symmetric monoidal (∞,1)-category of spectra
“Quantum algebra” is an overloaded term with several related meanings whose usage depends on the given author.
For many people, quantum algebra is a synonym for 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 or Clifford algebras. Often operator algebras of similar nature are also viewed as quantum algebras, at least in the respective community.
Large noncommutative algebras (say free associative algebras and alike) are typically not called quantum. Nevertheless, few authors view any noncommutative associative algebra as quantum.
Often a quantization of some classical algebra of observables, or of a related algebra of functions in classical mechanics or classical field theory is called the algebra of quantum observables or, somewhat more informally, the quantum algebra of observables (or an appropriate variant).
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.
Last revised on June 29, 2024 at 09:34:07. See the history of this page for a list of all contributions to it.