Equivalences of (co)module algebra structures over Hopf algebras

A duality theorem for weak multiplier Hopf algebra actions

International Journal of Mathematics ◽

10.1142/s0129167x1750032x ◽

2017 ◽

Vol 28 (05) ◽

pp. 1750032 ◽

Cited By ~ 2

Author(s):

Nan Zhou ◽

Shuanhong Wang

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Duality Theorem ◽

Smash Product ◽

Module Algebra ◽

Multiplier Hopf Algebras ◽

Multiplier Hopf Algebra ◽

Weak Hopf Algebras

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by Van Daele and Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids.

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Projectivity of Hopf algebras over subalgebras with semilocal central localizations

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is008001004jkt025 ◽

2008 ◽

Vol 2 (1) ◽

pp. 1-40 ◽

Cited By ~ 1

Author(s):

Serge Skryabin

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Module Algebra ◽

Finitely Generated ◽

Residually Finite ◽

Hopf Subalgebra ◽

Finite Dimensional ◽

Algebra Over A Field

AbstractThe purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

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Inner actions of weak Hopf algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501183 ◽

2017 ◽

Vol 16 (06) ◽

pp. 1750118

Author(s):

Dirceu Bagio ◽

Daiana Flôres ◽

Alveri Sant’ana

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Associative Ring ◽

Sufficient Conditions ◽

Product Formula ◽

Weak Hopf Algebra ◽

Smash Product ◽

Module Algebra ◽

Weak Hopf Algebras

Let [Formula: see text] be an associative ring and [Formula: see text] idempotent elements of [Formula: see text]. In this paper we introduce the notion of [Formula: see text]-invertibility for an element of [Formula: see text] and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra [Formula: see text] and an algebra [Formula: see text], we present sufficient conditions for [Formula: see text] to admit an inner action of [Formula: see text]. We also prove that if [Formula: see text] is a left [Formula: see text]-module algebra then [Formula: see text] acts innerly on the smash product [Formula: see text] if and only if [Formula: see text] is a quantum commutative weak Hopf algebra.

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Categorical methods in Hopf algebras

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.32.2001.343 ◽

2001 ◽

Vol 32 (4) ◽

pp. 281-292

Author(s):

Wang Dingguo ◽

Yang Shilin ◽

Ji Qingzhong

Keyword(s):

Hopf Algebras ◽

Module Algebra ◽

Injective Dimension ◽

Categorical Methods ◽

Relationship Of

In this paper, we first study the adjoint relationship of general functors between two Grothendieck categories. Then apply those results to obtain information about the connections of the categories $ A \# H $-mod and $ A^H $-mod, resp. $ {\cal M}^{C \rtimes H} $ and $ {\cal M}^{C^{coH}} $, where $ A $ is an $ H $-module algebra and $ C $ is an $ H $-comodule coalgebra. In particular, the relationships between the (co)generators, injectivity and projectivity of $ A \# H $-modules and the corresponding property of $ A^H $-modules are given, the relationships between the injective dimension $ A_A $ the corresponding dimension of $ A_{A^H}^H $ and $ A \# H_{A \# H} $ are also established.

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On the Semiprimitivity and Semiprimality Problems for Partial Smash Products

Algebra Colloquium ◽

10.1142/s1005386718000020 ◽

2018 ◽

Vol 25 (01) ◽

pp. 1-30

Author(s):

Rafael Cavalheiro ◽

Alveri Sant’Ana

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Irreducible Representations ◽

Module Algebra ◽

Perfect Field ◽

Locally Finite ◽

Semisimple Hopf Algebra ◽

Finite Dimensional ◽

Algebra Over A Field

In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let H be a semisimple Hopf algebra over a field 𝕜 and let A be a left partial H-module algebra. We study the H-prime and the H-Jacobson radicals of A and their relations with the prime and the Jacobson radicals of A#H, respectively. In particular, we prove that if A is H-semiprimitive, then A#H is semiprimitive provided that all irreducible representations of A are finite-dimensional, or A is an affine PI-algebra over 𝕜 and 𝕜 is a perfect field, or A is locally finite. Moreover, we prove that A#H is semiprime provided that A is an H-semiprime PI-algebra, generalizing to the setting of partial actions the known results for global actions of Hopf algebras.

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