113 research outputs found
The center of the affine nilTemperley-Lieb algebra
We give a description of the center of the affine nilTemperley-Lieb algebra
based on a certain grading of the algebra and on a faithful representation of
it on fermionic particle configurations. We present a normal form for
monomials, hence construct a basis of the algebra, and use this basis to show
that the affine nilTemperley-Lieb algebra is finitely generated over its
center. As an application, we obtain a natural embedding of the affine
nilTemperley-Lieb algebra on N generators into the affine nilTemperley-Lieb
algebra on N + 1 generators.Comment: 27 pages, 5 figures, comments welcom
Some Problems in the Representation Theory of Simple Modular Lie Algebras
The finite-dimensional restricted simple Lie algebras of characteristic p > 5
are classical or of Cartan type. The classical algebras are analogues of the
simple complex Lie algebras and have a well-advanced representation theory with
important connections to Kazhdan-Lusztig theory, quantum groups at roots of
unity, and the representation theory of algebraic groups. We survey progress
that has been made towards developing a representation theory for the
restricted simple Cartan-type Lie algebras, discuss comparable results in the
classical case, formulate a couple of conjectures, and pose a dozen open
problems for further study.Comment: References updated; a few minor changes made in this versio
The centroid of extended affine and root graded Lie algebras
We develop general results on centroids of Lie algebras and apply them to
determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody
Lie algebras, and Lie algebras graded by finite root systems.Comment: 35 page
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