2007/05/15 by Apoorva Khare, Khare, Apoorva
Mathematics · #16S40 #16W30 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16S40 #msc:16W30
paper · pdf · doi:10.48550/arxiv.0705.2067
This paper has been withdrawn, and replaced by arXiv:1601.04775, which supersedes and strengthens this paper
openalex publication_date 2007/05/15 · arxiv created 2016/01/20 · arxiv updated 2016/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have families of deformations that have been studied widely in the literature; examples include symplectic reflection algebras and infinitesimal Hecke algebras. We introduce a family of deformations of these smash product algebras for general A, and characterize the PBW property. We then characterize the Jacobi identity for "grouplike" algebras (that include group rings and the nilCoxeter algebra), and precisely identify the PBW deformations in the example where A is the nilCoxeter algebra. We end with the more prominent case - where A is a Hopf algebra. We show the equivalence of several versions of the "deformed" relations in the smash product, and identify the PBW deformations which are Hopf algebras as well.