nLab quantized function algebra

Description

A noncommutative algebra $A$ 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 $C^\ast$-algebra of functions on some compact Hausdorff topological space. Other good properties, e.g. being a Hopf algebra are often required.

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

Most well known approach to $\mathcal{O}(G_q)$ 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)

• Anthony 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.

category: algebra