Lie algebroids,Lie groupoids and TFT

We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.     

Compatible Lie algebroids and the periodic Toda lattice

We give the set of maps from to the structure of a Poisson manifold endowed with a pair of compatible Lie algebroids. A suitable reduction process, of the Marsden–Ratiu type, yields a smaller manifold with the same geometrical properties as the original manifold. Moreover, is a bi-Hamiltonian manifold and the flows naturally defined on it are the periodic Toda flows.     

Reduction of Lagrangian mechanics on Lie algebroids

We prove that if a surjective submersion which is a homomorphism of Lie algebroids is given, then there exists another homomorphism between the corresponding prolonged Lie algebroids and a relation between the dynamics on these Lie algebroid prolongations is established. We also propose a geometric reduction method for dynamics on Lie algebroids defined by a Lagrangian and the method is applied to regular Lagrangian systems with nonholonomic constraints.     

Pre-symplectic algebroids and their applications

In this paper, we introduce the notion of a pre-symplectic algebroid and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. Then we study exact pre-symplectic algebroids and show that they are classified by the third cohomology group of a left-symmetric algebroid. Finally, we study para-complex pre-symplectic algebroids. Associated with a para-complex pre-symplectic algebroid, there is a pseudo-Riemannian Lie algebroid. The multiplication in a para-complex pre-symplectic algebroid characterizes the restriction to the Lagrangian subalgebroids of the Levi–Civita connection in the corresponding pseudo-Riemannian Lie algebroid.     

Connections on Lie algebroids and on derivation-based noncommutative geometry

In this paper, we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two constructions, we emphasize the algebraic approach of connections on Lie algebroids, using a suitable differential calculus. Two examples allow this comparison: on the one hand, the Atiyah Lie algebroid of a principal fiber bundle and, on the other hand, the space of derivations of the algebra of endomorphisms of an SL(n,C)-vector bundle. Gauge transformations are also considered in this comparison.     

Nijenhuis and compatible tensors on Lie and Courant algebroids

We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.     

A characterization of strict Jacobi-Nijenhuis manifolds through the theory of Lie algebroids

We obtain a characterization of strict Jacobi-Nijenhuis structures using the equivalent notions of generalized Lie bialgebroid and Jacobi bialgebroid.     

Strong homotopy Lie algebras,homotopy Poisson manifolds and Courant algebroids

We study Maurer–Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson \(\mathfrak g\)-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term \(L_\infty \)-algebra is a homotopy Poisson manifold of degree \(n-1\), we obtain a Courant algebroid from a 2-term \(L_\infty \)-algebra \(\mathfrak g\) via the degree 2 symplectic NQ-manifold \(T^*[2]\mathfrak g^*[1]\). By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie-quasi-Poisson groupoid from a 2-term \(L_\infty \)-algebra, which is proposed to be the geometric structure on the dual of a Lie 2-algebra. These results lead to a construction of a new 2-term \(L_\infty \)-algebra from a given one, which could produce many interesting examples.     

局域模、简正模的李代数表示

本文介绍李代数的非简谐振子的表示。运用这样的表示可以方便地讨论局域模与简正模之关系及其半经典图象。     

Algebraic aspects of renormalization

This paper uses the formulation of renormalization theory in a pure algebraic way providing the notion of Hopf algebras as introduced by A. Connes and D. Kreimer [Commun. Math. Phys. 199 (1998) 203]. First an introduction to the Hopf algebra of rooted trees will be given. The second section explores how renormalization is achieved using the Hopf algebra of rooted trees. So far the paper reviews the results of D. Kreimer as published in the paper cited above. The review will end with a sketch of a category-theoretical interpretation which is under thorough investigation at the time of writing.     

Algebraic aspects of crystallography

Taking into account the fact that space groups are groups of transformations of Euclideann-dimensional space, non-equivalent systems of non-primitive translations are defined. They can be brought into one-to-one correspondence with the elements of the groupH ¹ (K, R ⁿ /Z ⁿ ) or with those of the groupH ¹ (K, Z ⁿ /kZ ⁿ )/H ¹ (K, Z ⁿ ). (K is a point group of orderk.) The consistency of these findings with the results of Part I is given by the isomorphisms $$H^2 (K,Z^n ) \cong H^1 (K,R^n /Z^n ) \cong H^1 (K,Z^n /kZ^n )/H^1 (K,Z^n ).$$ Theorems are proved giving the conditions for cohomology groupsH ^q (K, A) to be zero. These conditions are fulfilled in particular ifA=R ⁿ andK is a subgroup ofGL (n, R) that either is compact (thenq>0) or has a finite normal subgroup leaving no element ofR ⁿ invariant (thenq≧0). This implies that the affine, the Euclidean and the inhomogeneous Lorentz groups are the only extensions ofR ⁿ by the corresponding homogeneous groups. By way of illustration, the theory of this paper is applied to two 2-dimensional space groups.     

Lie Algebraic Solution of Nonlinear Fokker-Planck Equation

The algebraic structure for nonlinear Fokker-Planck equation is discussed. By using Lie algebraic techniques, the exponent operator equation can be decomposed. A time evolution solution of nonlinear Brownian motion is given and can be compared with other theories.     

Nonlinear equations with superposition principles and the theory of transitive primitive Lie algebras

The classification of systems of nonlinear ordinary differential equations with superposition principles is reduced to a classification of transitive primitive Lie algebras. Each system can be associated with the transitive primitive action of a Lie group G on a homogenous space G/H, where H is a maximal subgroup of G. The equations can have specific polynomial or rational nonlinearities.     

Graded Lie Algebra Generating of Parastatistical Algebraic Relations

A new kind of graded Lie algebra (We call it Z2,2 graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable Bose subspace of the Z2,2 graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.     

静电分析器非线性轨迹的Lie代数分析

介绍了Lie代数的方法，用Lie代数方法分析了静电分析器对束流传输过程的非线性影响，其计算结果分析到三级近似. 首先给出了静电分析器的哈密顿函数，然后将哈密顿函数展开为齐次多项式的和，再求Lie映射，最后得到粒子轨迹各级近似解.