Factor Representations of Diffeomorphism Groups

Solitons and nonsmooth diffeomorphisms in conformal nets

We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S^1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in S^1 \ {-1} which do not extend to the whole loop group. In the case of the U(1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms D^s(S^1), s > 2, and show that the positive-energ…

与无限维群相关的共形网的表示

Representations of conformal nets associated with infinite-dimensional groups

本文研究了无限维李群表示与相关共形网表示之间的关系，证明了共形网表示可诱导群的正能量表示且自动具有微分同胚协变性，其协变上循环满足自然性，即微分同胚在表示范畴上等变作用。

Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective unitary representation of Diff(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms D^s(S^1) for any real s>3, and in particular to C^k-diffeomorphisms Diff^k(S^1) with k>=4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to D^s(S^1), s > 3. Moreover every direct sum of irreducible representations of…

Poisson measures for topological groups and their representations

Gaussian quasi-invariant measures on groups of diffeomorphisms and loop groups G relative to dense subgroups G' were constructed. In the non-Archimedean case the wider class of measures was investigated, than in the real case. The cases of Riemann and non-Archimedean manifolds were considered. This article is related with unitary representations of G' associated with Poisson measures on $G^{\bf N}$ and uses quasi-invariant measures on G from the previous works. Several groups are considered: (1) (a) diffeomorphisms and (b) loop groups of real manifolds, (2) (a) diffeomorphisms and (b) loop gro…

紧支撑微分同胚群的广义正能表示

Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

受广义相对论启发，本文研究了光滑流形上紧支撑微分同胚群的广义正能射影酉表示，证明了在流形连通且维数大于1时该表示在单位分量上必然平凡，并确定了紧支撑向量场李代数的连续第二上同调。

关于直线上形式微分同胚群的若干注记

Some remarks on the group of formal diffeomorphisms of the line

本文研究严格正分次无限维实李代数的李群完备化问题，针对直线上的形式微分同胚群构造了具有波兰拓扑的典范稠密子群，并证明该子群恰好由系数具有次阶乘增长的形式级数组成。