Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping algebra $U_{q}(\SL)$. Using results of Alev--Dumas and Caldero related to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra $U(\g)$ of a complex semisimple Lie algebra $\g$, and also of an analogous result of Joseph--Letzter for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to its representation theory is the fact that $\U{+}$ is free over a larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$ to $\U{+}$ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

It is shown that every simple complex Lie algebra $\fg$ admits a 1-parameter family $\fg_q$ of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for $U_q(\fg)$-modules; here $U_q(\fg)$ is the quantized enveloping algebra of $\fg$. From this it follows that the multiplication on $\fg_q$ is $U_q(\fg)$-invariant. In the special case $\fg = {\ss}(2)$, the structure constants for the deformation ${\ss}(2)_q$ are obtained from the quantum Clebsch-Gordan formula applied to $V(2)_q \otimes V(2)_q$; here $V(2)_q$ is the simple 3-dimensional $U_q\bigl({\ss}(2)\bigr)$-module of highest weight $q^2$.