E(n)-coactions on Semisimple Clifford Algebras

Abstract

Let E(n) be the \(2^{n+1}\)-dimensional Hopf algebra generated by anti-commuting elements \(g,x_1, \ldots , x_n\), with g grouplike, each \(x_i\) skew-primitive, and \(g^2=1\), \(x_i^2=0\). In this article we prove that E(n)-coactions over a finite-dimensional algebra A are classified by tuples \((\varphi , d_1, \ldots , d_n)\) consisting of an involution \(\varphi \) and a family \((d_i)_{i=1,\ldots ,n}\) of \(\varphi \)-derivations satisfying appropriate conditions. Tuples of maps can be replaced by tuples of suitable elements \((c, u_1, \ldots , u_n)\), whenever A is a semisimple Clifford algebra.