Universal Similarity Factorization Equalities over Complex Clifford Algebras

Universal Similarity Factorization Equalities over Real Clifford Algebras

A variety of universal similarity factorization equalities over real Clifford algebras ${\cal R}_{p,q}$ are established. On the basis of these equalities, real, complex and quaternion matrix representations of elements in ${\cal R}_{p,q}$ can be explicitly determined.

Similarity and Consimilarity of Elements in Real Cayley-Dickson Algebras

A set of valuable universal similarity factorization equalities is established over complex Clifford algebras $\Cn.$ Through them matrix representations of complex Clifford algebras $\Cn$ can directly be derived, and their properties can easily be determined.

Factorization Results for Left Polynomials in Some Associative Real Algebras: State of the Art, Applications, and Open Questions

We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put particular emphasis on factorization results for quaternion, dual quaternion and split quaternion polynomials. A general algorithm ensures existence of a factorization for generic polynomials over division rings but we also consider factorizations for non-division rings. We explain the current state of the art, present some new results and provide examples …

Clifford deformations of Koszul Frobenius algebras and noncommutative quadrics

Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-graded algebra $E(θ)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $z\in E^!_2$. It turns out that the bounded derived category $D^b(\text{gr}_{\mathbb Z_2}E(θ))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(θ)$ is a semisimple …

弯曲DG模与非交换二次超曲面的矩阵分解

Curved DG Modules and Matrix Factorizations from Noncommutative Quadric Hypersurfaces

本文研究非交换二次超曲面范畴的对偶性，构造从分次模到弯曲DG模同伦范畴的忠实函子，并在一定条件下将其限制到矩阵分解稳定范畴，证明偶数克利福德代数与PBW形变同构。