Skew Hopf algebras, irreducible extensions and the pi-method

2007/01/15 by Lars Kadison, Kadison, Lars · 1 citation
Mathematics · #13B05 #16W30 #46L37 #81R15 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.math/0701427

openalex publication_date 2007/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To a depth two extension A | B, we associate the dual bialgebroids S := \End _BA_B and T := (A ø_B A)^B over the centralizer R=C_A(B). In the set-up where R is a subalgebra of B, which is quite common, two nondegenerate pairings of S and T will define an anti-automorphism τof the algebra S. Making use of a two-sided depth two structure, we prove that τis an antipode and S is a Hopf algebroid of a type we call skew Hopf algebra. A final section discusses how τand the nondegenerate pairings generalize to modules via the pi-method for depth two, and a certain derived mapping of cochain complexes is nullhomotopic.

Cited by

• Regular multiplier Hopf algebroids II. Integration on and duality of\n algebraic quantum groupoids

Related

• Dual Bialgebroids for Depth Two Ring Extensions
• Hopf algebroids and Galois extensions
• Codepth Two and Related Topics
• Anchor maps and stable modules in depth two
• Normal Hopf subalgebras, depth two and Galois extensions