Notes for AKT-170317/0:10:47

Here's a way to think about quantum groups: they are integrable quantum mechanical systems that live on Lie groups. To make sense of this let's first upgrade our 4d Lie algebra $g_0$ to a Lie group, call it $G_0$. In $GL_3$ it is the group of upper triangular matrices with ones on the top left and bottom right. I'm not sure it was mentioned in the course but $g_0$ is not just a Lie algebra, it is also a Lie bialgebra. (it has a compatible bracket on the dual the trivial one in the case of $g_0$). Translating this to $G_0$ makes $G_0$ a Poisson manifold. That means there is a Poisson bracket on the space of functions $F(G_0)$. Given such a Poisson bracket {.,.} and a Hamiltonian function $H\in F(G_0)$ is enough to write the equations of motion on $G_0$. They are $df/dt = {f,H}$ With all this in place we can first remark the fact that our r-matrix is closely related to the bialgebra structure and CYBE to be a condition for integrability of the classical mechanical system. Next we may quantize with respect to the Poisson structure to obtain a deformation of the algebra of functions $F_h(G_0)$. We would then like to think of this non-commutative algebra as the algebra of functions on the (non-existent) quantum group. Integrability survives the quantization and is now expressed in terms of the R-matrix satisfying Yang-Baxter. Dually one may also consider universal enveloping algebra and its deformation as a Hopf algebra.

Quantizing often seems a little ad-hoc but Kontsevich gave a general procedure for (deformation) quantizing with respect to any Poisson structure. [1]

A reference for such things would be A guide to quantum groups by Chari and Pressley. Roland