E(n)-coactions on semisimple Clifford algebras
Fabio Renda
TL;DR
This work classifies $E(n)$-coactions on finite-dimensional algebras by reducing coactions to $E(n)^{cop}$-actions and then to data consisting of an involution $\varphi$ and a family of $\varphi$-derivations $d_i$ with $d_i^2=0$, $\varphi d_i=-d_i\varphi$, and $d_id_j=-d_jd_i$. For semisimple Clifford algebras, the authors further show that these data are effectively encoded by tuples $(c,u_1,\dots,u_n)$ with $c^2$ in the center and the $u_i$ forming inner skew-derivations, leading to explicit formulas for the coaction $\rho$ in terms of the $d_P$ and $x_P$ generators of $E(n)$. They provide concrete descriptions in the odd and even $n$ cases, including the structure of the center, the role of the pseudoscalar, and the automorphism groups of the Clifford algebras, with detailed low-dimensional examples. The results connect $E(n)$-coactions to intrinsic Clifford-algebra data and illuminate how Hopf-symmetry actions interact with semisimple Clifford structures, offering a concrete toolkit for classifying such coactions.
Abstract
In this article we prove that $E(n)$-coactions over a finite-dimensional algebra $A$ are classified by tuples $(\varphi, d_1, ... , d_n)$ consisting of an involution $\varphi$ and a family $(d_i)_{i=1,...,n}$ of $\varphi$-derivations satisfying appropriate conditions. Tuples of maps can be replaced by tuples of suitable elements $(c, u_1, . . . , u_n)$, whenever $A$ is a semisimple Clifford algebra.