Modified $r$-matrices are solutions of the modified classical Yang-Baxter
equation, introduced by Semenov-Tian-Shansky, and play important roles in
mathematical physics. In this paper, first we introduce a cohomology theory for
modified $r$-matrices. Then we study three kinds of deformations of modified
$r$-matrices using the established cohomology theory, including algebraic
deformations, geometric deformations and linear deformations. We give the
differential graded Lie algebra that governs algebraic deformations of modified
$r$-matrices. For geometric deformations, we prove the rigidity theorem and
study when is a neighborhood of a modified $r$-matrix smooth in the space of
all modified $r$-matrix structures. In the study of trivial linear
deformations, we introduce the notion of a Nijenhuis element for a modified
$r$-matrix. Finally, applications are given to study deformations of complement
of the diagonal Lie algebra and compatible Poisson structures.Comment: 18 pages, to appear in JNC

Jiang, Jun

Sheng, Yunhe

arXiv.org e-Print Archive

English

Deformations of modified $r$-matrices and cohomologies of related
  algebraic structures

We establish a general mapping from one-dimensional non-Hermitian mosaic
models to their non-mosaic counterparts. This mapping can give rise to mobility
edges and even Lyapunov exponents in the mosaic models if critical points of
localization or Lyapunov exponents of localized states in the corresponding
non-mosaic models have already been analytically solved. To demonstrate the
validity of this mapping, we apply it to two non-Hermitian localization models:
an Aubry-Andr\'e-like model with nonreciprocal hopping and complex
quasiperiodic potentials, and the Ganeshan-Pixley-Das Sarma model with
nonreciprocal hopping. We successfully obtain the mobility edges and Lyapunov
exponents in their mosaic models. This general mapping may catalyze further
studies on mobility edges, Lyapunov exponents, and other significant quantities
pertaining to localization in non-Hermitian mosaic models.Comment: 9 pages, 2 figure

Jiang, Sheng-Lian

Lang, Li-Jun

Liu, Yanxia

General mapping of one-dimensional non-Hermitian mosaic models to
  non-mosaic counterparts: Mobility edges and Lyapunov exponents

Long Jiang

Sheng-Jun Fan

Comptes Rendus Mathématique

Numérisation de Documents Anciens Mathématiques

Uniqueness result for the BSDE whose generator is monotonic in <i>y</i> and uniformly continuous in <i>z</i>

Dai, Li-Yang

Jiang, Lei-Sheng

Shen, Chao

Yan, Jun

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