathrm{S}(\mathcal{K}u(X))={⨁}_{i,j\in \mathbb{Z}}\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{I}\mathrm{d},S_{\mathcal{K}u(X)}^i[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathrm{H}\mathrm{S}(\mathcal{K}u(X))$ when $X$ is a Fano hypersurface and establish its relation with Jacobian ring $\mathrm{J}\mathrm{a}\mathrm{c}(X)$. As an application, we prove a categorical Torelli theorem for Fano hypersurface $X\subset {\mathbb{P}}^n(n\ge 2)$ of degree $d$ if $\mathrm{g}\mathrm{c}\mathrm{d}(n+1,d)=1.$ In addition, we give a new proof of the $[Pir22,Theorem1.2]$ using a similar idea.