nLab quantized function algebra

A noncommutative algebra AA is called the quantum/quantized coordinate ring if it is a deformation of a (commutative) coordinate ring of some affine or, in graded case, projective variety. The expression quantized coordinate ring may be considered a synonym, but can also denote more generally a deformation of some other function algebra, say a C *C^\ast-algebra of functions on some compact Hausdorff topological space.

Most often these terms are used in quantum group theory. If GG is an algebraic group, then its coordinate ring is usually denoted by 𝒪(G)\mathcal{O}(G) or Fun(G)Fun(G) and is a Hopf algebra. There is a pairing between 𝒪(G)\mathcal{O}(G) and U(g)U(g) where gg is the Lie algebra of GG hence the consideration of the function algebra is essentially dual to the consideration of the universal enveloping algebra U(g)U(g) of gg. If qq is a deformation parameter, then there is a 1-parametric deformation 𝒪(G q)\mathcal{O}(G_q) (also denoted 𝒪 q(G)\mathcal{O}_q(G)) called the quantized function algebra on G qG_q (or quantized coordinate ring of G qG_q) which is also a Hopf algebra. One considers G qG_q as a quantum group; the approach via 𝒪(G q)\mathcal{O}(G_q) is dual to the approach via the quantized enveloping algebra U q(g)U_q(g).

Most well known approach to 𝒪(G q)\mathcal{O}(G_q) is the FRT construction

  • N. Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i analiz 1, 178 (1989) (Russian), English translation in Leningrad Math. J. 1.

and via the matrix elements of representations (using the analogue of the Peter-Weyl theorem)

  • A. Joseph, Quantum groups and their primitive ideals, Springer 1995.

Some special cases can be introduced via construction of universal coacting bialgebras, for example the quantum linear groups in the approach of

  • Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.

Last revised on June 12, 2023 at 11:55:28. See the history of this page for a list of all contributions to it.