Left-symmetric algebras, or pre-Lie algebras in geometry and physics
Left-symmetric algebras, or pre-Lie algebras in geometry and physics
复制标题
DOI：
10.2478/s11533-006-0014-9
复制
发表时间：
2005-09
影响因子：
--
通讯作者：
D. Burde
中科院分区：
文献类型：
--
作者：
D. Burde
本文讨论了左对称代数在几何学和物理学中的起源、理论和应用。最近，Connes， Kreimer和Kontsevich在数学物理（QFT和重整化理论）中引入了LSAs，其中pre-Lie代数的名称经常被使用。Cayley早在一百多年前就写过这样的代数。事实上，lsa出现在数学和物理的许多不同领域。我们试图对lsa发挥重要作用的领域进行调查。进一步研究了lsa的代数理论，如结构理论、自由基理论、上同调理论和简单lsa的分类。我们还讨论了忠实李代数表示的应用。
In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.