Cohomology Of Algebras Research Articles

Corrigendum to “Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements”

In this short note we correct the statement of the main result of [Trans. Amer. Math. Soc. 373 (2020), no. 3, 1909–1940]. That paper presented the rational cohomology ring of a toric arrangement by generators and relations. One of the series of relations given in the paper is indexed over the set circuits in the arrangement’s arithmetic matroid. That series of relations should however be indexed over all sets X X with | X | = rk ⁡ ( X ) + 1 \lvert X \rvert =\operatorname {rk}(X)+1 . Below we give the complete and correct presentation of the rational cohomology ring.

Turaev bicategories and generalized Yetter-Drinfel’d modules in 2-categories

We introduce Turaev bicategories and group braided Turaev bicategories. On one hand, the latter generalize the notions of braided Turaev categories, introduced at the turn of the millennium and originally called “braided crossed group categories” by Turaev himself, and on the other hand, the former generalize the notion of bicategories. For bimonads in 2-categories, which we defined in one of our previous papers, we introduce generalized Yetter-Drinfel’d modules in 2-categories. These generalize to the 2-categorical setting the generalized Yetter-Drinfel’d modules (over a field) of Panaite and Staic, and thus also in particular the anti Yetter-Drinfel’d modules, introduced by Hajac-Khalkhali-Rangipour-Sommerhauser as coefficients for the cyclic cohomology of Hopf algebras, defined by Connes and Moscovici. We construct a Turaev 2-category for bimonads in 2-categories as a Turaev extension of the 2-category of bimonads, generalizing the corresponding Turaev category of Panaite and Staic. We also prove in the 2-categorical setting their results on pairs in involution, which in turn go back to modular pairs in involution of Connes and Moscovici. Moreover, two examples of proper Turaev bicategories are constructed, one of which is group braided.

Cohomology of Algebras of Dihedral Type. V

Cohomology of Algebras of Dihedral Type. V

MOD-2 (CO)homology of an Abelian Group

It is known that for a prime $p\ne 2$ there is the following natural description of the homology algebra of an abelian group $H_*(A,\mathbb F_p)\cong \Lambda(A/p)\otimes \Gamma({}_pA)$ and for finitely generated abelian groups there is the following description of the cohomology algebra of $H^*(A,\mathbb F_p)\cong \Lambda((A/p)^\vee)\otimes {\sf Sym}(({}_pA)^\vee).$ We prove that there are no such descriptions for $p=2$ that `depend' only on $A/2$ and ${}_2A$ but we provide natural descriptions of $H_*(A,\mathbb F_2)$ and $H^*(A,\mathbb F_2)$ that `depend' on $A/2,$ ${}_2A$ and a linear map $\tilde \beta:{}_2A\to A/2.$ Moreover, we prove that there is a filtration by subfunctors on $H_n(A,\mathbb F_2)$ whose quotients are $\Lambda^{n-2i}(A/2)\otimes \Gamma^i({}_2A)$ and that for finitely generated abelian groups there is a natural filtration on $H^n(A,\mathbb F_2)$ whose quotients are $ \Lambda^{n-2i}((A/2)^\vee)\otimes {\sf Sym}^i(({}_2A)^\vee).$

Cohomology of Lie algebras on $${\mathbb{R}}$$ acting on trilinear differential operators

Cohomology of Lie algebras on $${\mathbb{R}}$$ acting on trilinear differential operators

Cohomology and deformations of hom-dendriform algebras and coalgebras

Hom-dendriform algebras are twisted analog of dendriform algebras and are splitting of hom-associative algebras. In this paper, we define a cohomology and deformation for hom-dendriform algebras. We relate this cohomology with the Hochschild-type cohomology of hom-associative algebras. We also describe similar results for the twisted analog of dendriform coalgebras.

On equivariant dendriform algebras

Dendriform algebras are certain associative algebras whose product splits into two binary operations and the associativity splits into three new identities. In this paper, we study finite group actions on dendriform algebras. We define equivariant cohomology for dendriform algebras equipped with finite group actions similar to the Bredon cohomology for topological $G$-spaces. We show that equivariant cohomology of such dendriform algebras controls equivariant one-parameter formal deformations.

The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in natural modules

The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in natural modules

Equivariant one-parameter formal deformations of Hom-Leibniz algebras

The aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which control deformations of Hom-Leibniz algebras. The cohomology and the associated deformation theory for Hom-Leibniz algebras as developed here are also extended to equivariant context, under the presence of finite group actions on Hom-Leibniz algebras.

One-Relator Maximal Pro-p Galois Groups and the Koszulity Conjectures

Abstract Let p be a prime number and let ${\mathbb{K}}$ be a field containing a root of 1 of order p. If the absolute Galois group $G_{\mathbb{K}}$ satisfies $\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p)\lt\infty$ and $\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for ${\mathbb{K}}$. Also, under the above hypothesis, we show that the $\mathbb{F}_p$-cohomology algebra of $G_{\mathbb{K}}$ is the quadratic dual of the graded algebra ${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{\mathbb{K}}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s elementary type conjecture.

Computation of cohomology of vertex algebras

We review cohomology theories corresponding to the chiral and classical operads. The first one is the cohomology theory of vertex algebras, while the second one is the classical cohomology of Poisson vertex algebras (PVA), and we construct a spectral sequence relating them. Since in “good” cases the classical PVA cohomology coincides with the variational PVA cohomology and there are well-developed methods to compute the latter, this enables us to compute the cohomology of vertex algebras in many interesting cases. Finally, we describe a unified approach to integrability through vanishing of the first cohomology, which is applicable to both classical and quantum systems of Hamiltonian PDEs.

On Leibniz cohomology

In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with adjoint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of the dual of a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.

Book Review: Hochschild cohomology for algebras

Book Review: Hochschild cohomology for algebras

Cohomology of BiHom-associative algebras

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.

On reconstructing subvarieties from their periods

We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then verifies the results. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. A highlight of the method is that the Picard group computations are proved to be correct despite the fact that the Picard numbers of our examples are not extremal.

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras . These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.

Lie algebra fermions

We define a supersymmetric quantum mechanics of fermions that take values in a simple Lie algebra . We summarize what is known about the spectrum and eigenspaces of the Laplacian which corresponds to the Koszul differential d. Firstly, we concentrate on the zero eigenvalue eigenspace which coincides with the Lie algebra cohomology. We provide physical insight into useful tools to compute the cohomology, namely Morse theory and the Hochschild–Serre spectral sequence. We list explicit generators for the Lie algebra cohomology ring. Secondly, we concentrate on the eigenspaces of the supersymmetric quantum mechanics with maximal eigenvalue at given fermion number. These eigenspaces have an explicit description in terms of abelian ideals of a Borel subalgebra of the simple Lie algebra. We also introduce a model of Lie algebra valued fermions in two dimensions, where the spaces of maximal eigenvalue acquire a cohomological interpretation. Our work provides physical interpretations of results by mathematicians, and simplifies the proof of a few theorems. Moreover, we recall that these mathematical results play a role in pure supersymmetric gauge theory in four dimensions, and observe that they give rise to a canonical representation of the four-dimensional chiral ring.

The last formula of Jean-Louis Koszul

Since the first international conference in France on the Information Geometry, GSI2013, Jean-Louis Koszul’s interest in Information Geometry went increasing. Motivated by the impact of the cohomology of Koszul-Vinberg algebras on the Information Geometry and moved by some issues raised by Albert Nijenhuis, Jean-Louis Koszul undertook another rewriting of the Brut formula of the coboundary operator of the KV complex. The source of other motivations of Jean-Louis Koszul was the relationships between the theory of KV cohomology and the theory of deformation of locally flat manifolds.In 2015 Jean-Louis Koszul sent me his Last formula of the KV boundary operator. In that Last formula Jean-Louis Koszul dealt with the case where spaces of coefficients are trivial modules of KV algebras. A part of the present work is devoted to extending the Last formula of Jean-Louis Koszul to KV cochain complexes whose spaces of coefficients are non-trivial two-sided modules of KV algebras. At another side, I also aim to highlight other significant impacts of the theory of KV cohomology of Koszul-Vinberg algebras. In particular I will use the KV cohomology to widely revisit the theory of statistical models of measurable sets. The reader will see why the source of the theory of statistical models is of homological nature.I also intend to highlight several impacts of the KV cohomology on the quantitative differential topology. I am particularly concerned with problems regarding the existence of Riemannian foliations, the existence of symplectic foliations as well as the existence of multi-dimensional webs. The homological theory of statistical models is presented as branches of rooted trees whose roots are weakly Jensen random cohomology classes.

Spectral sequences for commutative Lie algebras

AbstractWe construct some spectral sequences as tools for computing commutative cohomology of commutative Lie algebras in characteristic 2. In a first part, we focus on a Hochschild-Serre-type spectral sequence, while in a second part we obtain spectral sequences which compare Chevalley--Eilenberg-, commutative- and Leibniz cohomology. These methods are illustrated by a few computations.

Right-angled Artin groups and enhanced Koszul properties

Abstract Let 𝔽 {\mathbb{F}} be a finite field. We prove that the cohomology algebra H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} with coefficients in 𝔽 {\mathbb{F}} of a right-angled Artin group G Γ {G_{\Gamma}} is a strongly Koszul algebra for every finite graph Γ. Moreover, H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} is a universally Koszul algebra if, and only if, the graph Γ associated to the group G Γ {G_{\Gamma}} has the diagonal property. From this, we obtain several new examples of pro-p groups, for a prime number p, whose continuous cochain cohomology algebra with coefficients in the field of p elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro-p Galois groups of fields formulated by J. Mináč et al.