Depth One Extensions of Semisimple Algebras and Hopf Subalgebras

Burciu, Sebastian

doi:10.1007/s10468-012-9371-1

Cited by 2 publications

(3 citation statements)

References 19 publications

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“…If ker π = 0, then there is 0 = x ∈ R such that π(x) = 0. In connection with the proof of the next proposition, we state Burciu's result in [7]: if B ⊆ A is a subalgebra pair of semisimple complex algebras, then d(B, A) = 1 if and only if the centers satisfy Z(B) ⊆ Z(A). (We have seen in Section 1 that ⇒ is more generally true.)…”

Section: Uniquely Separable Frobenius Extensionsmentioning

confidence: 97%

“…We show in Theorem 3.1 that A has a unique separability element over B if and only if the centralizer A B = Z(A). If K is an algebraically closed field of characteristic zero, it follows from the trivial observation Z(B) ⊆ A B = Z(A) and Burciu's characterization of depth one in [7] that A is centrally projective over B: B A B ⊕ * ∼ = n • B B B , or A as a natural B-bimodule is isomorphic to a direct summand of a finite direct sum of copies of B. It is shown in [25] that such extensions automatically satisfy B B B ⊕ * ∼ = B A B , so that a centrally projective extension A ⊇ B is characterized by the bimodules B A B and B B B being similar [1].…”

Section: Introductionmentioning

confidence: 99%

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Uniquely separable extensions

Kadison¹

2020

Expositiones Mathematicae

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The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only the H-depth is one. Similarly, a split extension has a bimodule projection; this idempotent is full if and only if the ring extension has depth 1. The depth one Hopf algebroids are derived explicitly. If the separable idempotent is unique for some reason, then the separable extension is called uniquely separable. For example, a Frobenius extension with invertible E-index is uniquely separable if the centralizer equals the center of the over-ring. It is also shown that a uniquely separable extension of semisimple complex algebras with invertible E-index has depth 1. Earlier results for subalgebra pairs of group algebras are recovered, with corollaries on depth 1 over fields of characteristic zero and more general ground rings.1991 Mathematics Subject Classification. 12F10, 13B02, 16D20, 16H05, 16S34.

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Section: Uniquely Separable Frobenius Extensionsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Uniquely separable extensions

Kadison¹

2020

Expositiones Mathematicae

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“…More recently in the early two thousands the idea of depth of a ring extension was further studied in the context of Galois coring structures [17], and to characterize structure properties involving self duality, Forbenius extensions and normality such as in [20], [6] and [22]. Moreover, fair amount of research regarding combinatorial aspects of finite group extensions has been done recently as well, we point out [3] and [2].…”

Section: Introductionmentioning

confidence: 97%

Minimum depth of double cross product extensions

Alvarado¹

2020

Preprint

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In this paper we explore minimum odd and minimum even depth subalgebra pairs in the context of double cross products of finite dimensional Hopf algebras. We start by defining factorization algebras and outline how subring depth in this context relates with the module depth of the regular left module representation of the given subalgebra. Next we study minimum odd depth for double cross product Hopf subalgebras and determine their value in terms of their related module depth, we conclude that minimum odd depth of Drinfel'd double Hopf subalgebras is 3. Finaly we produce a necessary and sufficient condition for depth 2 in double cross product Hopf subalgebra extensions. This sufficient condition is then used to prove results regarding minimum depth 2 in Drinfel'd double Hopf subalgebras, particularly in the case of finite Group Hopf algebras. Lastly we provide formulas for the centralizer of a normal Hopf subalgebra in a double cross product scenario.

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Depth One Extensions of Semisimple Algebras and Hopf Subalgebras

Cited by 2 publications

References 19 publications

Uniquely separable extensions

Uniquely separable extensions

Minimum depth of double cross product extensions

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