Hochschild cohomology

Hochschild cohomology is a mathematical tool used to study how algebraic structures like rings or algebras can be deformed or changed gradually. It captures information about the symmetries, extensions, and possible "twists" of these structures, providing insight into their internal properties. Think of it as a way to measure the ways an algebra can be subtly altered without losing its fundamental nature, revealing deep connections between algebraic operations and their higher-order relationships. It has applications in deformation theory, algebraic geometry, and mathematical physics.