2014/02/24 by Partha Sarathi Chakraborty, Satyajit Guin, Chakraborty, Partha Sarathi +1
Mathematics · #16E45 #46L87 #58B34 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1402.5735
openalex publication_date 2014/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the quadruples (A,\mathbbV,D,γ) where A is a unital, associative \mathbbK -algebra represented on the \mathbbK -vector space \mathbbV, D∈ End(\mathbbV), γ\inEnd(\mathbbV) is a ℤ2-grading operator which commutes with A and anticommutes with D. We prove that the collection of such quadruples, denoted by \widetildeSpec , is a monoidal category. We consider the monoidal subcategory \widetildeSpecsub of objects of \widetildeSpec for which γ∈π(A). We show that there is a covariant functor G:\widetildeSpec\longrightarrow\widetildeSpecsub . Let ΩD^\bullet be the differential graded algebra defined by Connes ([Con2]) and DGA denotes the category of differential graded algebras over the field \mathbbK . We show that F:\widetildeSpecsub\longrightarrow DGA , given by (A,\mathbbV,D,γ)\longmapstoΩD^\bullet(A), is a monoidal functor. To show that F\circG is not trivial we explicitly compute it for the cases of compact manifold and the noncommutative torus along with the associated cohomologies.