Anchor Maps and Stable Modules in Depth Two

Abstract

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.

Finite depth and Jacobson–Bourbaki correspondence

“…which are the total algebras of the left R-and V -bialgebroids in depth two theory [13][14][15]. Proof.…”

“…Depth two theory is a type of Galois theory for noncommutative ring extensions, where the Galois group in field theory is replaced by a Hopf algebroid (with or perhaps without antipode). The Galois theory of depth two ring extensions has been studied in a series of papers by the author [14,15,16,17] in collaboration with Nikshych [12,11], Szlachányi [13], and Külshammer [10], with a textbook treatment by Brzeziński and Wisbauer [2]. There are a number of issues that remain unexplored or unanswered in full including chirality [14,15], normality [10,16], a Galois inverse problem and a Galois correspondence problem [25].…”

Semisimple Hopf algebras and their depth two Hopf subalgebras

“…Let k be a field. A weak bialgebra W is a finite dimensional k-algebra and kcoalgebra (W, ∆, ε) such that the comultiplication ∆ : W → W ⊗ k W is linear and multiplicative, ∆(ab) = ∆(a)∆(b), and the counit is linear just as for bialgebras; however, one of the change from Hopf algebra is the weakening of the axioms that ∆ and ε be unital, ∆(1) = 1 ⊗ 1 and ε(1 W ) = 1 k , but must satisfy (12) 1…”

“…Ocneanu saw in the late eighties that especially depth two has extraordinary algebraic properties, which he phrased in terms of paragroups. A realization of this project in algebra occured in stages starting with Szymanski and others, and proceeding with a score of papers, of which [19,14,16,10,9,11,12] are somewhat representative. Critical input in the shaping of this algebraic theory came from results in Hopf-Galois extensions in the eighties and early nineties, the development of weak Hopf algebras and Hopf algebroids in the mid-to late nineties (and the change of definition of antipode coming from consideration of depth two Frobenius extension by Böhm-Szlachanyi).…”

Depth two Hopf subalgebras of a semisimple Hopf algebra

“…Ocneanu saw in the late eighties that especially depth two has extraordinary algebraic properties, which he phrased in terms of paragroups. A realization of this project in algebra occured in stages starting with Szymanski and others, and proceeding with a score of papers, of which [9][10][11][12]14,16,23] are somewhat representative. Critical input in the shaping of this algebraic theory came from results in Hopf-Galois extensions in the eighties and early nineties, the development of weak Hopf algebras and Hopf algebroids in the mid-to late nineties and a change in the definition of antipode coming from consideration of depth two Frobenius extension by Böhm-Szlachanyi.…”