Parametrizations of infinite biconvex sets in affine root systems

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dc.creatorIto, Ken
dc.date1999-11-26
dc.date2003-02-19
dc.date.accessioned2026-07-07T05:31:57Z
dc.date.available2026-07-07T05:31:57Z
dc.descriptionWe investigate in detail relationships between the set ${\mathfrak B}^\infty$ of all infinite ``biconvex'' sets in the positive root system $Δ_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and the set ${\mathcal W}^\infty$ of all infinite ``reduced word'' of the Weyl group of ${\mathfrak g}$. The study is applied to the classification of ``convex orders'' on $Δ_+$ (cf. \cite{kI}), which are indispensable to construct ``convex bases'' of Poincaré-Birkhoff-Witt type of the upper triangular subalgebra $U_q^+$ of the quantized universal enveloping algebra $U_q({\mathfrak g})$. We construct a set $\boldsymbol{\mathcal P}$ by using data of the underlying finite-dimensional simple Lie algebra, and bijective mappings $\nabla\colon\boldsymbol{\mathcal P}\to{\mathfrak B}^\infty$ and $χ\colon\boldsymbol{\mathcal P}\to W^\infty$ such that $\nabla=\varPhi^\infty\circχ$, where $W^\infty$ is an quotient set of ${\mathcal W}^\infty$ and $\varPhi^\infty\colon W^\infty\to{\mathfrak B}^\infty$ is a natural injective mapping.
dc.descriptionLaTeX2e, 21 pages
dc.identifierhttps://arxiv.org/abs/math/9911214
dc.identifierhttp://arxiv.org/abs/math/9911214
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/79488
dc.subjectQuantum Algebra
dc.subjectRepresentation Theory
dc.subject17B37; 17B67
dc.titleParametrizations of infinite biconvex sets in affine root systems
dc.typetext

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