Subring Depth, Frobenius Extensions, and Towers

Kadison, Lars

doi:10.1155/2012/254791

Cited by 11 publications

(39 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we define depth of algebras and coalgebras in tensor categories. When applied to algebras and coalgebras in a bimodule tensor category, this definition recovers minimum odd depth defined in [7] and h-depth defined in [30]. In particular, a coalgebra in bimodule tensor category is a coring, with depth defined in [16].…”

Section: Depth Of Algebras and Coalgebras In Tensor Categoriesmentioning

confidence: 88%

Algebra depth in tensor categories

Kadison¹

2016

Bull. Belg. Math. Soc. Simon Stevin

Self Cite

View full text Add to dashboard Cite

Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It is shown in a last section that the depth, Bratteli diagram and relative cyclic homology of algebra extensions are Morita invariants.1991 Mathematics Subject Classification. 16D20, 16D90, 16T05, 18D10, 20C05.

show abstract

Section: Depth Of Algebras and Coalgebras In Tensor Categoriesmentioning

confidence: 88%

Algebra depth in tensor categories

Kadison¹

2016

Bull. Belg. Math. Soc. Simon Stevin

Self Cite

View full text Add to dashboard Cite

show abstract

“…(1) for some n ∈ N [25]. This in turn is equivalent to there being f i ∈ Hom( B A B , B B B ) and r i ∈ A B such that id A = i f i (−)r i , the classical central projectivity condition [32].…”

Section: 2mentioning

confidence: 99%

“…for some positive integer q[4,24,25]. Note that if B ⊆ A has H-depth 2n − 1, the subalgebra has (left or right) depth 2n by restriction of modules.…”

mentioning

confidence: 99%

Uniquely separable extensions

Kadison¹

2020

Expositiones Mathematicae

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Proof. The hypothesis of surjectivity is equivalent to: A B (or B A) is a generator [34,Lemma 4.1]. We have elaborated on Miyashita's theory of Morita equivalence of ring extensions in [36,Section 5], where it is shown that Frobenius extension is an invariant notion of this equivalence.…”

Section: Core Hopf Ideals Of Hopf Subalgebrasmentioning

confidence: 99%

“…Remark 4.10. The viewpoint of interior algebras and induced algebra of Puig and [13] is related to the approach of the endomorphism ring tower of a Frobenius extension A ⊆ B with Frobenius homomorphism E : A → B and dual bases tensor [34,Section 4.1] for a complete set of tower equations). Then (*)…”

Section: Core Hopf Ideals Of Hopf Subalgebrasmentioning

confidence: 99%

An In-Depth Look at Quotient Modules

Kadison

2018

Algebr Represent Theor

Self Cite

View full text Add to dashboard Cite

The coset G-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization Q to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that Q has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of Q and its tensor powers in terms of annihilator ideals, sigma categories, trace ideals, Burnside ring formulas, and when dealing with semisimple Hopf algebras, the depth of Q in terms of the McKay quiver and the Green ring.1991 Mathematics Subject Classification. 16D20, 16D90, 16T05, 18D10, 20C05.

show abstract

Subring Depth, Frobenius Extensions, and Towers

Cited by 11 publications

References 25 publications

Algebra depth in tensor categories

Algebra depth in tensor categories

Uniquely separable extensions

An In-Depth Look at Quotient Modules

Contact Info

Product

Resources

About