Abstract
This chapter introduces the basic notion of a differential algebra—an algebra equipped with a bimodule of 1-forms and an exterior derivative. Also the exterior algebra, cohomology, quantum metrics, Laplacians, and many of the key examples that will feature throughout the book. These include matrix algebras and enveloping algebras of Lie algebras as noncommutative spaces, group algebras, the algebra of functions on groups and more generally on graphs, the noncommutative torus and the fuzzy sphere. Some mathematical physics applications at the end cover electromagnetism on the group of permutations on three elements, and an interpretation of stochastic calculus.
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