Cocycle Deformations of Algebraic Identities and R-matrices

TL;DR

This paper introduces a general framework for cocycle deformations of algebraic identities and R-matrices, linking deformation theory with knot invariants through explicit examples and calculations.

Contribution

It provides a universal construction of 2-cocycle conditions for algebraic identities, connecting deformation theory to knot invariants and explicit algebraic examples.

Findings

Constructed a general 2-cocycle condition for algebraic identities

Applied the framework to canceling pairings and copairings

Discussed relations to the Kauffman bracket and knot invariants

Abstract

For an arbitrary identity L=R between compositions of maps L and R on tensors of vector spaces V, a general construction of a 2-cocycle condition is given. These 2-cocycles correspond to those obtained in deformation theories of algebras. The construction is applied to a canceling pairings and copairings, with explicit examples with calculations. Relations to the Kauffman bracket and knot invariants are discussed.