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Extension, deformation and categorification of $\text{AssDer}$ pairs

Apurba Das, Ashis Mandal

Abstract

In this paper, we consider associative algebras equipped with derivations. A pair consisting of an associative algebra and a distinguished derivation is called an AssDer pair. We study central extensions and formal one-parameter deformations of AssDer pairs in terms of cohomology. Finally, we define $2$-derivations on associative $2$-algebras and show that the category of associative $2$-algebras with $2$-derivations is equivalent to the category of $2$-term $A_\infty$-algebras with homotopy derivations.

Extension, deformation and categorification of $\text{AssDer}$ pairs

Abstract

In this paper, we consider associative algebras equipped with derivations. A pair consisting of an associative algebra and a distinguished derivation is called an AssDer pair. We study central extensions and formal one-parameter deformations of AssDer pairs in terms of cohomology. Finally, we define -derivations on associative -algebras and show that the category of associative -algebras with -derivations is equivalent to the category of -term -algebras with homotopy derivations.

Paper Structure

This paper contains 11 sections, 29 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. AssDer pairs
  3. Representations and cohomology of AssDer pairs
  4. Relation with LieDer pairs
  5. Central extensions of AssDer pairs
  6. Extensions of a pair of derivations
  7. Deformations
  8. Extensions of finite order deformation
  9. Automorphisms of the deformed AssDer pair
  10. Homotopy derivations on $2$-term $A_\infty$-algebras
  11. Categorification of AssDer pairs

Key Result

Proposition 2.2

Let $A$ be an unital, commutative associative algebra. There is a one-to-one correspondence between derivations on $A$ and Lie-Rinehart algebra structures on $(A, A)$.

Theorems & Definitions (69)

  • Example 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 59 more