Cohomology Of Algebras Research Articles

Formality of -objects

We show that a$\mathbb{P}$-object and simple configurations of$\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra

In this contribution, we provide the results on the low-dimensional algebraic cohomology with values in the trivial and the adjoint module of the Witt and the Virasoro algebra over a field with . A sketch of the various proofs is given, the details can be found in previous articles of the authors. More precisely, we give a summary of results known concerning the zeroth, first and second algebraic cohomology of the Witt and the Virasoro algebra, and we present some details about the more recent results related to the third algebraic cohomology. It has to be noted that we are dealing entirely with algebraic cohomology, meaning that our results are valid for any concrete realization of the Witt and the Virasoro algebra. Moreover, our results are independent of any underlying topology chosen.

On the ℓ q,p cohomology of Carnot groups

We study the simplicial l q,p cohomology of Carnot groups G. We show vanishing and non-vanishing results depending of the range of the (p, q) gap with respect to the weight gaps in the Lie algebra cohomology of G.

Group invariance of integrable Pfaffian systems

Let $${\mathcal {S}}$$ be an integrable Pfaffian system. When it is invariant under a transversally free infinitesimal action of a finite-dimensional real Lie algebra g and consequently invariant under the local action of a Lie group G, we show that the vertical variational cohomology of $${\mathcal {S}}$$ is equal to the Lie algebra cohomology of g with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Batalin-Vilkovisky operators on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson structure is pseudo-unimodular. The relation between modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.

Singular Hochschild cohomology and algebraic string operations

Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex \mathcal{D}^{*}(A,A) , called the Tate–Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex \mathcal{D}^*(A,A) computes the singular Hochschild cohomology of A . We construct a cyclic (or Calabi–Yau) A -infinity algebra structure, which extends the classical Hochschild cup and cap products, and an L -infinity algebra structure extending the classical Gerstenhaber bracket, on \mathcal{D}^*(A,A) . Moreover, we prove that the cohomology algebra H^*(\mathcal{D}^*(A,A)) is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.

Resolutions for twisted tensor products

We build resolutions for general twisted tensor products of algebras. These bimodule and module resolutions unify many constructions in the literature and are suitable for computing Hochschild (co)homology and more generally Ext and Tor for (bi)modules. We analyze in detail the case of Ore extensions, consequently obtaining Chevalley-Eilenberg resolutions for universal enveloping algebras of Lie algebras (defining the cohomology of Lie groups and Lie algebras). Other examples include semidirect products, crossed products, Weyl algebras, Sridharan enveloping algebras, and Koszul pairs.

Cohomology of finite p-groups of fixed nilpotency class

Let p be a prime number and let c,d be natural numbers. Then, the number of possible isomorphism types for the mod p cohomology algebra of a d-generated finite p-group of nilpotency class c is bounded by a function depending only on p, c and d.

The Hochschild cohomology of square-free monomial complete intersections

We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular we provide an explicit formula for the cup product which gives the cohomology module an algebra structure and then we describe this structure in terms of generators and relations. In addition, we compute the Hilbert series of the Hochschild cohomology of these algebras.

Topological Hopf algebras and their Hopf-cyclic cohomology

A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of coefficients (AYD modules) over a topological Lie algebra and those over its universal enveloping (Hopf) algebra are isomorphic. For topological Hopf algebras, the category of coefficients is identified with the representation category of a topological algebra called the anti-Drinfeld double. Finally, a topological van Est type isomorphism is detailed, connecting the Hopf-cyclic cohomology to the relative Lie algebra cohomology with respect to a maximal compact subalgebra.

An n-dimensional Klein bottle

AbstractAn n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

L∞-algebras and their cohomology

We give an overview of different characterisations of L∞-structures in terms of symmetric brackets and (co)differentials on the symmetric (co)algebra. We then do the same for their representations (up to homotopy) and approach L∞-algebra cohomology using the commutator bracket on the space of coderivations of the symmetric coalgebra. This leads to abelian extensions of L∞-algebras by 2-cocycles.

On the characteristic rank of vector bundles over oriented Grassmannians

We study the cohomology algebra of the Grassmann manifold $\widetilde G_{k,n}$ of oriented $k$-dimensional subspaces in $\mathbb R^{n+k}$ via the characteristic rank of the canonical vector bundle $\widetilde\gamma_{k,n}$ over $\widetilde G_{k,n}$ (denote

OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension ${\leqslant}3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension ${\geqslant}4$ that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general.

Hochschild cohomology of polynomial representations of GL2

We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of GL2 over an algebraically closed field of characteristic p>2 , that is, of any block whose number of simple modules is a power of p>2. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.

Volumes of SLn(ℂ)–representations of hyperbolic 3–manifolds

Let $M$ be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows the strategy of Reznikov's rigidity when $M$ is closed, in particular we use Fuks' approach to variations by means of Lie algebra cohomology. When $n=2$, we get back Hodgson's formula for variation of volume on the space of hyperbolic Dehn fillings. Our formula also yields the variation of volume on the space of decorated triangulations obtained by Bergeron-Falbel-Guillou and Dimofte-Gabella-Goncharov.

Computation methods of logarithmic vector fields associated to semi-weighted homogeneous isolated hypersurface singularities

Methods for computing logarithmic vector fields along a semi-weighted homogeneous hypersurface with an isolated singularity are considered in the context of symbolic computation. The main idea of our approach is based on the concept of polar variety and of algebraic local cohomology. New algorithms are introduced for computing a set of generators of the modules of logarithmic vector fields. The keys of the resulting algorithms are a notion of parametric syzygy system and that of parametric local cohomology system.

Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields

AbstractIn this paper, we explore a relationship between the topology of the complex hyperplane complements ℳBCn(ℂ) in type B/C and the combinatorics of certain spaces of degree-n polynomials over a finite field Fq. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras H*(ℳBCn(ℂ);ℂ), and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over Fq with non-zero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FIW-algebras finitely generated inFIW-degree 2, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.

Oriented Algebras and the Hochschild Cohomology Group

Koam and Pirashivili developed the equivariant version of Hochschild cohomology by mixing the standard chain complexes computing group with associative algebra cohomologies to obtain the bicomplex C ˜ G * ( A , X ). In this paper, we form a new bicomplex F ˘ G * ( A , X ) by deleting the first column and the first row and reindexing. We show that H ˘ G 1 ( A , X ) classifies the singular extensions of oriented algebras.

FREE ACTIONS OF SOME COMPACT GROUPS ON MILNOR MANIFOLDS

AbstractIn this paper, we investigate free actions of some compact groups on cohomology real and complex Milnor manifolds. More precisely, we compute the mod 2 cohomology algebra of the orbit space of an arbitrary free ℤ2 and $\mathbb{S}^1$-action on a compact Hausdorff space with mod 2 cohomology algebra of a real or a complex Milnor manifold. As applications, we deduce some Borsuk–Ulam type results for equivariant maps between spheres and these spaces. For the complex case, we obtain a lower bound on the Schwarz genus, which further establishes the existence of coincidence points for maps to the Euclidean plane.