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The existence or nonexistence of a complex structure on S 6 was a long standing unsolved problem. There is a well-known orbit O (Λ) in G 2 which is diffeomorphic to S 6 and used by Gábor Etesi in an effort to find a complex structure. Etesi suggested to give a complex structure in S 6 through this orbit. In Daniel Guan's earlier paper, he proved that the orbit can not be a complex submanifold. Since there was not a clear description of the map from O (Λ) to S 6 in that paper, we give another clearer, simpler and explicit proof of that result in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the notions of a compatible Lie triple system and its representation. We construct a bidifferential graded Lie algebra whose Maurer–Cartan elements are compatible Lie triple systems. We also obtain the bidifferential graded Lie algebra which controls deformations of a compatible Lie triple system. Then we investigate the cohomology theory of compatible Lie triple systems and consider the connection between the cohomology group of compatible Lie triple systems and the cohomology group of Lie triple systems. Furthermore, we develop the 1-parameter formal deformation theory of compatible Lie triple systems and prove that it is governed by the cohomology groups. At last, we study abelian extensions of compatible Lie triple systems and classify them by the second cohomology group. [ABSTRACT FROM AUTHOR]

A relative Rota-Baxter algebra is a generalization of a Rota-Baxter algebra. Relative Rota-Baxter algebras are closely related to dendriform algebras. In this paper, we introduce bimodules over a relative Rota-Baxter algebra that fits with the representations of dendriform algebras. We define the cohomology of a relative Rota-Baxter algebra with coefficients in a bimodule and then study abelian extenfsions of relative Rota-Baxter algebras in terms of the second cohomology group. Finally, we consider homotopy relative Rota-Baxter algebras and classify skeletal homotopy relative Rota-Baxter algebras in terms of the above-defined cohomology. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the notion of an HLieDer triple, which includes a Hom-Lie algebra and a derivation. We define a cohomology theory for HLieDer triples with coefficients in a representation. We study central extensions of an HLieDer triple. Finally, we consider homotopy derivations on HLieb∞ algebras and 2-derivations on Hom-Lie 2-algebras, and we prove that the category of 2-term HLieb∞ algebras with homotopy derivations and the category of Hom-Lie 2-algebras with 2-derivations are equivalent. [ABSTRACT FROM AUTHOR]

In this paper, we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras. Explicitly, we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras. Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras. In particular, we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang (2001) in their study of bi-Hamiltonian structures. Finally, we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras. [ABSTRACT FROM AUTHOR]

Let Γ be an arithmetic subgroup of SU (d , 1) with cusps, and let X Γ be the associated locally symmetric space. In this paper we investigate the pre-image of Γ in the covering groups of SU (d , 1) . Let H ! ∙ (X Γ , C) be the inner cohomology, i.e. the image in H ∙ (X Γ , C) of the compactly supported cohomology. We prove that if the first inner cohomology group H ! 1 (X Γ , C) is non-zero then the pre-image of Γ in each connected cover of SU (d , 1) is residually finite. At the end of the paper we give an example of an arithmetic subgroup Γ satisfying the criterion H ! 1 (X Γ , C) ≠ 0 . [ABSTRACT FROM AUTHOR]

In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space. [ABSTRACT FROM AUTHOR]

In this paper, first we introduce the notion of a nonabelian embedding tensor, which is a nonabelian generalization of an embedding tensor. Then, we introduce the notion of a Leibniz–Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor, and can also be viewed as a nonabelian generalization of a Leibniz algebra. Next using the derived bracket, we construct a differential graded Lie algebra, whose Maurer–Cartan elements are exactly nonabelian embedding tensors. Consequently, we obtain the differential graded Lie algebra that governs deformations of a nonabelian embedding tensor. Finally, we define the cohomology of a nonabelian embedding tensor and use the second cohomology group to characterize linear deformations. [ABSTRACT FROM AUTHOR]

It was conjectured in Glover (Trans Am Math Soc 267:423–434, 1981) that for a complex flag manifold F every endomorphism φ : H ∗ (F ; Z) → H ∗ (F ; Z) is either a grading endomorphism or a projective endomorphism. In this paper, we verify this conjecture for a new class of complex flag manifolds that captures all cases for which the conjecture was previously known to be true. This allows us to calculate the noncoincidence index (invariant that naturally generalizes the fixed-point property) for these manifolds. [ABSTRACT FROM AUTHOR]

It is shown that the lamplighter Lie algebra over the field of rational numbers, introduced by S. Ivanov, R. Mikhailov, and A. Zaikovskii, is isomorphic to the infinite-dimensional naturally graded Lie algebra of maximal class . Y. Félix and A. Murillo proved that its -dimensional homology is infinite-dimensional. However, they failed to completely calculate the spaces , . In this paper, an infinite basis of the bigraded homology is explicitly constructed using the results of D. Millionshchikov and A. Fialowski on the cohomology . [ABSTRACT FROM AUTHOR]

In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber’s formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation. [ABSTRACT FROM AUTHOR]

Recently, Skjelnes and Smith classified which Hilbert schemes on projective space are smooth in terms of integer partitions λ = (λ 1 , ... , λ r) with r = 0 , λ = (n + 1) , or n ⩾ λ 1 ⩾ ⋯ ⩾ λ r ⩾ 1 . In particular, they found there to be seven families of smooth Hilbert schemes: one with r = 0 or λ = (n + 1) , one with Hilbert schemes on the projective line or plane, 4 families with λ r = 1 , and one with λ r ⩾ 2 . In this paper, we compute the sum of the Betti numbers for all of these families of smooth Hilbert schemes over projective space except the case λ r ⩾ 2 . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce αk-derivations of multiplicative Hom-Lie conformal algebras and study their properties. [ABSTRACT FROM AUTHOR]

It is well known that the set of isomorphism classes of extensions of groups with abelian kernel is characterized by the second cohomology group. In this paper we generalise this characterization of extensions to a natural class of extensions of monoids, the cosetal extensions. An extension is cosetal if for all g , g ′ ∈ G in which e (g) = e (g ′) , there exists a (not necessarily unique) n ∈ N such that g = k (n) g ′ . These extensions generalise the notion of special Schreier extensions, which are themselves examples of Schreier extensions. Just as in the group case where a semidirect product could be associated to each extension with abelian kernel, we show that to each cosetal extension (with abelian group kernel), we can uniquely associate a weakly Schreier split extension. The characterization of weakly Schreier split extensions is combined with a suitable notion of a factor set to provide a cohomology group granting a full characterization of cosetal extensions, as well as supplying a Baer sum. [ABSTRACT FROM AUTHOR]

The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $$h_k(X)$$ for an arbitrary simplicial complex X, and any $$k\ge 0$$ . In this paper we investigate the value of $$h_1(\Delta ^{[n]})$$ -the first Cheeger constant of a simplex with n vertices. It is known, due to the pioneering work of Meshulam and Wallach [12], that and that the equality $$h_1(\Delta ^{[n]})=n/3$$ is achieved when n is divisible by 3. Here we expand on these results. First, we show that So the sharp equality holds on a set whose density goes to 1. Second, we show that In other words, as n goes to infinity, the value $$h_1(\Delta ^{[n]})-n/3$$ is either 0 or goes to 0 very rapidly. Our methods include recasting the original question in purely graph-theoretic language, followed by a detailed investigation of a specific graph family, the so-called staircase graphs. These are defined by associating a graph to every partition, and appear to be especially suited to gain information about the first Cheeger constant of a simplex. [ABSTRACT FROM AUTHOR]

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic (0, q)-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we estimate the dimension of cohomology of semipositive line bundles on q-convex manifolds, pseudo-convex domains, weakly 1-complete manifolds and complete manifolds. We also obtain the estimate of cohomology on compact manifolds with semipositive line bundles endowed with a Hermitian metric with analytic singularities and related results. [ABSTRACT FROM AUTHOR]

In the predecessor to this paper Dempwolff (Comm Algebra 34(3):1077–1131, 2006), group-theoretic methods were used to solve automorphism and equivalence questions for (certain) ordinary bent functions, i.e., bent functions over F 2 . Here, we consider the same problems for p-ary bent functions, p an odd prime and solve these questions for functions analogous to those which appear in Dempwolff (Comm Algebra 34(3):1077–1131, 2006). Although our general analysis is similar to the approach of Dempwolff (Comm Algebra 34(3):1077–1131, 2006), it turns out that the odd characteristic leads to simplifications: Often, the double derivative can be computed (cf. Lemma 2.10) and factorizations of the automorphism group (cf. Lemma 2.3) can be established resulting in restrictions for automorphisms and equivalence maps. [ABSTRACT FROM AUTHOR]

In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) Künneth formula; (v) local and global Verdier duality. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the k-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic 0. [ABSTRACT FROM AUTHOR]

We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincaré dualities for them and prove a stabilization property for the infinite series of reflection groups. [ABSTRACT FROM AUTHOR]

Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra 𝔤 and a root of unity ζ. When L, L′ are irreducible Uζ-modules having regular highest weights, the dimension of Ext U ζ n (L , L ′) can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of Uζ. This paper shows for L, L′ irreducible modules in a singular block that dim Ext U ζ n (L , L ′) is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra. [ABSTRACT FROM AUTHOR]

Let G be a semisimple, simply connected, linear algebraic group over an algebraically closed field k. Donkin (In J. Algebra, 307, 570–613 2007), Donkin gave a recursive description for the characters of the cohomology of line bundles on the three dimensional flag variety in prime characteristic. The recursion involves not only line bundles but also certain natural rank 2 bundles associated to two dimensional B −modules Nα(λ), where λ in an integral weight and α is a simple root. In this paper we compute the cohomology of these rank 2 bundles and simplify the recursion in Donkin (In J. Algebra, 307, 570–613 2007). We also compute the socle of Nα(λ) and give a rank 2 version of Kempf's vanishing theorem. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study cohomological theory of Hom-associative H-pseudoalgebras and Hom–Lie H-pseudoalgebras. We define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative H-pseudoalgebra. Furthermore, the symmetric Schouten product and alternating Schouten product are studied. Using the Gerstenhaber bracket and alternating Schouten product, differential graded Lie algebra are constructed on the space of multilinear mappings of Hom-associative H-pseudoalgebra and Hom-Lie H-pseudoalgebras. [ABSTRACT FROM AUTHOR]

In this paper we construct arbitrarily large families of smooth projective varieties and closed Riemannian manifolds that share many algebraic and analytic invariants. For instance, every non-arithmetic, closed hyperbolic 3-manifold admits arbitrarily large collections of non-isometric finite covers which are strongly isospectral, length isospectral, and have isomorphic integral cohomology where the isomorphisms commute with restriction and co-restriction. We can also construct arbitrarily large collections of pairwise non-isomorphic smooth projective surfaces where these isomorphisms in cohomology are natural with respect to Hodge structure or as Galois modules. In particular, the projective varieties have isomorphic Picard and Albanese varieties, and they also have isomorphic effective Chow motives. Our construction employs an integral refinement of the Gassman–Sunada construction that has recently been utilized by D. Prasad. One application of our work shows the non-injectivity of the map from the Grothendieck group of varieties over Q ¯ to the Grothendieck group of the category of effective Chow motives. We also answer a question of D. Prasad. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an L ∞ -algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This leads to cohomology associated to a twisted Rota–Baxter operator. This cohomology can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota–Baxter operators by means of the above-defined cohomology. Application is given to Reynolds operators. In the second part, we consider NS-algebras of Leroux that are related to twisted Rota–Baxter operators in the same way dendriform algebras are related to Rota–Baxter operators. We define cohomology of NS-algebras using non-symmetric operads and study their deformations in terms of the cohomology. [ABSTRACT FROM AUTHOR]

This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We provide an overview of the theory and explore the structure properties such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of generalized representation of a 3-Lie algebra, by which we obtain a generalized semidirect product 3-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product 3-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements. [ABSTRACT FROM AUTHOR]

We completely calculate the Fitting ideal of the classical p-ramified Iwasawa module for any abelian extension K/k of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results where we had to assume that only p-adic places may ramify in K/k. One of the important ingredients is the computation of some complexes in appropriate derived categories. [ABSTRACT FROM AUTHOR]

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups ϒ? κ: Hκ( A, F) → Hκ( Lχ, Fχ), called the localization map, where Lχ is the adjoint algebra at χ ∈ M. The main result in this paper is that if M is simply connected, or H0( Lχ, Fχ) is trivial, then ϒ1 is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases. [ABSTRACT FROM AUTHOR]

Let Vect (R) be the Lie algebra of smooth vector fields on R and D τ , λ , μ ; ν be the space of trilinear differential operators acting on weighted densities. The main topic of this paper is two folds. First, we compute the first differential cohomology group of the Lie algebra sl (2) with coefficients in D τ , λ , μ ; ν . Second, we classify sl (2) -invariant linear differential operators from Vect (R) to D τ , λ , μ ; ν vanishing on sl (2) . This result allows us to compute the first differential sl (2) -relative cohomology of Vect (R) with coefficients in D τ , λ , μ ; ν . This work is the simplest generalization of a result by Bouarroudj (Int J Geom Methods Mod Phys 2(1):23–40, 2005). [ABSTRACT FROM AUTHOR]

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D( G) u( $$\mathfrak{g}$$ ) containing u( $$\mathfrak{g}$$ ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D( G r) u( $$\mathfrak{g}$$ ) have also been constructed, and the results are stated in this setting. [ABSTRACT FROM AUTHOR]

Let $$F\subseteq {\mathbb {P}^{3}}$$ be a smooth determinantal quartic surface which is general in the Nöther-Lefschetz sense. In the present paper we give a complete classification of locally free sheaves $${\mathcal E}$$ of rank 2 on F such that $$h^1(F,{\mathcal E}(th))=0$$ for $$t\in \mathbb {Z}$$ . [ABSTRACT FROM AUTHOR]

Let $$F\subseteq {\mathbb {P}^{3}}$$ be a smooth quartic surface and let $${\mathcal {O}}_F(h):={\mathcal {O}}_{{\mathbb {P}^{3}}}(1)\otimes {\mathcal {O}}_F$$ . In the present paper we classify locally free sheaves $${\mathcal {E}}$$ of rank 2 on F such that $$c_1({\mathcal {E}})={\mathcal {O}}_F(2h), c_2({\mathcal {E}})=8$$ and $$h^1\big (F,{\mathcal {E}}(th)\big )=0$$ for $$t\in \mathbb {Z}$$ . We also deal with their stability. [ABSTRACT FROM AUTHOR]

Suppose V is an irreducible $${\mathbb F}_pG$$ -module where p is a prime and G is a finite p-solvable group. Then by a result of Gaschütz $$\dim _FH^1(G,V)=s_G(V)$$ where $$F=\text {End}_G(V)$$ and $$s_G(V)$$ is the multiplicity of V as a split chief factor in any chief series of G. One also knows that $$\dim _FH^1(G,V)\le \dim _FH^2(G,V)$$ , and the object of the present paper is to explain this inequality in terms of Gaschütz's result. [ABSTRACT FROM AUTHOR]

Currently, modelling hysteresis losses in superconductors is most often based on the H-formulation of the eddy current model (ECM) solved using the finite element method (FEM). In the H-formulation, the problem is expressed using the magnetic field intensity H and discretized using edge elements in the whole domain. Even though this approach is well established, it uses unnecessary degrees of freedom (DOFs) and introduces modelling error such as currents flowing in air regions due to finite air resistivity. In this paper, we present a modelling tool utilizing another H-oriented formulation of the ECM, making use of cohomology of the air regions. We constrain the net currents through the conductors by fixing the DOFs related to the so-called cohomology basis functions. As air regions will be truly non-conducting, DOFs and running times of these nonlinear simulations are reduced significantly as compared to the classical H-formulation. This fact is demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ : G → G be a strict endomorphism (i.e., the subgroup G( σ) of σ-fixed points is finite). Also, let G be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group H m( G( σ), L) for any irreducible k G( σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups $ Ext^{m}_{G(\sigma )}(L,L^{\prime })$ between irreducible k G( σ)-modules $L,L^{\prime }$, with a similar result holding for G. In these Ext m results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p. We also show that one can find bounds independent of the prime for the Cartan invariants of G( σ) and G, and even for the lengths of the underlying PIMs. [ABSTRACT FROM AUTHOR]

We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre's diagonalization for cubical cells. An implementation in C++ of the introduced algorithms is available at together with some examples. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it. [ABSTRACT FROM AUTHOR]

A Lie algebra morphism triple is a triple (g , h , ϕ) consisting of two Lie algebras g , h and a Lie algebra homomorphism ϕ : g → h . We define representations and cohomology of Lie algebra morphism triples. As applications of our cohomology, we study some aspects of deformations, abelian extensions of Lie algebra morphism triples and classify skeletal sh Lie algebra morphism triples. Finally, we consider the cohomology of Lie group morphism triples and find a relation with the cohomology of Lie algebra morphism triples. [ABSTRACT FROM AUTHOR]

We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials. [ABSTRACT FROM AUTHOR]

In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Fröhlich and Furtado-Coelho's general theory on Baer invariants in varieties of Ω-groups and Modi's theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the last two decades. We show how to use Baer invariants in order to connect these various homology theories. First, we express the low-dimensional Baer invariants of precrossed modules in terms of a new non-abelian tensor product of a precrossed module. This expression is used to analyze the connection between the Baer invariants and the homological invariants of precrossed modules defined by Conduché and Ellis. Specifically we prove that the second homological invariant of Conduché and Ellis is in general a quotient of the first component of the Baer invariant we consider. The definition of classical Baer invariants is generalized using homological methods. These generalized Baer invariants of precrossed modules are applied to the construction of five term exact sequences connecting the generalized Baer invariants with the cohomology theory of crossed modules considered by Carrasco, Cegarra and R.-Grandjeán and the cohomology theory of precrossed modules. [ABSTRACT FROM AUTHOR]

Hom-Lie algebra (superalgebra) structure appeared naturally in q-deformations, based on σ-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of αk-derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted osp(1, 2) superalgebra and q-deformed Witt superalgebra. [ABSTRACT FROM AUTHOR]

Let ( M, M, N) be three symplectic manifolds and suppose that we can do the symplectic connected sum of M and M along their submanifold N to obtain M M. In this paper, we consider the bilinear and cubic forms of H*( M M, ℤ) when dim M M = 4, 6. Under some conditions, we get some relations of the bilinear and the cubic forms between M M and M∐ M. [ABSTRACT FROM AUTHOR]

Based on the basis theorem of Bruhat–Chevalley (in Algebraic Groups and Their Generalizations: Classical Methods, Proceedings of Symposia in Pure Mathematics, vol. 56 (part 1), pp. 1–26, AMS, Providence, ) and the formula for multiplying Schubert classes obtained in (Duan, Invent. Math. 159:407–436, ) and programmed in (Duan and Zhao, Int. J. Algebra Comput. 16:1197–1210, ), we introduce a new method for computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). The method and results of this paper have been extended in (Duan and Zhao, and ) to obtain the integral cohomology rings of all complete flag manifolds, and to construct the integral cohomologies of Lie groups in terms of Schubert classes. [ABSTRACT FROM AUTHOR]

Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of “discrete differential forms” built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes. [ABSTRACT FROM AUTHOR]

The space of linear polyvector fields on $$\mathbb{R}^d$$ is a Lie subalgebra of the (graded) Lie algebra $$T_{\rm poly}(\mathbb{R}^d)$$ , equipped with the Schouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in $$T_{\rm poly}(\mathbb{R}^d)$$ , restricting ourselves to the case of cochains defined with purely aerial Kontsevich’s graphs, as in Pac. J. Math. 218(2):201–239, 2005. We find a result which is very similar to the cohomology for the vector case Pac. J. Math. 229(2):257–292, 2007. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to give an elementary method for the computation of the cohomology groups $$ H^{q} {\left( {X,\Omega ^{p}_{X} {\left( L \right)}} \right)} $$ , (0 ≤ q ≤ n) of an n–dimensional non–primary Hopf manifold X with arbitrary fundamental group. We use the method of Zhou to generalize the results for primary Hopf manifolds and non–primary Hopf manifold with an Abelian fundamental group. [ABSTRACT FROM AUTHOR]