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We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in Yang and Zhao (The cohomological Hall algebra of a preprojective algebra. arXiv: 1407.7994v5, 2015) for any quiver Q and any one-parameter formal group G. In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra gQ associated to Q, whose quantization comes from the formal group G. We prove, when the group G is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian. [ABSTRACT FROM AUTHOR]

R. W. Carey and J. Pincus in [] proposed an index theory for non-Fredholm bounded operators T on a separable Hilbert space such that is in the trace class. We showed in [] using Dirac-type operators acting on sections of bundles over that we could construct bounded operators T satisfying the more general condition that the operator is in the trace class. We proposed there a 'homological index' for these Dirac-type operators given by . In this paper we show that the index introduced in [] represents the result of a paring between a cyclic homology theory for the algebra generated by T and and its dual cohomology theory. This leads us to establish the homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations. [ABSTRACT FROM AUTHOR]

In this paper we transport Steenrod's cup-iproducts,i ≥ 0, from the singular cochains on the free loop space Maps(S1,BG) to Hochschild's original cochain complex Homk(k[G]⊗*,k[G]) defining Hochschild cohomology. HereGis a discrete group,kan arbitrary (commutative) coefficient ring, andBGthe classifying space ofG. This induces a natural action of the (mod 2) Steenrod algebra on the Hochschild cohomology of a group ring. For cochains supported onBG, we prove that Gerstenhaber's cup product agrees with the simplicial cup product and Gerstenhaber's pre-Lie product agrees with Steenrod's cup-one product. As a consequence, for cocyclesfandgsupported onBG, the Gerstenhaber bracket [f,g] = 0 inHH*(k[G]; k[G]). This is interpreted in terms of the Batalin–Vilkovisky structure onHH*(k[G]; k[G]). [ABSTRACT FROM PUBLISHER]

Based on Kadeishvili's original theorem inducing A ∞ {A_{\infty}} -algebra structures on the homology of dg-algebras, several directions of algorithmic research in A ∞ {A_{\infty}} -algebras have been pursued. In this paper, we survey the work done on calculating explicit A ∞ {A_{\infty}} -algebra structures from homotopy retractions, in group cohomology and in persistent homology. [ABSTRACT FROM AUTHOR]

Persistent homology with coefficients in a field $$\mathbb {F}$$ coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substantially the amount of matrix operations. In addition, we propose a heuristic to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with state-of-the-art software for persistent homology and cohomology. [ABSTRACT FROM AUTHOR]

In this paper, we study an excision theorem of the dihedral and reflexive (co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dihedral cohomology. Also, we present and prove the relation between cyclic and dihedral cohomology of algebras and some examples. [ABSTRACT FROM AUTHOR]

In this paper, the authors develop an approach to construct a q-deformed Heisenberg-Virasoro algebra which is a Hom-Lie algebra, and investigate its central extensions and second cohomology group. Finally, quantum deformations of the Heisenberg-Virasoro algebra which provide a non-trivial Hopf structure are presented. [ABSTRACT FROM AUTHOR]

We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber (resp. Batalin-Vilkovisky) algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In case the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to be zero in cohomology and hence the interesting structure is not given by this e2- algebra structure but rather by the resulting e3-algebra structure, which is expressed in terms of the cup product and B. [ABSTRACT FROM AUTHOR]

It is proved that every topologically pure extension of Fréchet algebras 0 → I → A → A/I → 0 such that I is strongly H-unital has the excision property in continuous (co)homology of the following types: bar, naive-Hochschild, Hochschild, cyclic, and periodic cyclic. In particular, the property holds for every extension of Fréchet algebras such that I has a left or right bounded approximate identity. [ABSTRACT FROM PUBLISHER]

In the context of commutative differential graded algebras over ℚ , we show that an iteration of "odd spherical fibration" creates a "total space" commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its "fibrations" with "fiber" of finite cohomological dimension, the induced map on cohomology is injective. [ABSTRACT FROM AUTHOR]

We present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved for any quadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example. [ABSTRACT FROM AUTHOR]

We study the homological algebra of bimodules over involutive associative algebras. We show that Braun's definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the ℤ/2-invariants intersected with the centre. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of ℤ/2-coinvariants and abelianization. [ABSTRACT FROM PUBLISHER]

We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld (Anal i Prilozhen 20(1):69-70, ). We generalize in this way, the Dirac cohomology theory for Lusztig's graded affine Hecke algebras defined in Barbasch et al. (Acta Math 209(2):197-227, ) and further developed in Barbasch et al. (Acta Math 209(2):197-227, ), Ciubotaru et al. (J Inst Math Jussieu 13(3):447-486, ), Ciubotaru (J Reine Angew Math 671:199-222, ), Ciubotaru and He (Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index, Adv. Math. 283:1-50, ), Chan (On a twisted Euler-Poincaré pairing for graded affine Hecke algebras, preprint , ). We apply these constructions to the case of the symplectic reflection algebras defined by Etingof and Ginzburg (Invent Math 147:243-348, ), particularly to rational Cherednik algebras for real or complex reflection groups. As applications, we give criteria for unitarity of modules in category $${\mathcal {O}}$$ and we show that the 0-fiber of the Calogero-Moser space admits a description in terms of a certain 'Dirac morphism' originally defined by Vogan for representations of real reductive groups. [ABSTRACT FROM AUTHOR]

We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras. [ABSTRACT FROM AUTHOR]

We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety X. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo-Mumford regularity of the sheaf of differential p-forms on X is bounded by p( em+1) D, where e, m, and D are the maximal codimension, dimension, and degree, respectively, of all irreducible components of X. It follows that, for a union V of generic hyperplane sections in X, the algebraic de Rham cohomology of X∖ V is described by differential forms with poles along V of single exponential order. By covering X with sets of this type and using a Čech process, we obtain a similar description of the de Rham cohomology of X, which allows its efficient computation. Furthermore, we give a parallel polynomial time algorithm for testing whether a projective variety is smooth. [ABSTRACT FROM AUTHOR]

We consider two preenveloping classes of left R-modules [This symbol cannot be presented in ASCII format], [This symbol cannot be presented in ASCII format] such that Inj ⊂ 7 ⊂ [This symbol cannot be presented in ASCII format], and a left R-module N. For any left R-module M and n ≥ we define the relative extension modules [This symbol cannot be presented in ASCII format](M, N) and prove the existence of an exact sequence connecting these modules and the modules Ext[This symbol cannot be presented in ASCII format](M, N) and Ext[This symbol cannot be presented in ASCII format](M, N). We show that there is a long exact sequence of [This symbol cannot be presented in ASCII format](M,—) associated with a Hom(—, [This symbol cannot be presented in ASCII format]) exact sequence 0 ↵ N′ ↵ N ↵ N″ ↵ 0 and a long exact sequence of [This symbol cannot be presented in ASCII format](—, N) associated with a Hom(—, [This symbol cannot be presented in ASCII format]) exact sequence 0 ↵ M′ ↵ M ↵ M″ ↵ 0. Using these properties we prove that for two complete hereditary cotorsion theories ([This symbol cannot be presented in ASCII format], [This symbol cannot be presented in ASCII format]), ([This symbol cannot be presented in ASCII format], [This symbol cannot be presented in ASCII format]) we have [This symbol cannot be presented in ASCII format]Proj(M, N) ≅ [This symbol cannot be presented in ASCII format]Inj(M, N) for any left R modules M, N and for any n ≥ 1, where [This symbol cannot be presented in ASCII format]Proj(M, N) are the generalized Tate cohomology modules (see Section I for the definition). So in this case we have an occurrence of balance, i.e. the generalized Tate cohomology can be computed either using a left [This symbol cannot be presented in ASCII format]-resolution and a projective resolution of M or using a right [This symbol cannot be presented in ASCII format]-resolution and an injective resolution of N. [ABSTRACT FROM AUTHOR]