Article
Factorized solutions of Temperley-Lieb qKZ equations on a segment
Advances in Theoretical and Mathematical Physics. 2010. Vol. 14. No. 3. P. 795-877.
We study the q-deformed Knizhnik–Zamolodchikov (qKZ) equation in
path representations of the Temperley–Lieb algebras. We consider two
types of open boundary conditions, and in both cases we derive factorized
expressions for the solutions of the qKZ equation in terms of Baxterized
Demazurre–Lusztig operators. These expressions are alternative to
known integral solutions for tensor product representations. The factorized
expressions reveal the algebraic structure within the qKZ equation,
and effectively reduce it to a set of truncation conditions on a single scalar
function. The factorized expressions allow for an efficient computation
of the full solution once this single scalar function is known. We further
study particular polynomial solutions for which certain additional
factorized expressions give weighted sums over components of the solution.
In the homogeneous limit, we formulate positivity conjectures in the spirit of Di Francesco and Zinn-Justin. We further conjecture relations
between weighted sums and individual components of the solutions for
larger system sizes.
Research target:
Philosophy, Ethics, and Religious Studies
Language:
English
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