Narrow Results

For a foliation ${ℱ}$ defined on a smooth complex manifold $M$ we introduce the category of vertex operator algebra $V$ bundles with sections provided by vectors of elements of the space of algebraically extended $V$-module $W$-valued differentials. An intrinsic coordinate-independent formulation for such bundles is given. Finally, we identify the cohomology of the spaces of sections for a vertex operator algebra $V$ bundle with vertex operator algebra cohomology of the holonomy groupoid $\mathrm{\text{Hol}}(M,{ℱ})$.

A twistor framework for foundations of physics is outlined, in which (1) spacetime is secondary, derived from twistor constructions; (2) the primary physical agents are described by elements of sheaf cohomology defined on twistor space, which encode all information about the spacetime entities (fields and particles); (3) the dynamics of spacetime entities is encoded in the holomorphic structures of twistor space and revealed through Penrose transform and its generalizations (twistor diagrams); (4) the nonlinearity of gravity and nonabelian gauge interactions is a manifestation of the deformation of twistor space effected by the self-interacting twistor agents. A consistent framework of quantum gravity is attainable within twistor formalism because not only can gravity be derived from the primary physical agents of the formalism but also, and most importantly, the formalism itself can be proven intrinsically quantal without the necessity of imposing a quantum postulate upon it extrinsically through a quantization procedure. Indeed, the Planck constant and the whole quantum edifice built thereupon can be derived from formalism. Thus, the conflicts between the principles of general relativity and quantum theory are dissolved because both theories are consequences of twistor formalism, and the century-long debate concerning the possibility of a consistent framework of quantum gravity is closed. Many technical and conceptual issues must be clarified before the twistor formalism can be turned into a practical research framework. But the foundational pillars of the framework provided by twistor formalism are rock solid.

In this paper, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.

In this paper, we give cohomologies and deformations theory, as well as abelian extensions for compatible pre-Lie superalgebras. Explicitly, we first introduce the notation of a compatible pre-Lie superalgebra and its representation. We also construct a new bidfferential graded Lie superalgebra whose Maurer–Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie superalgebra which controls deformations of a compatible pre-Lie superalgebra. Then, we introduce a cohomology of a compatible pre-Lie superalgebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie superalgebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group ${{\mathscr{H}}}^2(𝔤;𝔤)$ is trivial, then the compatible pre-Lie superalgebra is rigid. Additionally, we give a cohomology of a compatible pre-Lie superalgebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie superalgebras using this cohomology. We show that abelian extensions are classified by the second cohomology group. Finally, we study the relation between compatible Lie superalgebras and compatible pre-Lie superalgebras and we give some examples of compatible pre-Lie superalgebras.

In this paper, first we give the notion of a crossed homomorphism on a $3$-Lie algebra with respect to an action on another $3$-Lie algebra, and characterize it using a homomorphism from a 3-Lie algebra to the semidirect product 3-Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota–Baxter operators of weight $1$ on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on $3$-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an $L_{\infty }$-algebra whose Maurer–Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted $L_{\infty }$-algebra that controls deformations of a given crossed homomorphism on $3$-Lie algebras.

The Lie conformal algebra ${𝒲}(a,b)$ with two parameters $a,b\in ℂ$ is a free $ℂ[\partial ]$-module generated by $L$ and $W$ satisfying $[L_{\lambda }L]=(\partial +2\lambda )L,[L_{\lambda }W]=(\partial +a\lambda +b)W$ and $[W_{\lambda }W]=0$, which is the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we determine the basic and reduced cohomology groups of ${𝒲}(a,b)$ for the case of $a\in \{a\le 2|a\in \mathbb{Q}\}\cup (ℂ-\mathbb{Q})$ with coefficients in its module. In particular, the low-dimensional basic cohomology groups with trivial coefficients are completely determined.

In this paper, we first define twisted ${𝒪}$-operator families on Leibniz algebras indexed by a semigroup $\mathrm{\Omega }$. Then we introduce and study NS-Leibniz family algebras as the underlying structures of twisted ${𝒪}$-operator families. We show that an NS-Leibniz family algebra induces an ordinary NS-Leibniz algebra on the tensor product with the semigroup algebra. Finally, we investigate the cohomology of a twisted ${𝒪}$-operator family. This cohomology can also be seen as the cohomology of a certain $\mathrm{\Omega }$-Leibniz algebra with coefficients in a suitable representation. As an application, we study the formal deformations of twisted ${𝒪}$-operator families on Leibniz algebras.

The concept of Rota–Baxter family algebra is a generalization of Rota–Baxter algebra. It appears naturally in the algebraic aspects of renormalizations in quantum field theory. Rota–Baxter family algebras are closely related to dendriform family algebras. In this paper, we first construct an $L_{\infty }$-algebra whose Maurer–Cartan elements correspond to Rota–Baxter family algebra structures. Using this characterization, we define the cohomology of a given Rota–Baxter family algebra. As an application of our cohomology, we study formal and infinitesimal deformations of a given Rota–Baxter family algebra. Next, we define the notion of a homotopy Rota–Baxter family algebra structure on a given $A_{\infty }$-algebra. We end this paper by considering the homotopy version of dendriform family algebras and their relations with homotopy Rota–Baxter family algebras.

The aim of this work is to introduce representations of BiHom-left-symmetric algebras and develop its cohomology theory. As applications, we study linear deformations of BiHom-left-symmetric algebras, which are characterized by its second cohomology group with the coefficients in the adjoint representation. The notion of a Nijenhuis operator on a BiHom-left-symmetric algebra is introduced. We will prove that it can generate trivial linear deformations of a BiHom-left-symmetric algebra.

We consider the $1|1$-dimensional real superspace ${ℝ}^{1|1}$ endowed with its standard contact structure defined by the $1$-form $\alpha$. The conformal Lie superalgebra ${𝒦}(1)$ acts on ${ℝ}^{1|1}$ as the Lie superalgebra of contact vector fields. We compute the first differential cohomology of ${𝒦}(1)$ with coefficients in superspace ${𝔇}_{\lambda ,\mu ;\nu }$ of bilinear differential operators acting on weighted densities. This work is the simplest super analog of the problems solved in Ref. 5.

In this paper, we give a unified approach to study the controlling algebras and cohomology theories of various types of operators on $3$-Lie algebras, including relative Rota–Baxter operators of weight $0$, relative Rota–Baxter operators of nonzero weight $\lambda$, twisted Rota–Baxter operators, Reynolds operators, derivations, crossed homomorphisms and homomorphisms. The main ingredient is deformation maps of quasi-twilled $3$-Lie algebras. More precisely, we introduce two types of deformation maps of a quasi-twilled $3$-Lie algebra. Left deformation maps unify relative Rota–Baxter operators of weight $0$, relative Rota–Baxter operators of nonzero weight $\lambda$, twisted Rota–Baxter operators and Reynolds operators. Right deformation maps unify derivations, crossed homomorphisms and $3$-Lie algebra homomorphisms. Furthermore, we give controlling algebras and cohomologies of these two types of deformation maps, respectively, which not only recover some existing results, but also lead to some new results, e.g. the controlling algebras for twisted Rota–Baxter operators and Reynolds operators on 3-Lie algebras.

We give an example of a compact connected Lie group of the lowest rank such that the mod 2 cohomology ring of its classifying space has a nonzero nilpotent element.

Let ${ℱ}$ be a foliation on a connected manifold $M$. Denote by $T{ℱ}$ the tangent bundle to ${ℱ}$ and ${𝒱}=TM/T{ℱ}$ its normal bundle. We say that ${ℱ}$ is developable if there exists a normal $\mathrm{\Gamma }$-covering $\pi :\widehat{M}\to M$ such that the pull-back $\widehat{{ℱ}}$ of ${ℱ}$ by $\pi$ is a locally trivial fibration $D:\widehat{M}\to W$ on which $\mathrm{\Gamma }$ acts by automorphisms (and then on $W$). The fiber of $D$ is denoted $\widehat{L}$. Suppose $M$ compact. We prove: (i) if $H^1(\widehat{L})=0$, the space $H_{{ℱ}}^1(M,{𝒱})$ of infinitesimal deformations of $(M,{ℱ})$ is naturally identified to the first cohomology space $H^1(\mathrm{\Gamma },𝔛(W))$ of the discrete group $\mathrm{\Gamma }$ with values in the $\mathrm{\Gamma }$-module $𝔛(W)$ of smooth vector fields on $W$; (ii) if $\widehat{L}$ has trivial cohomology, the foliated cohomology $H_{{ℱ}}^{\ast }(M,{𝒱})$ of $(M,{ℱ})$ with values in ${𝒱}$ is isomorphic to the cohomology $H^{\ast }(\mathrm{\Gamma },𝔛(W))$ of $\mathrm{\Gamma }$ with values in $𝔛(W)$. Some examples and explicit computations are given.

A way to characterize the space of leaves of a foliation in terms of connections is proposed. A particular example of vertex algebra cohomology of codimension one foliations on complex curves is considered. Multiple applications in Mathematical Physics are revealed.

We study the algebraic conditions leading to the chain property of complexes for vertex operator algebra n-point functions (with their convergence assumed) with differential being defined through reduction formulas. The notion of the reduction cohomology of Riemann surfaces is introduced. Algebraic, geometrical, and cohomological meanings of reduction formulas are clarified. A counterpart of the Bott–Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven. It is shown that the reduction cohomology is given by the cohomology of n-point connections over the vertex operator algebra bundle defined on a genus g Riemann surface ${\mathrm{\Sigma }}^{(g)}$. The reduction cohomology for a vertex operator algebra with formal parameters identified with local coordinates around marked points on ${\mathrm{\Sigma }}^{(g)}$ is found in terms of the space of analytical continuations of solutions to Knizhnik–Zamolodchikov equations. For the reduction cohomology, the Euler–Poincaré formula is derived. Examples for various genera and vertex operator cluster algebras are provided.

For a foliation ${ℱ}$ of a smooth complex manifold, we introduce the category ${𝒞}$ of $V$-structures associated to a vertex operator algebra $V$ and the category of its modules. The main result consists of the construction of $V$-structures and canonicity proof of ${𝒞}$ on ${ℱ}$.

Using the Schottky procedure of forming a genus $g$ Riemann surface by attaching handles to the Riemann sphere, we construct coboundary operators and corresponding cohomology for the double complexes of rational functions associated to a particular example of the rank two bosonic vertex operator algebra $V$.

Suppose that S is a Hopf C*-algebra. Further to our study of the category of completely bounded S-comodules in [14], we will compare the Ext-functors for (the category of counital S-comodules) as well as for (the category of counital S-bicomodules) with the cohomology theories defined in [10]. The particular case of compact quantum groups as well as the cases of S=C₀(G) and S = C*(G) (where G is a locally compact group) will be considered in more details. Moreover, we give some relations between the vanishing of the first dual cohomologies of C₀(G) (respectively, C*(G)) and the injectivity of the bi-comodule U(G) (respectively, U(C*(G))).

Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225–255] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

We continue studying net bundles over partially ordered sets (posets), defined as the analogues of ordinary fiber bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, giving versions of classical Hurewicz theorems. Focusing our attention on Hilbert net bundles, we define the K-theory of a poset and introduce functions over the homotopy groupoid satisfying the same formal properties as Chern classes. As when the given poset is a base for the topology of a space, our results apply to the category of locally constant bundles.