Split Extensions and Actions of Bialgebras and Hopf Algebras

AbstractLet A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

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CLASSIFYING COALGEBRA SPLIT EXTENSIONS OF HOPF ALGEBRAS

Journal of Algebra and Its Applications ◽

10.1142/s0219498812502271 ◽

2013 ◽

Vol 12 (05) ◽

pp. 1250227

Author(s):

A. L. AGORE ◽

C. G. BONTEA ◽

G. MILITARU

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Infinite Family ◽

Crossed Products ◽

Primitive Elements ◽

Characteristic P ◽

Generators And Relations ◽

Split Extensions ◽

Groups Of Automorphisms

For a given Hopf algebra A we classify all Hopf algebras E that are coalgebra split extensions of A by H4, where H4is the Sweedler's four-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras A# H4by computing explicitly two classifying objects: the cohomological "group" [Formula: see text] and CRP (H4, A) ≔ the set of types of isomorphisms of all crossed products A# H4. All crossed products A# H4are described by generators and relations and classified: they are parameterized by the set [Formula: see text] of all central primitive elements of A. Several examples are worked out in detail: in particular, over a field of characteristic p ≥ 3 an infinite family of non-isomorphic Hopf algebras of dimension 4p is constructed. The groups of automorphisms of these Hopf algebras are also described.

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Unified products and split extensions of Hopf algebras

Hopf Algebras and Tensor Categories - Contemporary Mathematics ◽

10.1090/conm/585/11613 ◽

2013 ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

A. Agore ◽

G. Militaru

Keyword(s):

Hopf Algebras ◽

Split Extensions

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Split Extensions of Hopf Algebras and Semi-Tensor Products.

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-10963 ◽

1970 ◽

Vol 26 ◽

pp. 17 ◽

Cited By ~ 7

Author(s):

Larry Smith

Keyword(s):

Hopf Algebras ◽

Tensor Products ◽

Split Extensions

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A note on split extensions of bialgebras

Forum Mathematicum ◽

10.1515/forum-2017-0016 ◽

2018 ◽

Vol 30 (5) ◽

pp. 1089-1095 ◽

Cited By ~ 2

Author(s):

Xabier García-Martínez ◽

Tim Van der Linden

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Closed Field ◽

Split Extension ◽

Algebraically Closed Field ◽

Split Extensions

AbstractWe prove a universal characterization of Hopf algebras among cocommutative bialgebras over an algebraically closed field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.

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