Combinatorics and Representation Theory

本文主要研究组合结构上的李代数、李群、李超代数、量子代数、Hopf代数和算子的结合代数的表示，具体内容包括：（1）下-上代数（推广了偏序集、图、结合模式等上的下和上算子的代数）; (2)通勤行动;（3）有限根系阶化的李代数;（4）与李代数和Jordan代数有关的量子结构;（5）晶体基;（6）Hochschild上同调和Clifford理论。研究这些学科的动力大部分来自于它们与组合学、不变量理论、群论、环理论和数学中的可解晶格模型以及自旋模型、孤子方程、物理学中的共形场论。这个提议试图理解某些偏序集合（集合中的一个对象可能小于另一个对象，或者对象可能不相关）。向下运算符将对象映射到其正下方的对象，向上运算符将其映射到其正上方的对象。此类运算符在以下操作中起着重要作用：物理学，他们称之为湮灭和创造算子。调查将集中在这些算子的组合和代数性质以及它们的各种推广。 也要研究对称性和它们的相关数学。 这是一个对粒子物理学、化学晶体研究和数学研究产生巨大影响的领域，自从数学家Issai Schur和物理学家Hermann Weyl的开创性工作以来。 目前的许多活动和许多悬而未决的问题证明了它的持续活力。 例如，晶体基底对应于物理学中的绝对零度温度不可解晶格模型。这样的基地有显着的属性。 一个典型的问题是当对称算子表现出一定的行为时如何构造这样的基。

9970119The primary focus of this proposal is on the representations of Liealgebras, Lie groups, Lie superalgebras, quantum envelopingalgebras, Hopf algebras, and associativealgebras of operators on combinatorial structures.The specific topics to be investigated are (1) down-up algebras (which generalize the algebrascoming from the down and up operators on partially ordered sets, graphs, association schemes etc.); (2) commuting actions; (3) Lie algebrasgraded by finite root systems; (4) quantum structuresrelated to Lie and Jordan algebras; (5) crystal bases; (6) Hochschild cohomology and Clifford theory. Much of the impetus to study these subjectscomes from their close connections with combinatorics, invariant theory, group theory, ring theory, and solvable lattice models in mathematics, and spin models,soliton equations, and conformal field theory in physics.This proposal seeks to understand certain setshaving a partial order (one object in the set may be less than another or the objects may not be related).The down operator maps an object to the ones directly below it and the up operator maps it to the ones directly above it.Such operators play an important role in physics,where they termed annihilation and creation operators.The investigations will focus on the combinatoricsand algebraic properties of these operators and variousgeneralizations of them. Also to be studied are symmetries and their related mathematics. This is an area that has had an enormous impact on particle physics, the study of crystals in chemistry, and mathematical research ever since the pioneering work of mathematician Issai Schur and physicist Hermann Weyl. Its continuing vitality is evidenced by much current activity and many open problems. For example, crystalbases correspond to absolute zero temperature insolvable lattice models in physics. Such baseshave remarkable properties. One typical problem to be addressed is to construct such baseswhen the symmetry operators exhibit certain behavior.