  We give a selfcontained introduction to the theory of quantum groups
according to Drinfeld highlighting the formal aspects as well as the
applications to the Yang-Baxter equation and representation theory.
Introductions to Hopf algebras, Poisson structures and deformation quantization
are also provided. After having defined Poisson-Lie groups we study their
relation to Lie-bi algebras and the classical Yang-Baxter equation. Then we
explain in detail the concept of quantization for them. As an example the
quantization of $sl_2$ is explicitly carried out. Next we show how quantum
groups are related to the Yang-Baxter equation and how they can be used to
solve it. Using the quantum double construction we explicitly construct the
universal $R$-matrix for the quantum $sl_2$ algebra. In the last section we
deduce all finite dimensional irreducible representations for $q$ a root of
unity. We also give their tensor product decomposition (fusion rules) which is
relevant to conformal field theory.
