Depth Two, Normality, and a Trace Ideal Condition for Frobenius Extensions

Kadison, Lars; Külshammer, Burkhard

doi:10.1080/00927870600650291

Cited by 29 publications

(54 citation statements)

References 16 publications

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“…Examples of D2 extensions include centrally projective algebra extensions (such as finite dimensional or finite projective algebras), H-separable extensions as well as pseudo-, group-, and Hopf-Galois extensions. For example, if A and B are the complex group algebras corresponding to a subgroup H < G of a finite group, then A | B is D2 if and only if H is a normal subgroup in G [11]. Also finitely generated weak Hopf-Galois extensions are left and right D2 [10].…”

Section: Preliminaries On D2 Extensionsmentioning

confidence: 99%

The endomorphism ring theorem for Galois and depth two extensions

Kadison

2006

Journal of Algebra

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Let S be the left bialgebroid End B A B over the centralizer R of a right depth two (D2) algebra extension A | B, which is to say that its tensor-square is isomorphic as A-B-bimodules to a direct summand of a finite direct sum of A with itself. We prove that its left endomorphism algebra is a left S-Galois extension of A op . As a corollary, endomorphism ring theorems for D2 and Galois extensions are derived from the D2 characterization of Galois extension. We note the converse that a Frobenius extension satisfying a generator condition is D2 if its endomorphism algebra extension is D2.

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Section: Preliminaries On D2 Extensionsmentioning

confidence: 99%

The endomorphism ring theorem for Galois and depth two extensions

Kadison

2006

Journal of Algebra

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“…S and T. Thus the notion of depth two algebra extension recovers classical depth two for subfactors ( [17,19] for further details). Below we examine a pairing between S and T that becomes [19, 8.9] the Szymaǹski nondegenerate pairing of C A1 (B) and C A2 (A) in [18, (14)], which transfers the algebra structure of one centralizer to a coalgebra structure on the other when R is trivial.…”

Section: Preliminariesmentioning

confidence: 92%

“…The counit is given by ε(t) = μ(t)(1) = t 1 t 2 . We note that the representation μ is the module algebra R T and studied in [10,17] as a generalized Miyashta-Ulbrich action.…”

Section: Bialgebroids In Terms Of Anchor Mapsmentioning

confidence: 99%

Anchor Maps and Stable Modules in Depth Two

Kadison

2007

Appl Categor Struct

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An algebra extension A | B is right depth two if its tensor-square A ⊗ B A is in the Dress category Add A A B . We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids S = End B A B and T = End A A ⊗ B A A over the centralizer R = C A (B) are the modules S R and R T studied in Kadison (J. Alg. & Appl., 2005, preprint), Kadison (Contemp. Math., 391: 149-156, 2005), and Kadison and Külshammer (Commun. Algebra, 34: 3103-3122, 2006), which provide information about the bialgebroids and the extension (Kadison, Bull. Belg. Math. Soc. Simon Stevin, 12: 275-293, 2005). The anchor maps for the Hopf algebroids in Khalkhali and Rangipour (Lett. Math. Phys., 70: 259-272, 2004) and Kadison (2005, preprint) reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in Van Oystaeyen and Panaite (Appl. Categ. Struct., 2006, in press). We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in Kadison (Proc. Am. Math. Soc., 131: 2993-3002, 2003 Prop. 1.1) to any A-module. We observe that Schneider's coGalois theory in Schneider (Isr. J. Math., 72: 167-195, 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.

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“…For instance, even in the case of group algebra extensions these conditions cannot be translated in only group theoretical terms. Although the depth two subgroups are precisely the normal subgroups [8], a similar description is not yet known for any higher depth. For example, in [4] a sufficient condition for depth three was given but it is not yet known if that condition is necessary.…”

Section: Introductionmentioning

confidence: 98%

Subgroups of odd depth—a necessary condition

Burciu

2013

Czech Math J

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Depth Two, Normality, and a Trace Ideal Condition for Frobenius Extensions

Cited by 29 publications

References 16 publications

The endomorphism ring theorem for Galois and depth two extensions

The endomorphism ring theorem for Galois and depth two extensions

Anchor Maps and Stable Modules in Depth Two

Subgroups of odd depth—a necessary condition

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