Cohomology Groups Research Articles

Thompson’s semigroup and the first Hochschild cohomology

Abstract In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson’s group $\mathcal {F}$ . We introduce the notion of unique factorization semigroup which contains Thompson’s semigroup $\mathcal {S}$ and the free semigroup $\mathcal {F}_n$ on n ( $\geq 2$ ) generators. Let $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal {S}$ and $\mathcal {F}_n$ , respectively. We prove that all derivations on $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ are automatically continuous, and every derivation on $\mathfrak {B}(\mathcal {S})$ is induced by a bounded linear operator in $\mathcal {L}(\mathcal {S})$ , the weak-operator closed Banach algebra consisting of all bounded left convolution operators on $l^2(\mathcal {S})$ . Moreover, we prove that the first continuous Hochschild cohomology group of $\mathfrak {B}(\mathcal {S})$ with coefficients in $\mathcal {L}(\mathcal {S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompson’s semigroup.

Computing cohomology groups that classify bundles of strongly self-absorbing C∗-algebras

Locally trivial bundles of [Formula: see text]-algebras with fiber [Formula: see text] for a strongly self-absorbing [Formula: see text]-algebra [Formula: see text] over a finite CW-complex [Formula: see text] form a group [Formula: see text] that is the first group of a cohomology theory [Formula: see text]. In this paper, we compute these groups by expressing them in terms of ordinary cohomology and connective [Formula: see text]-theory. To compare the [Formula: see text]-algebraic version of [Formula: see text] with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological [Formula: see text]-theory.

Separation of homogeneous connected locally compact spaces

We prove that any region Γ \Gamma in a homogeneous n n -dimensional and locally compact separable metric space X X , where n ≥ 2 n\geq 2 , cannot be irreducibly separated by a closed ( n − 1 ) (n-1) -dimensional subset C C with the following property: C C is acyclic in dimension n − 1 n-1 and there is a point b ∈ C ∩ Γ b\in C\cap \Gamma having a special local base B C b \mathcal B_C^b in C C such that the boundary of each U ∈ B C b U\in \mathcal B_C^b is acyclic in dimension n − 2 n-2 . In case X X is strongly locally homogeneous, it suffices to have a point b ∈ C ∩ Γ b\in C\cap \Gamma with an ordinary base B C b \mathcal B_C^b satisfying the above condition. The acyclicity means triviality of the corresponding Čech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces.

A vanishing theorem for varieties with finitely many solvable group orbits

AbstractWe reprove and generalize a result proved by Yavin that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a connected linear solvable group acts, including all spherical varieties.

Deformations and abelian extensions of compatible pre-Lie superalgebras

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 [Formula: see text] 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.

Generalized Reynolds Operators on Hom-Lie Triple Systems

In this paper, we first introduce the notion of generalized Reynolds operators on Hom-Lie triple systems associated to a representation and a 3-cocycle. Then, we develop a cohomology of generalized Reynolds operators on Hom-Lie triple systems. As applications, we use the first cohomology group to classify linear deformations and we study the obstruction class of an extendable order n deformation. Finally, we introduce and investigate Hom-NS-Lie triple system as the underlying structure of generalized Reynolds operators on Hom-Lie triple systems.

Quandles with one nontrivial column

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and [Formula: see text] quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.

Cohomologies of difference Lie groups and the van Est theorem

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group, and establish the relation between representations of difference Lie groups and representations of difference Lie algebras via differentiation and integration. Then we introduce a cohomology theory for difference Lie groups and justify it via the van Est theorem. Finally, we classify abelian extensions of difference Lie groups using the second cohomology group as applications.

The four operations on perverse motives

Let k be a field of characteristic zero with a fixed embedding \sigma:k\hookrightarrow \mathbf{C} into the field of complex numbers. Given a k -variety X , we use the triangulated category of étale motives with rational coefficients on X to construct an abelian category \mathscr{M}(X) of perverse mixed motives. We show that over \mathop{\mathrm{Spec}}(k) the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories \mathrm{D}^{\mathrm{b}}(\mathscr{M}(X)) are equipped with the four operations of Grothendieck (for morphisms of quasi-projective k -varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo–Migliorini and provides an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.

Lie theory and cohomology of relative Rota–Baxter operators

AbstractIn this paper, we establish a local Lie theory for relative Rota–Baxter operators of weight 1. First we recall the category of relative Rota–Baxter operators of weight 1 on Lie algebras and construct a cohomology theory for them. We use the second cohomology group to study infinitesimal deformations of relative Rota–Baxter operators and modified ‐matrices. Then we introduce a cohomology theory of relative Rota–Baxter operators on a Lie group. We construct the differentiation functor from the category of relative Rota–Baxter operators on Lie groups to that on Lie algebras, and extend it to the cohomology level by proving the Van Est theorem between the two cohomology theories. We integrate a relative Rota–Baxter operator of weight 1 on a Lie algebra to a local relative Rota–Baxter operator on the corresponding Lie group, and show that the local integration and differentiation are adjoint to each other. Finally, we give two applications of our integration of Rota–Baxter operators: one is to give an explicit formula for the factorization problem, and the other is to provide an integration for matched pairs.

Deformations of commutative Artinian algebras

The paper is devoted to the study of deformations of Artinian algebras and zero-dimensional germs of varieties. In particular, an approach is developed to solving the open problem about the nonexistence of rigid Artinian algebras; it is based essentially on the use of the canonical duality in the cotangent complex. Thus, it is shown that there are no rigid Gorenstein Artinian algebras and rigid almost complete intersections. The proof of the latter statement is based on the properties of the torsion functors. More precisely, the tensor product of the conormal and canonical modules of the corresponding Artinian algebra is calculated. In this case, the homology and cohomology groups of higher degrees are also found. Among other things, some estimates are obtained for the dimension of the spaces of the first lower and upper cotangent functors of Artinian algebras, and the relationship between them is described. In conclusion, several examples of nonsmoothable Artinian noncomplete intersections are examined, and some unusual properties of such algebras are discussed.

Cohomology of modified Rota–Baxter Leibniz algebra of weight λ

Rota–Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota–Baxter operators on Leibniz algebras. We investigate modified Rota–Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota–Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.

2-term averaging L∞-algebras and non-abelian extensions of averaging Lie algebras

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L∞-algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context.

Cohomology of the Morava stabilizer group through the duality resolution at 𝑛=𝑝=2

We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava E E -theory, H ∗ ( G 2 , E t ) H^*(\mathbb {G}_2, \mathbf {E}_t) , at p = 2 p=2 , for 0 ≤ t > 12 0\leq t > 12 , using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the d 3 d_3 -differentials in the homotopy fixed point spectral sequence for the K ( 2 ) K(2) -local sphere spectrum. These cohomology groups and differentials play a central role in K ( 2 ) K(2) -local stable homotopy theory.

Deformation cohomology for cyclic groups acting on polynomial rings

We examine the Hochschild cohomology governing graded deformations for finite cyclic groups acting on polynomial rings. We classify the infinitesimal graded deformations of the skew group algebra S⋊G for a cyclic group G acting on a polynomial ring S. This gives all graded deformations of the first order. We are particularly interested in the case when the characteristic of the underlying field divides the order of the acting group, which complicates the determination of cohomology.

Generalizations of quasielliptic curves

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all characteristics and having higher genera. This relies on the study of certain infinitesimal group schemes acting on the affine line and certain compactifications. The group schemes are defined in terms of invertible additive polynomials over rings with nilpotent elements, and the compactification is constructed with the theory of numerical semigroups. The existence of regular twisted forms relies on Brion's recent theory of equivariant normalization. Furthermore, extending results of Serre from the realm of group cohomology, we describe non-abelian cohomology for semidirect products, to compute in special cases the collection of all twisted forms.

Cohomology of line bundles on the incidence correspondence

For a finite dimensional vector space V V of dimension n n , we consider the incidence correspondence (or partial flag variety) X ⊂ P V × P V ∨ X\subset \mathbb {P}V \times \mathbb {P}V^{\vee } , parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X X in characteristic p > 0 p>0 . If n = 3 n=3 then X X is the full flag variety of V V , and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0 0 , the cohomology groups are described for all V V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n = 3 n=3 , we recover the recursive description of characters from recent work of Linyuan Liu, while for general n n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

Cohomology and Deformations of Relative Rota–Baxter Operators on Lie-Yamaguti Algebras

In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.

Local analog of the Deligne-Riemann-Roch isomorphism for line bundles in relative dimension 1

We prove a local analog of the Deligne-Riemann-Roch isomorphism in the case of line bundles and relative dimension $1$. This local analog consists in computation of the class of $12$th power of the determinant central extension of a group ind-scheme $\mathcal G$ by the multiplicative group scheme over $\mathbb Q$ via the product of $2$-cocyles in the second cohomology group. These $2$-cocycles are the compositions of the Contou-Carrère symbol with the $\cup$-product of $1$-cocycles. The group ind-scheme $\mathcal{G}$ represents the functor which assigns to every commutative ring $A$ the group that is the semidirect product of the group $A((t))^*$ of invertible elements of $A((t))$ and the group of continuous $A$-automorphisms of $A$-algebra $A((t))$. The determinant central extension naturally acts on the determinant line bundle on the moduli stack of geometric data (proper quintets). A proper quintet is a collection of a proper family of curves over $\operatorname{Spec} A$, a line bundle on this family, a section of this family, a relative formal parameter at the section, a formal trivialization of the bundle at the section that satisfy further conditions.

Algebraic de Rham theorem and Baker-Akhiezer function

For the case of algebraic curves (compact Riemann surfaces), it is shown that de Rham cohomology group $H^1_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$ of the Riemann surface $X$ has a natural structure of a symplectic vector space. Every choice of a non-special effective divisor $D$ of degree $g$ on $X$ defines a symplectic basis of $H^1_{\mathrm{dR}}(X,\mathbb{C})$ consisting of holomorphic differentials and differentials of the second kind with poles on $D$. This result, which is the algebraic de Rham theorem, is used to describe the tangent space to Picard and Jacobian varieties of $X$ in terms of differentials of the second kind, and to define a natural vector fields on the Jacobian of the curve $X$ that move points of the divisor $D$. In terms of the Lax formalism on algebraic curves, these vector fields correspond to the Dubrovin equations in the theory of integrable systems, and the Baker-Akhierzer function is naturally obtained by the integration along the integral curves.