Quantum group notation

Question

I was jumping into the deep end and reading a few papers and lectures on quantum groups. My knowledge on Lie algebras is a bit thin but I was just wondering the notation used in the starting of this document:

One should have $U_\hbar(g) = U(g)[[\hbar]]$ as a vector space [...]

My question is, is the $\hbar$ the Cartan subalgebra $\mathfrak{h}$ I have been seeing in other papers, and what does the $U(g)[[\hbar]]$ mean precisely? Usually I see the quantum group notated as $U_q(g)$, so the sudden change in notation concerned me. Thanks in advance!

Answer 1

In this case $\hbar$ is just a parameter. Frequently, one also denotes $U_{\hbar}(g)$ as $U_q(g)$.

$U(g)[[\hbar]]$ denotes the space of formal power series over $U(g)$, i.e. if $f\in U(g)[[\hbar]]$, then $$f=\sum_{n\in\mathbb{Z}}a_n \hbar^n$$ with $a_n\in U(g)$ for every $n$. (Sometimes people reindex so that the sum is over $\mathbb{N}$ or some other countable set.)

Answer 2

Usually $\hbar$ is simply the reduced Planck constant $h/(2\pi)$.

I have sadly no idea why it is featured such prominently in your context.