论文标题
循环编织组的Hecke代数的概括
Generalisations of Hecke algebras from Loop Braid Groups
论文作者
论文摘要
我们介绍了由Loop Braid Group $ LB_N $和BURAU代表的扩展名，介绍了普通Hecke代数的概括$ LH_N $。普通的Hecke代数具有许多显着的算术和表示理论特性，以及许多应用。我们表明$ lh_n $具有其中几个属性的类似物。特别是我们介绍的是考虑从（非函数）burau代表的序列和（函数）deguchi {\ em et al} -kauffman-saleur-saleur-saleur-rittenberg表示Burau-Rittenberg表示的一类局部（张量空间/函子）表示。在其最超对称的情况下，发生异常的神秘取消，因此Burau-Rittenberg的代表性扩展到循环的Burau-Rittenberg表示。这是通过$ lh_n $的因素。令$ sp_n $表示相应的商代数，$ k $ the Ground Ring和$ t \ in K $ the loop-hecke参数。我们证明了以下内容： 1）$ lh_n $在一个字段上是有限维度。 2）天然包含$ lb_n \ rightArrow lb_ {n+1} $传递给包含$ sp_n \ rightarrow sp_ {n+1} $。 3）超过$ k = \ mathbb {c} $，$ sp_n / rad $通常是帕斯卡（Pascal）三角形给出的简单矩阵代数（和bratteli图）的简单矩阵代数的总和。 4）我们确定$ sp_n $表示理论的其他基本不变性：cartan分解矩阵；和Quiver，是A型。 5）$ sp_n $的结构独立于参数$ t $，除了$ t = 1 $。 \对于$ t^2 \ neq 1 $ then $ lh_n \ cong sp_n $至少至少排名$ n = 7 $（对于$ t = -1 $，它们不是同构的，对于$ n> 2 $；对于$ t = 1 $，它们不是同构成$ n> 1 $）。 最后，我们讨论了拓扑，表示理论和组合学中这种结构引起的其他一些有趣的观点。
We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we %introduce consider a class of local (tensor space/functor) representations of the braid group derived from a meld of the (non-functor) Burau representation and the (functor) Deguchi {\em et al}-Kauffman--Saleur-Rittenberg representations here called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t \in k$ the loop-Hecke parameter. We prove the following: 1) $LH_n$ is finite dimensional over a field. 2) The natural inclusion $LB_n \rightarrow LB_{n+1}$ passes to an inclusion $SP_n \rightarrow SP_{n+1}$. 3) Over $k=\mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. 4) We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. 5) The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. \item For $t^2 \neq 1$ then $LH_n \cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.
