caa a22 4500 606160914 CHVBK 20210128100632.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s10468-015-9521-3 doi (NATIONALLICENCE)springer-10.1007/s10468-015-9521-3 Oriented Hopf Algebras and their Actions [Elektronische Daten] [Chris Plyley, David Riley] 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 ℤ 2 $\mathbb {Z}_{2}$ -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. Springer Science+Business Media Dordrecht, 2015 Hopf algebras nationallicence Actions by Hopf algebras nationallicence Graded algebras nationallicence Algebras with involution nationallicence Automorphisms nationallicence Anti-automorphisms nationallicence Derivations nationallicence Anti-derivations nationallicence Invariants nationallicence Skew invariants nationallicence Plyley Chris Department of Mathematics, Western University, N6A 5B7, London, Canada aut Riley David Department of Mathematics, Western University, N6A 5B7, London, Canada aut Algebras and Representation Theory Springer Netherlands 18/4(2015-08-01), 895-906 1386-923X 18:4<895 2015 18 10468 https://doi.org/10.1007/s10468-015-9521-3 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10468-015-9521-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Plyley Chris Department of Mathematics, Western University, N6A 5B7, London, Canada aut NATIONALLICENCE 700 1- Riley David Department of Mathematics, Western University, N6A 5B7, London, Canada aut NATIONALLICENCE 773 0- Algebras and Representation Theory Springer Netherlands 18/4(2015-08-01), 895-906 1386-923X 18:4<895 2015 18 10468