nLab quantum algebra (Rev #1, changes)

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Idea

Quantum algebra is an overloaded term with several related meanings and usages really depend on an author.

Quantum algebras as a class of algebras

For many people, a quantum algebra is a synonym for a quantum group or some related algebra (e.g. a deformed oscillator algebra, quantum homogeneous space, quantum Schubert variety, Yangian, quantum nilpotent algebra etc.). One usually assumes that quantum algebras are formal deformations of (typically finitely generated) commutative algebras or other standard algebras like the Weyl and Clifford algebras. Often operator algebras of similar nature are also viewed as quantum algebras, at least in the community studying operator algebras.

Large noncommutative algebras (say free associative algebras and alike) are typically not called quantum. Nevertheless, few authors view any noncommutative associative algebra as quantum.

Quantum algebras in quantizations

Often a quantization of some classical algebra of observables, or of a related algebra of functions in classical mechanics or field theory is called the algebra of quantum observables or, somewhat more informally, the quantum algebra of observables (or an appropriate variant).

Quantum algebras as a subject

By “quantum algebras” one often means a wider subject which does not only include the study of quantum algebras in the above sense but also related mathematics, e.g. noncommutative Hopf algebras in general, associative bialgebroids, braided monoidal categories, low dimensional topological invariants (see quantum topology) like the knot and link invariants which stem say from research on R-matrices and related quantum algebras. It is recognized as one of the subject classes at arXiv.