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1. CMB Online first
Affine actions of $U_q(sl(2))$ on polynomial rings We classify the affine actions of $U_q(sl(2))$ on commutative
polynomial rings in $m \ge 1$ variables.
We show that, up to scalar multiplication, there are two possible
actions.
In addition, for each action, the subring of invariants is a
polynomial ring in either $m$ or $m-1$ variables,
depending upon whether $q$ is or is not a root of $1$.
Keywords:skew derivation, quantum group, invariants Categories:16T20, 17B37, 20G42 |
2. CMB 2005 (vol 48 pp. 587)
Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$ 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.
Categories:17B37, 16W35, 17B10, 16D60 |
3. CMB 2000 (vol 43 pp. 79)
Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of
cyclotomic Schur algebras and of cyclotomic Hecke algebras, over
discrete valuation rings.
Categories:20G05, 20C20, 16G30, 17B37, 57M25 |
4. CMB 1997 (vol 40 pp. 143)
Quantum deformations of simple Lie algebras 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 $q^2$.
Categories:17B37, 17A01 |