2013/09/13 by Hossein Abbaspour, Abbaspour, Hossein
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA) #math.AT #math.KT #math.QA
paper · pdf · doi:10.48550/arxiv.1309.3384
Many corrections have been made and the signs are now given explicily
arxiv created 2015/06/27 · arxiv updated 2015/06/30
We prove that the shifted Hochschild chain complex C_*(A,A)[m] of a symmetric open Frobenius algebra A of degree m has a natural homotopy coBV-algebra structure. As a consequence HH_*(A,A)[m] and HH^*(A,A^\vee)[-m] are respectively coBV and BV algebras. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural homotopy BV-algebra structure on C_*(A,A)[m] hence a BV-structure on HH_*(A,A)[m]. Moreover we prove that the product and coproduct on HH_*(A,A)[m] satisfy the Frobenius compatibility condition i.e. HH_*(A,A)[m] is an open Frobenius algebras. If A is commutative, we also introduce a natural BV structure on the shifted relative Hochschild homology \widetildeHH_*(A)[m-1]. We conjecture that the product of this BV structure is identical to the Goresky-Hingston\citeGH product on the cohomology of free loop spaces when A is a commutative cochain algebra model for M.