Cohomology Groups Research Articles

On some K3 surfaces with order sixteen automorphism

AbstractWe consider K3 surfaces of Picard rank 14 which admit a purely non‐symplectic automorphism of order 16. The automorphism acts on the second cohomology group with integer coefficients and we compute the invariant sublattice for the action. We show that all of these K3 surfaces admit an elliptic fibration and we compute the invariant lattices in a geometric way by using special curves of the elliptic fibration. The computation of these lattices plays an important role when one wants to study moduli spaces and mirror symmetry for lattice polarized K3 surfaces.

The de Rham Cohomology Group of a Hemi-Slant Submanifold in Metallic Riemannian Manifolds

In this study, we answer the question under what conditions a hemi-slant submanifold of locally decomposable metallic Riemannian manifolds admits a well defined canonical de Rham cohomology class. Firstly, we give the integrability and minimality conditions of the distributions arose from its definition. Later, we find some necessary conditions depending on the above-named concepts of the associated distributions for such a type of submanifold to define a de Rham cohomology class. Furthermore, we analyzed the non-triviality of this cohomology class In the end, we construct two examples which enable better expressing the main results.

3-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on 3-Lie algebras

In this paper, first we introduce the notions of relative Rota-Baxter operators of nonzero weight on 3-Lie algebras and 3-post-Lie algebras. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weight induces a 3-post-Lie algebra naturally. Conversely, a 3-post-Lie algebra gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Then we construct an L∞-algebra whose Maurer-Cartan elements are relative Rota-Baxter operators of nonzero weight. Consequently, we obtain the twisted L∞-algebra that controls deformations of a given relative Rota-Baxter operator of nonzero weight on 3-Lie algebras. Finally, we introduce a cohomology theory for a relative Rota-Baxter operator of nonzero weight on 3-Lie algebras and use the second cohomology group to classify infinitesimal deformations.

Deformations and cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights

A cohomology theory of Rota-Baxter 3-Lie algebras of arbitrary weights is introduced. Formal deformations, abelian extensions, skeletal Rota-Baxter 3-Lie 2-algebras and crossed modules of Rota-Baxter 3-Lie algebras are interpreted by using lower degree cohomology groups.

Extensions and Well’s type exact sequence of skew braces

In this paper, we describe the split exact sequences of skew left braces. We define a free action of the second cohomology group of a skew left brace [Formula: see text] by [Formula: see text] on [Formula: see text] and show that this action becomes transitive if [Formula: see text] is a trivial skew brace. We also generalize the Well’s type exact sequence for extensions by the trivial skew brace.

Deformations, cohomologies and integrations of relative difference Lie algebras

In this paper, first using the higher derived brackets, we give the controlling algebra of relative difference Lie algebras, which are also called crossed homomorphisms or differential Lie algebras of weight 1 when the action is the adjoint action. Then using Getzler's twisted L∞-algebra, we define the cohomology of relative difference Lie algebras. In particular, we define the regular cohomology of difference Lie algebras by which infinitesimal deformations of difference Lie algebras are classified. We also define the cohomology of difference Lie algebras with coefficients in arbitrary representations, and using the second cohomology group to classify abelian extensions of difference Lie algebras. Finally, we show that any relative difference Lie algebra can be integrated to a relative difference Lie group globally in a functorial way.

On Farrell–Tate cohomology of GL3 over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on mod ℓ Farrell–Tate cohomology for arithmetic groups. We deal with ℓ-rank 1 cases different from PSL2. The conjugacy classification of cyclic subgroups of order ℓ is reduced to the classification of modules of Cℓ-group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell–Tate cohomology computations, we discuss in detail the homological torsion in PGL3 over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

Some uniform bounds for elliptic curves over ℚ

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E/\mathbb Q$. We consider in particular the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of $E$, and the Kummer extensions generated by points of infinite order in $E(\mathbb Q)$.

Tame and strongly étale cohomology of curves

For a curve C over a perfect field k of characteristic p > 0 we study the tame cohomology of $$\mathcal{C}$$ = Spa(C, k) introduced in [Hüb21]. We prove that the tame cohomology groups of $$\mathcal{C}$$ with p-torsion coefficients satisfy cohomological purity (which is not true in full generality for the étale cohomology). Using purity we show Poincaré duality for the tame cohomology of $$\mathcal{C}$$ with p-torsion coefficients.

Residual finiteness of extensions of arithmetic subgroups of {{,mathrm{SU},}}(d,1) with cusps

Let Gamma be an arithmetic subgroup of {{,mathrm{SU},}}(d,1) with cusps, and let X_Gamma be the associated locally symmetric space. In this paper we investigate the pre-image of Gamma in the covering groups of {{,mathrm{SU},}}(d,1). Let H^bullet _!(X_Gamma ,mathbb {C}) be the inner cohomology, i.e. the image in H^bullet (X_Gamma ,mathbb {C}) of the compactly supported cohomology. We prove that if the first inner cohomology group H^1_!(X_Gamma ,mathbb {C}) is non-zero then the pre-image of Gamma in each connected cover of {{,mathrm{SU},}}(d,1) is residually finite. At the end of the paper we give an example of an arithmetic subgroup Gamma satisfying the criterion H^1_!(X_Gamma ,mathbb {C}) ne 0.

Symmetric cohomology of groups and Poincaré duality

We prove a Poincaré duality theorem for finite groups, where the (co)homologies involved are a variant of classical group cohomology due to Zarelua and Staic, denoted by Hλ⁎. In particular, we show that in certain cases (but not always), for a finite group G of order n and a G-module M, we have the isomorphismHλn−i−1(G,M)→≅Hiλ(G,Mtw), where Mtw is a twisting of M defined in Section 5.

Weight filtrations on GKZ-systems

If $\beta\in\CC^d$ is integral but not a strongly resonant parameter for the homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the associated GKZ-system carries a naturally defined mixed Hodge module structure. We study here in the normal case the corresponding weight filtration by computing the intersection complexes with respective multiplicities on the associated graded parts. We do this by computing the weight filtration of a Gauss-Manin system with respect to a locally closed embedding of a torus inside an affine space. We then produce a result, based on a Fourier-Sato transforms, that allows to port an MHS structure on a monodromic module through a Fourier-Laplace transform, from the Gauss-Manin system to the GKZ-system. Our results show that these data, which we express in terms of intersection cohomology groups on induced toric varieties, are purely combinatorial, and not arithmetic, in the sense that they only depend on the polytopal structure of the cone of $A$ but not on the semigroup itself. As a corollary we get a purely combinatorial formula for the length of the underlying (regular) holonomic GKZ-system, irrespective of homogeneity. In dimension up to three, and for simplicial semigroups, we give explicit generators of the weight filtration.

PONTRYAGIN DUALITY FOR VARIETIES OVER p-ADIC FIELDS

AbstractWe define cohomological complexes of locally compact abelian groups associated with varieties over p-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.

Macaulay matrix for Feynman integrals: linear relations and intersection numbers

We elaborate on the connection between Gel’fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman Integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for mathcal{A} -hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.

The elasticity complex: compact embeddings and regular decompositions

We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex.

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Let A be the classical, singly-graded Steenrod algebra over the prime order field F2 and let P⊗h:=F2[t1,…,th] denote the polynomial algebra on h generators, each of degree 1, viewed as a module over A. Write GLh for the usual general linear group of rank h over F2. As well known, the (mod 2) cohomology groups of the Steenrod algebra, ExtAh,h+⁎(F2,F2) is still largely mysterious for all homological degrees h≥6. The h-th algebraic transferTrhA:(F2⊗GLhAnnA‾[P⊗h]⁎)n⟶ExtAh,h+n(F2,F2), defined by William Singer [28], is a helpful tool to describe that Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains open for any h≥5. There is currently no information on the conjecture for h=6. In this paper, we verify Singer's conjecture for homological degree h=6 and the internal degree of the general form ns:=6(2s−1)+10⋅2s. This result is important, since it tells us that the non-zero elements h22g1=h4Ph2∈ExtA6,6+n1(F2,F2), and D2∈ExtA6,6+n2(F2,F2) are not in the image of the sixth algebraic transfer. This Note is a continuation of our previous one [22], which will refer to as Part I. Our approach is based on explicitly solving the hit problem for the Steenrod algebra in the case h=6 and degree ns. This extends a result in Mothebe, Kaelo and Ramatebele [13]. At the same time, we also use this obtained result to establish the dimension result for the space of the unhit elements, F2⊗AP⊗7 in a certain general degree.

Curvature criterion for vanishing of group cohomology

We introduce a new geometric criterion for vanishing of cohomology for BN-pair groups. In particular, this new criterion yields a sharp vanishing of cohomology result for all BN-pair groups acting on non-thin affine buildings.

On the cohomology of the Ree groups and kernels of exceptional isogenies

Let G be a simple, simply connected algebraic group over an algebraically closed field k of characteristic p>0. Let σ:G→G be a surjective endomorphism of G such that the fixed point set G(σ) is a Suzuki or Ree group. Then, let Gσ denote the scheme-theoretic kernel of σ. Using methods of Jantzen and Bendel–Nakano–Pillen, we compute the 1-cohomology for the Frobenius kernels with coefficients in the induced modules, H1(Gσ,H0(λ)), and extensions between the simple modules, ExtGσ1(L(λ),L(μ)). When G(σ) is a Ree group of type F4, these results are used to improve the known bounds for identifying extensions of simple modules in defining characteristic ExtG(σ)1(L(λ),L(μ)) with those for the algebraic group.

Adiabatic cycles of quantum spin systems

Motivated by the $\mathrm{\ensuremath{\Omega}}$-spectrum proposal of unique gapped ground states by Kitaev, we study adiabatic cycles in gapped quantum spin systems from various perspectives. We give a few exactly solvable models in one and two spatial dimensions and discuss how nontrivial adiabatic cycles are detected. For one spatial dimension, we study the adiabatic cycle in detail with the matrix product state and show that the symmetry charge can act on the space of matrices without changing the physical states, which leads to nontrivial loops with symmetry charges. For generic spatial dimensions, based on the Bockstein isomorphism ${H}^{d}(G,U(1))\ensuremath{\cong}{H}^{d+1}(G,\mathbb{Z})$, we study a group cohomology model of the adiabatic cycle that pumps a symmetry-protected topological phase on the boundary by one period. It is shown that the spatial texture of the adiabatic Hamiltonian traps a symmetry-protected topological phase in one dimension lower.

Solid locally analytic representations of 𝑝-adic Lie groups

We develop the theory of locally analytic representations of compact p p -adic Lie groups from the perspective of the theory of condensed mathematics of Clausen and Scholze. As an application, we generalise Lazard’s isomorphisms between continuous, locally analytic and Lie algebra cohomology to solid representations. We also prove a comparison result between the group cohomology of a solid representation and of its analytic vectors.