Transgressive loop group extensions

李群扩张的路群构造

The path group construction of Lie group extensions

通过推广Mickelsson对loop群的处理方法和Losev-Moore-Nekrasov-Shatashvili对current群的处理方法，给出了利用路群的交换扩张实现无限维李群的交换扩张的一个显式方法.我们将我们的方法应用到当前李代数上的耦合上圈和发散自由向量场的李代数上的Lichnerowicz上圈。

A loop group extension of the odd Chern character

We show that the universal odd Chern form, defined on the stable unitary group $U$, extends to the loop group $LU$ in a way that is closed with respect to an equivariant-type differential. This provides an odd analogue to the Bismut-Chern form. We also describe the associated transgression form, the so-called Bismut-Chern-Simons form, and explicate some properties it inherits as a differential form on the space of maps of a cylinder into the stable unitary group. As a corollary, we obtain the Chern character homomorphism from odd K-theory to the periodic cohomology of the free loop space, repr…

环积与环余积的推广、对径路径空间及闭测地线的共振

Extensions of the loop product and coproduct, the space of antipodal paths and resonances of closed geodesics

本文推广弦拓扑中的环积与环余积至对合确定的路径空间，并在偶数维球面上利用 Morse-Bott 方法完整计算，进而证明实射影空间闭测地线的共振定理。

P流和KdV流中量子环的模：作为欧拉弹性体观点扩展的超椭圆曲线的研究I

On the Moduli of a quantized loop in P and KdV flows: Study of hyperelliptic curves as an extension of Euler's perspective of elastica I

量化需要评估量化对象的所有状态，而不是其相对于其能量的静止状态。本文研究了黎曼球面$\PP$上的量子化弹性体的模量$\CMeP$.然后证明了它的模空间被分解为一组由服从KdV（Korteweg-de Vries）族的流所确定的能量守恒的等价类。由于服从KdV层次的流具有自然拓扑，因此它在模空间$\CMeP$中导出拓扑。利用拓扑结构对$\CMeP$进行了分类。拓扑空间$\Top$范畴中的loop空间的研究已经很成熟，它的上同调性质也是众所周知的。由于量子化弹性体的模空间可以看作微分几何范畴$\DGeom$中的一个环空间，我们还利用诱导拓扑证明了与{ \bf Top}中的一个环空间相关的三角形范畴与$\DGeom$中的三角形范畴之间存在函子.由于欧拉通过在$\CC$上观察经典弹性曲线的形状来研究椭圆积分及其模量，本文致力于超椭圆曲线与$\PP$上的量子化弹性曲线之间的关系，作为欧拉对弹性曲线观点的扩展。

CMC hierarchy I: Commuting symmetries and loop algebra

We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loop algebra valued Gauß$\,$ map, the CMC hierarchy is obtained by the assembly of a pair of Adler-Kostant-Symes bi-Hamiltonian hierarchies to the original CMC system. The infinite sequence of higher-order conservation laws of the CMC system admits the corresponding extension, and w…

Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most $9$-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.

循环群和字符串拓扑

Loop groups and string topology

关于循环群及其表示的调查文章，继2005年6月27日至7月13日在哥廷根乔治-奥古斯特大学暑期学校“代数群“举行的三次讲座之后。我们讨论循环群，它们的中心扩展，和正能量表示。