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1. CMB Online first
On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras |
On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras We describe of all finite
dimensional uniserial representations of a commutative associative
(resp. abelian Lie) algebra over a perfect (resp. sufficiently
large perfect) field. In the Lie case the size of the field
depends on the answer to following question, considered and solved
in this paper. Let $K/F$ be a finite separable field extension
and
let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some
non-zero elements $\alpha,\beta\in F$?
Keywords:uniserial module, Lie algebra, associative algebra, primitive element Categories:17B10, 13C05, 12F10, 12E20 |
2. CMB 2007 (vol 50 pp. 603)
Construction of Generalized Harish-Chandra Modules with Arbitrary Minimal $\mathfrak k$-Type Let $\mathfrak g$ be a semisimple complex Lie algebra and $\k\subset\g$ be
any algebraic subalgebra reductive in $\mathfrak g$. For any simple
finite dimensional $\mathfrak k$-module $V$, we construct simple
$(\mathfrak g,\mathfrak k)$-modules $M$ with finite dimensional $\mathfrak k$-isotypic
components such that $V$ is a $\mathfrak k$-submodule of $M$ and the Vogan
norm of any simple $\k$-submodule $V'\subset M, V'\not\simeq V$, is
greater than the Vogan norm of $V$. The $(\mathfrak g,\mathfrak k)$-modules
$M$ are subquotients of the fundamental series of
$(\mathfrak g,\mathfrak k)$-modules.
Categories:17B10, 17B55 |
3. 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 |