CATEGORICAL ASPECTS OF VIRTUALITY AND SELF-DISTRIBUTIVITY
CATEGORICAL ASPECTS OF VIRTUALITY AND SELF-DISTRIBUTIVITY
复制标题
DOI：
10.1142/s0218216513500454
复制
发表时间：
2013-09
影响因子：
0.5
通讯作者：
V. Lebed
中科院分区：
文献类型：
--
作者：
V. Lebed
本文围绕两个主要结果。首先，我们根据对称类别（SC）中的“局部”编织对象提出虚拟辫子组和正虚拟辫子幺半群的“hom-set”类型分类。这种“双编织”方法提供了丰富的表示来源，并为虚拟机架和扭曲的 Burau 表示提供了自然的分类解释。其次，我们在任意 SC 中定义自分配（SD）结构。 SD 结构被证明可以在 SC 中产生编织物体。例如，我们将 SC 中的结合性和雅可比恒等式解释为广义自分配性，从而赋予结合性和莱布尼茨代数以（预）编织。分类 SD 结构的同源理论是使用 [Lebed，通过编织和量子洗牌的代数结构同源性，出现在《代数杂志》中的“编织”技术开发的，概括了机架、杆、莱布尼兹和其他熟悉的复合体。
This paper revolves around two main results. First, we propose a "hom-set" type categorification of virtual braid groups and positive virtual braid monoids in terms of "locally" braided objects in a symmetric category (SC). This "double braiding" approach provides a rich source of representations, and offers a natural categorical interpretation for virtual racks and for the twisted Burau representation. Second, we define self-distributive (SD) structures in an arbitrary SC. SD structures are shown to produce braided objects in a SC. As for examples, we interpret the associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a (pre-)braiding. A homology theory of categorical SD structures is developed using the "braided" techniques from [Lebed, Homologies of algebraic structures via braidings and quantum shuffles, to appear in J. Algebra], generalizing rack, bar, Leibniz and other familiar complexes.