Quantum enveloping algebra

Summary

The quantum enveloping algebra $U_q(\mathfrak{g})$ is also called the quantized universal enveloping algebra or Drinfeld–Jimbo quantum group. It is a one-parameter deformation of the universal enveloping algebra of a semisimple Lie algebra $\mathfrak{g}$. It is a Hopf algebra given by generators $E_i, F_i, K_i^{\pm 1}$ and relations read off from the Cartan matrix of $\mathfrak{g}$. For $q$ not a root of unity its finite-dimensional representations closely mirror those of $\mathfrak{g}$. It is the main example of a quantum group, and suitable versions of it are quasitriangular.