Generalized complex and Dirac structures on homogeneous spaces

The aim of this paper is to study generalized complex geometry (Hitchin, 2002) [6] and Dirac geometry (Courant, 1990) [3], (Courant and Weinstein, 1988) [4] on homogeneous spaces. We offer a characterization of equivariant Dirac structures on homogeneous spaces, which is then used to construct new examples of generalized complex structures. We consider Riemannian symmetric spaces, quotients of compact groups by closed connected subgroups of maximal rank, and nilpotent orbits in sln(R). For each of these cases, we completely classify equivariant Dirac structures. Additionally, we consider equivariant Dirac structures on semisimple orbits in a semisimple Lie algebra. Here equivariant Dirac structures can be described in terms of root systems or by certain data involving parabolic subalgebras.     

Generalized Calabi-Yau manifolds and the chiral de Rham complex

We show that the chiral de Rham complex of a generalized Calabi-Yau manifold carries N=2 supersymmetry. We discuss the corresponding topological twist for this N=2 algebra. We interpret this as an algebroid version of the super-Sugawara or Kac-Todorov construction.     

Generalized shear deformation theory for bending analysis of functionally graded plates

In this study, the static response is presented for a simply supported functionally graded rectangular plate subjected to a transverse uniform load. The generalized shear deformation theory obtained by the author in other recent papers is used. This theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transversal shear correction factors are needed because a correct representation of the transversal shearing strain is given. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The equilibrium equations of a functionally graded plate are given based on a generalized shear deformation plate theory. The numerical illustrations concern bending response of functionally graded rectangular plates with two constituent materials. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, and volume fraction distributions are studied. The results are verified with the known results in the literature.     

Reduction of Courant algebroids and generalized complex structures

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on CP².     

Automorphisms of Generalized Down-Up Algebras

A generalization of down-up algebras was introduced by Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 ( 1 ): 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]), the so-called “generalized down-up algebras”. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) such that r is not a root of unity, listing explicitly the elements of the group. In the last section, we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(α, β, γ) for which the roots of the polynomial X ² ? α X ? β are not both roots of unity.     

Vertex F-algebra structures on the complex oriented homology of H-spaces

We give a topological construction of graded vertex F-algebras by generalizing Joyce's vertex algebra construction to complex-oriented homology. Given an H-space X with a $B\mathrm{U}(1)$-action, a choice of K-theory class, and a complex oriented homology theory E, we build a graded vertex F-algebra structure on $E_{?}(X)$ where F is the formal group law associated with E.     

Algebraic Entropy of Generalized Shifts on Direct Products

For a set Γ, a function λ: Γ → Γ and a nontrivial abelian group K, the \emphgeneralized shift σ_λ: K ^Γ → K ^Γ is defined by (x _i ) _i∈Γ ? (x _λ(i)) _i∈Γ [3 Ayatollah Zadeh Shirazi , F. , Heidari Ardi , F. , Karami Kabir , N. ( 2008 ). A note on shift theory . Math. Pannon. 19 : 187 – 195 . [Google Scholar]]. In this article we compute the algebraic entropy of σ_λ; it is either zero or infinite, depending exclusively on the properties of λ. This solves two problems posed in [2 Akhavin , M. , Ayatollah Zadeh Shirazi , F. , Dikranjan , D. , Giordano Bruno , A. , Hosseini , A. ( 2009 ). Algebraic entropy of shift endomorphisms on abelian groups . Quaest. Math. 32 : 529 – 550 .[Crossref], [Web of Science ®] , [Google Scholar]].     

4维流形的混合型广义复结构

分析了4维流形M,在其子集N上是类型2且其余上是类型0的混合型广义复结构．特别地,对该广义复结构是由一个特殊的纯旋量φ=φ0 φ2 φ4定义,而且N=Ker(φ0)和dφ0|N是处处非零的情况尤其感兴趣．将发现N是余维数为2的子流形和c1(N)=0且局部定义该结构的条件可以简化．最后将给出具有这种结构的一个例子．     

A differential graded approach to the silting theorem

A silting theorem was established by Buan and Zhou as a generalisation of the classical tilting theorem of Brenner and Butler. In this paper, we give an alternative proof of the theorem by using differential graded algebras.     

Generalized modular forms

The theory of “generalized modular forms,” initiated here, grows naturally out of questions inherent in rational conformal field theory. The latter physical theory studies q-series arising as trace functions (or partition functions), which generate a finite-dimensional SL(2,Z)-module. It is a natural step to investigate whether these q-series are in fact modular forms in the classical sense. As it turns out, the existence of the module does not, of itself, guarantee that this is so. Indeed, our Theorem 1 shows that such q-series of necessity behave like modular forms in every respect, with the important exception that the multiplier system need not be of absolute value one. The Supplement to Theorem 1 shows that such q-series are classical modular forms exactly when the scalars relating the q-series generators of the module have absolute value one. That is, the SL(2,Z)-module in question is unitary. (There is the further restriction that the associated representation is monomial.) We prove as well that there exist generalized modular forms which are not classical modular forms. (Hence, as asserted above, the q-series need not be classical modular forms.)Beyond Theorem 1 and its Supplement, which serve to relate our generalized modular forms to classical modular forms (and thus justify the name), this work develops a number of their fundamental properties. Among these are a basic result relating generalized modular forms to classical modular forms of weight 2 and so, as well, to abelian integrals. Further, we prove two general existence results and a complete characterization of weight k generalized modular forms in terms of generalized modular forms of weight 0 and classical modular forms of weight k.     

Lie Bialgebras of Generalized Loop Virasoro Algebras

The first cohomology group of generalized loop Virasoro algebras with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is used to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. The authors generalize the results to generalized map Virasoro algebras.     

Generalized isochronous centers for complex systems

In this paper, the definition of generalized isochronous center is given in order to study unitedly real isochronous center and linearizability of polynomial differential systems. An algorithm to compute generalized period constants is obtained, which is a good method to find the necessary conditions of generalized isochronous center for any rational resonance ratio. Its two linear recursive formulas are symbolic and easy to realize with computer algebraic system. The function of time-angle difference is introduced to prove the sufficient conditions. As the application, a class of real cubic Kolmogorov system is investigated and the generalized isochronous center conditions of the origin are obtained.     

Primitive Ideals of Noetherian Generalized Down-Up Algebras

We classify the primitive ideals of noetherian generalized down-up algebras.     

Realising Smashing Localisations as Morphisms of DG Algebras

For a smashing localisation L of the derived category of a differential graded (dg) algebra A we construct a dg algebra A _L and a morphism of dg algebras A→A _L that induces the canonical map in cohomology. As a first application we obtain a localisations of a dg algebra A with graded commutative homology at a prime ideal in the homology H ^* A, namely a morphism of dg algebras. As a second application we can use results of Keller to “model” every smashing localisation of compactly generated algebraic triangulated categories by a morphism of dg algebras.      

广义矩阵代数上的李导子

研究了广义矩阵代数上的一类李导子,证明了广义矩阵代数上李导子可以表示成一个导子和一个中心映射之和,并将这个结果应用到全矩阵代数上.     

On the Image of the Totaling Functor

Let A be a differential graded (DG) algebra with a trivial differential over a commutative unital ring. This paper investigates the image of the totaling functor, defined from the category of complexes of graded A-modules to the category of DG A-modules. Specifically, we exhibit a special class of semifree DG A-modules which can always be expressed as the totaling of some complex of graded free A-modules. As a corollary, we also provide results concerning the image of the totaling functor when A is a polynomial ring over a field.     

On the Generalized Enveloping Algebra of a Color Lie Algebra

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle , such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L. Supported by the EC project Liegrits MCRTN 505078.     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.