Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1044.17009
Leclerc, B.
Imaginary vectors in the dual canonical basis of $U_q({\germ n})$. (English)
[J] Transform. Groups 8, No. 1, 95-104 (2003). ISSN 1083-4362

If $\frak{g}$ is a simple Lie algebra over $\Bbb C$, $\germ{n}$ a maximal nilpotent subalgebra of $\frak{g}$, let $\bold B$ be the canonical basis of $U_q(\frak{n})$ and ${\bold B}^{\ast}$ the dual basis with respect to the natural scalar product in $U_q(\frak{n})$. Berenstein and Zelevinsky had conjectured that the product $b_1b_2$ is of the form $q^mb$, for $b_1,b_2,b\in{\bold B}^{\ast}$ if and only if $b_1$ and $b_2$ $q$-commute. This would imply that $b_1^2$ is always of the form $q^mb$, for $b_1 \in {\bold B}^{\ast}$. Such vectors are called {\it real}, otherwise they are called {\it imaginary}. The paper shows that there are imaginary vectors except when $\frak{g}$ if of type $A_1,A_2,A_3,A_4,B_2$. The author uses this to exhibit an explicit irreducible representation $V$ for $U_q(\hat{sl}_N)$ such that $V\otimes V$ is not irreducible.
[Stefano Capparelli (Roma)]

MSC 2000:: *17B37 Quantum groups and related deformations

Keywords: canonical basis

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]