Anchor maps and stable modules in depth two. (English) Zbl 1170.16030
Summary: An algebra extension \(A\mid B\) is right depth two if its tensor-square \(A\otimes_BA\) is in the Dress category \(\mathbf{Add}{_AA_B}\). 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 \(S=\text{End}{_BA_B}\) and \(T=\text{End}{_AA \otimes_BA_A}\) over the centralizer \(R=C_A(B)\) are the modules \(_SR\) and \(R_T\) studied by L. Kadison [J. Algebra Appl. 6, No. 3, 505-526 (2007; Zbl 1140.16017), Contemp. Math. 391, 149-156 (2005; Zbl 1102.16023)], and L. Kadison and B. Külshammer [Commun. Algebra 34, No. 9, 3103-3122 (2006; Zbl 1115.16020)], which provide information about the bialgebroids and the extension [L. Kadison, Bull. Belg. Math. Soc. - Simon Stevin 12, No. 2, 275-293 (2005; Zbl 1105.16037)]. The anchor maps for the Hopf algebroids of M. Khalkhali and B. Rangipour [Lett. Math. Phys. 70, No. 3, 259-272 (2004; Zbl 1067.58007)] and L. Kadison [Trends in Mathematics, 247-264 (2008; Zbl 1167.16033)] reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism of F. Van Oystaeyen and F. Panaite [Appl. Categ. Struct. 14, No. 5-6, 627-632 (2006; Zbl 1107.16039)]. We sketch a theory of stable \(A\)-modules and their endomorphism rings and generalize the smash product decomposition of L. Kadison [Proc. Am. Math. Soc. 131, No. 10, 2993-3002 (2003; Zbl 1033.16017), Proposition 1.1] to any \(A\)-module. We observe that H.-J. Schneider’s coGalois theory [in Isr. J. Math. 72, No. 1/2, 167-195 (1990; Zbl 0731.16027)] 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.
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 16D90 | Module categories in associative algebras |
| 16L60 | Quasi-Frobenius rings |
| 16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
| 13B05 | Galois theory and commutative ring extensions |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
depth two extensions; anchor maps; stable modules; codepth two coextensions; algebra extensions; finite dimensional algebras; right bialgebroids; Hopf algebroids; Hopf algebras; endomorphism rings; smash products; coalgebrasCitations:
Zbl 1140.16017; Zbl 1102.16023; Zbl 1115.16020; Zbl 1105.16037; Zbl 1067.58007; Zbl 1167.16033; Zbl 1107.16039; Zbl 1033.16017; Zbl 0731.16027References:
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