Crossed products and inner actions of Hopf algebras

Abstract

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if 7r : H-► H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #" H, a crossed product of H with the left Hopf kernel A of it. Given any crossed product A #CT H with H (weakly) inner on A, then A #CT if is isomorphic to a twisted product AT[H] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #" H; in particular this is true if the (weak) action of H is inner.

Heisenberg doubles of quantized Poincaré algebras

“…In the deformed case, the relation stays the same: the noncommutative space-time is a Hopf module algebra over the quantum symmetry group. The noncommutative space-time algebra and quantum group can be described (unified) as one larger algebra called the smash (crossed) product algebra [14], [15]. It is interesting from the physical standpoint that such a deformed (twisted) smash product algebra is isomorphic to the undeformed one.…”

“…In the quantum group approach, we start by deforming relativistic symmetries and investigate the implications of this assumption, for example, in the theory of gravity. This standpoint allows unifying the space-time coordinates with quantum group generators via the smash (crossed) product construction (see, e.g., [14], [15] for the mathematical details). This new (smash product) algebra turns out to be obtained by a nonlinear change of the generators from the undeformed (nonquantized) algebra [1].…”

Stable Jacobson Radicals and Semiprime Smash Products

“…Many special cases of Question 2 have been answered: for example, if H = kG (see [9]), if H = (kG) * (see [6]), or if H is semisolvable [21]. Other results are known with additional assumptions on R (for example, a result of [5] says it is true when R is semisimple Artinian), or on the type of action of H on R (in [3] it is shown for inner actions, and in [20] for Q H -inner actions).…”

The structure of bialgebra with a Hopf module

“…The crossed product algebra and the crossed coproduct coalgebra were introduced and studied in [1]- [3], [5]- [9], [13] as a generalization of the smash product algebra and the smash coproduct coalgebra; these notions can be viewed as motivated by the semidirect product construction in the theory of groups and affine group schemes, respectively. Blattner, Cohen, and Montgomery (see [3]) showed that an algebra A ⊂ B is an H-cleft extension if and only if B is isomorphic to the crossed product A# σ H. Dually, Dˇascˇalescu, Militaru, and Raiann (see [5]) showed that a coalgebra C is an H-cleft coextension of C/CH + if and only if C is isomorphic to the crossed coproduct C/CH + α H. E. S. Kim, Y. S. Park, and S. B. Yoon gave necessary and sufficient conditions for a crossed product and a crossed coproduct to be a bialgebra called bicrossproduct in [8] generalizing Radford's biproduct (see [10], [12]).…”

“…Blattner, Cohen, and Montgomery (see [3]) showed that an algebra A ⊂ B is an H-cleft extension if and only if B is isomorphic to the crossed product A# σ H. Dually, Dˇascˇalescu, Militaru, and Raiann (see [5]) showed that a coalgebra C is an H-cleft coextension of C/CH + if and only if C is isomorphic to the crossed coproduct C/CH + α H. E. S. Kim, Y. S. Park, and S. B. Yoon gave necessary and sufficient conditions for a crossed product and a crossed coproduct to be a bialgebra called bicrossproduct in [8] generalizing Radford's biproduct (see [10], [12]). Y. Bespalov and B. Drabant (see [1], [2]) investigated all sorts of cross product bialgebras deeply and developed a fully-fledged formulation in terms of (co-)modular and (co-)cyclic conditions (called Hopf Data).…”