Cohomology Of Algebras Research Articles

Wells-type exact sequence and crossed extensions of algebras with bracket

Abstract We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight-term exact sequence in the cohomology of algebras with bracket.

2-Step nilpotent L∞-algebras and hypergraphs

We describe a procedure to attach a nilpotent strong homotopy Lie algebra to every simple hypergraph and prove that two hypergraphs are isomorphic if and only if the corresponding strong homotopy Lie algebras are isomorphic. As an application, we characterize hypergraphs admitting a system of distinct representatives in terms of symplectic forms on the corresponding strong homotopy Lie algebra. We conclude with a combinatorial description of the cohomology of these strong homotopy Lie algebras in low degree.

Equivariant formality of the isotropy action on (ℤ2 ⊕ ℤ2)-symmetric spaces

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalization, it is even known that their isotropy action is equivariantly formal. In this paper, we show that [Formula: see text]-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

On the braided Connes–Moscovici construction

In 1998, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In 2010, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra H in a braided category \mathcal{B} , they associate a paracocyclic object in \mathcal{B} . In this paper, we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for H and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with H to that associated with an H -module coalgebra via a categorical version of the Connes–Moscovici trace.

Reductive cohomology associated with vertex algebras

We review the notion of the reduction cohomology of vertex algebras. The algebraic conditions leading to the chain property for complexes of vertex operator algebra n-point functions (with their convergence assumed) with a coboundary operator defined through reduction formulas are studied. Algebraic, geometrical, and cohomological meanings of reduction formulas and chain condition are clarified. The reduction cohomology for vertex operator algebras associated to Jacobi forms is computed. A counterpart of the Bott-Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven.

Nijenhuis operators on Leibniz algebras

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As applications of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras.

Cohomologies of relative Rota-Baxter Lie algebras with derivations and applications

The purpose of the present paper is to investigate cohomologies of relative Rota-Baxter Lie algebras with derivations and applications. First, we introduce a notion of relative Rota-Baxter LieDer pairs. Moreover, we construct cohomologies of relative Rota-Baxter LieDer pairs. Finally, we discuss abelian extensions, crossed modules and skeletal relative Rota-Baxter Lie 2-Der pairs in terms of cohomology groups.

Dirac cohomology for the BGG category [formula omitted

We study Dirac cohomology HDg,h(M) for modules belonging to category O of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of M⊗S, with S being a spin module of h⊥. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for HDg,h(M) while we show that in the case of a Hermitian symmetric pair (g,k) and an irreducible unitary module M∈O, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in M. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for M∈O.

Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

We obtain a new proof of Shlyakhtenko's result which states that if G is a sofic, finitely presented group with vanishing first \ell^2 -Betti number, then L(G) is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.

On the cohomology of solvable Leibniz algebras

This paper is a sequel to a previous paper of the authors in which the cohomology of semi-simple Leibniz algebras was computed by using spectral sequences. In the present paper we generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we establish the Leibniz analogue of a non-vanishing theorem of Dixmier for nilpotent Leibniz algebras. In addition, we prove a Fitting lemma for Leibniz bimodules

Deformations of Yang-Baxter operators via n-Lie algebra cohomology

We introduce a cohomology theory of n-ary self-distributive objects in the tensor category of vector spaces that classifies their infinitesimal deformations. For n-ary self-distributive objects obtained from n-Lie algebras we show that (n-ary) Lie cohomology naturally injects in the self-distributive cohomology and we prove, under mild additional assumptions, that the map is an isomorphism of second cohomology groups. This shows that the self-distributive deformations are completely classified by the deformations of the Lie bracket. This theory has important applications in the study of Yang-Baxter operators as the self-distributive deformations determine nontrivial deformations of the Yang-Baxter operators derived from n-ary self-distributive structures. In particular, we show that there is a homomorphism from the second self-distributive cohomology to the second cohomology of the associated Yang-Baxter operator. Moreover, we prove that when the self-distributive structure is induced by a Lie algebra with trivial center, we get a monomorphism. We construct a deformation theory based on simultaneous deformations, where both the coalgebra and self-distributive structures are deformed simultaneously. We show that when the Lie algebra has trivial cohomology (e.g. for semi-simple Lie algebras) the simultaneous deformations might still be nontrivial, producing corresponding Yang-Baxter operator deformations. We provide examples and computations in low dimensions, and we completely characterize 2-cocycles for the self-distributive objects obtained from all the nontrivial real Lie algebras of dimension 3, i.e. the Bianchi I-IX, and all the nontrivial complex Lie algebras of dimension 3.

Diassociative family algebras and averaging family operators

Algebraic structures relative to a semigroup Ω (also called family algebraic structures) first appeared in the work of renormalization in quantum field theory. In this paper, we first consider Ω-diassociative algebras and diassociative family algebras as family analogues of diassociative algebras. We define the cohomologies of these algebras generalizing the cohomology of diassociative algebras introduced by Frabetti. We also introduce (relative) averaging family operators as the family analogue of averaging operators and show that they induce diassociative family algebras and Ω-diassociative algebras. Next, we define the cohomology of a (relative) averaging family operator as the cohomology of the induced Ω-diassociative algebra with coefficients in a suitable representation. Finally, as an application of our cohomology, we study formal one-parameter deformations of relative averaging family operators.

A Friedlander-Suslin Theorem over a Noetherian Base Ring

AbstractLet k be a noetherian commutative ring and let G be a finite flat group scheme over k. Let G act rationally on a finitely generated commutative k-algebra A. We show that the cohomology algebra $$H^*(G,A)$$ H ∗ ( G , A ) is a finitely generated k-algebra. This unifies some earlier results: If G is a constant group scheme, then it is a theorem of Evens (Trans. Amer. Math. Soc. 101, 224–239, 1961, Theorem 8.1), and if k is a field of finite characteristic, then it is a theorem of Friedlander and Suslin (Invent. Math. 127, 209–270, 1997). If k is a field of characteristic zero, then there is no higher cohomology, so then it is a theorem of invariant theory.

Cohomology algebra of mapping spaces between quaternion Grassmannians

Let Gk,n(ℍ) for 2≤k<n denote the quaternion Grassmann manifold of k-dimensional vector subspaces of ℍn. In this paper we compute, in terms of the Sullivan models, the rational cohomology algebra of the component of the inclusion i: Gk,n(ℍ) → Gk,n+r(ℍ) in the space of mappings from Gk,n(ℍ) to Gk,n+r(ℍ) for r≥1 and, more generally, we show that the cohomology of Map(Gk,n(ℍ),Gk,n+r(ℍ);i) contains a truncated algebra ℚ[x]x4r+n+k^{2}-nk for n≥4.

Cohomology and deformations of compatible Hom-Lie algebras

In this paper, we consider compatible Hom-Lie algebras as a twisted version of compatible Lie algebras. Compatible Hom-Lie algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-Lie algebras generalizing the recent work of Liu, Sheng and Bai. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-Lie algebras.

On integral cohomology algebra of some oriented Grassmann manifolds

The integral cohomology algebra of G˜6,3 has been determined in the recent work of Kalafat and Yalçınkaya. We completely determine the integral cohomology algebra of G˜n,3 for n=8 and n=10. The main method used to describe these algebras is the Leray–Serre spectral sequence. We also illustrate this method by determining the integral cohomology algebra of G˜n,2 for n odd.

Smooth subvarieties of Jacobians

We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible. The classes that we consider are minimal cohomology classes on Jacobians of very general curves. Our main tool is complex cobordism.

On the Goldman-Millson theorem for A∞-algebras in arbitrary characteristic

Complete filtered A∞-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the A∞ analogs of several key theorems from the Maurer-Cartan theory for L∞-algebras. In contrast with the L∞ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the A∞-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set N•(A) to an A∞-algebra A, sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of N•(A) in terms of the cohomology algebra H(A), and its group of quasi-invertible elements. Finally, we return to the characteristic zero case and show that the nerve of A is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator L∞-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in [26].

Relative quantum cohomology

We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov–Witten invariants of a Lagrangian submanifold L \subset X with a bounding chain. Simultaneously, we define the quantum cohomology algebra of X relative to L and prove its associativity. We also define the relative quantum connection and prove it is flat. A wall-crossing formula is derived that allows the interchange of point-like boundary constraints and certain interior constraints in open Gromov–Witten invariants. Another result is a vanishing theorem for open Gromov–Witten invariants of homologically non-trivial Lagrangians with more than one point-like boundary constraint. In this case, the open Gromov–Witten invariants with one point-like boundary constraint are shown to recover certain closed invariants. From open WDVV and the wall-crossing formula, a system of recursive relations is derived that entirely determines the open Gromov–Witten invariants of (X,L) = (\mathbb{C} P^n, \mathbb{R} P^n) with n odd, defined in previous work of the authors. Thus, we obtain explicit formulas for enumerative invariants defined using the Fukaya–Oh–Ohta–Ono theory of bounding chains.

Cohomology algebra of orbit spaces of free involutions on the product of projective space and 4-sphere

Let X be a finitistic space with the mod 2 cohomology of the product space of a projective space and a 4 -sphere. Assume that X admits a free involution. In this paper we study the mod 2 cohomology algebra of the quotient of X by the action of the free involution and derive some consequences regarding the existence of \mathbb{Z}_2 -equivariant maps between such X and an n -sphere.