Oriented Hopf Algebras and their Actions

Let H be a finite-dimensional Hopf algebra acting on an algebra A. Then A is called an H-algebra whenever H acts on products in A via the coproduct. For example, when H=K[G] is a group Hopf algebra, then A is an H-algebra precisely when the group G acts by automorphisms on A; hence, if G acts by both automorphisms and anti-automorphisms on A, then A is not an H-algebra. In order to study such H-actions, we introduce the notion of an oriented H-algebra. An oriented Hopf algebra is a Hopf algebra with a specified ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {Z}_{2}$\end{document}-grading, H = H+ ⊕ H−, while A is an oriented H-algebra whenever H+ acts on products via the coproduct and H− acts on products via the ‘anti-coproduct’. This leads to an oriented convolution operation, ⋆. First, we characterize when ⋆ is associative. Then, when H is isomorphic to the dual of a group algebra of a finite abelian group, we provide certain duality theorems between oriented H-algebra actions on A and grading properties of the associated vector space decomposition of A. For example, a Lie algebra A is an oriented H-algebra if and only if A is quasigroup-graded with respect to ⋆. We show that there exists a quasigroup-grading of a Lie algebra that cannot be realized as a semigroup-grading, which was once thought impossible.