Gerstenhaber-Schack Bialgebras

2024/01/31 by Ronald Umble, Umble, Ronald
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2401.17771

Abstract

A *Gerstenhaber-Schack (G-S) bialgebra* consists of a graded Hopf algebra H together with multilinear operations \ω¹₃,ω²₂,ω³₁\⊂ \Hom^-1(H^{⊗ m},H^{⊗ n}): m+n=4\, whose sum is the degree -1 component of a 2-cocycle in the G-S complex of H. A *G-S extension* of a graded Hopf algebra H is a G-S bialgebra containing H. G-S extensions of H are classified up to isomorphism by the degree -1 component of the G-S cohomology group H_GS²(H;H). We exhibit a space X and a non-trivial topologically induced G-S bialgebra structure on H^∗( ΩX;ℤ₂) .

Related

• G-Structure on the cohomology of Hopf algebras
• Cyclic Gerstenhaber-Schack cohomology
• Gerstenhaber-Schack Cohomology for Hom-bialgebras and Deformations
• Gerstenhaber bracket on the Hochschild cohomology via an arbitrary\n resolution
• Higher structures on homology groups