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Search: MSC category 17B10 ( Representations, algebraic theory (weights) )

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1. CMB Online first

Bavula, V. V.; Lu, T.
Classification of simple weight modules over the Schrödinger algebra
A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple $C_{\mathcal{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $\mathcal{S}$ (over the centre). It is proved that some (prime) factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull minimal.

Keywords:weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra
Categories:17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50

2. CMB 2014 (vol 57 pp. 735)

Cagliero, Leandro; Szechtman, Fernando
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

3. CMB 2007 (vol 50 pp. 603)

Penkov, Ivan; Zuckerman, Gregg
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

4. CMB 2005 (vol 48 pp. 587)

Lopes, Samuel A.
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

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