We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, the relative semi-infinite cohomology of the loop algebra has a structure, wh...

Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose severe restrictions on the cohomology algebra of a smooth compact Kähler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions o...

We establish results about the second cohomology with coefficients in trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over commutative associative without unit. These provide simple unified approach to number questions treated earlier completely separated ways: periodization semisimple algebras (Anna Larsson), derivation algebras, prescribed part, ni...

let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

This paper is an outgrowth of the notes for a lecture given at the Mathematics Department of Queen’s University in December of 1988. I would like to take this opportunity to thank the members of that department for their hospitality during my visit. I would also like to thank Bruno Kahn for reading early versions of this manuscript and making several helpful remarks. The basic idea of the lectu...

If X is a smooth complex projective n-fold, Hodge-Lefschetz theory provides a filtration on the (primitive) cohomology H(X,C)0 by complex subspaces, satisfying certain compatiblity conditions with a bilinear form Q on cohomology. The Torelli problem, proposed in this form by Griffiths, is to decide whether given X, Y in a certain class C of varieties, the existence of an isomorphism H(X,C)0 ∼= ...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

Let V be a vector space of dimension n+1 over a field k, and consider the scheme X = PV ∼= Pk ∼= Proj(k[x0, . . . , xn]). Consider F a quasi-coherent sheaf over X. We can examine its Cech cohomology which coincides with its sheaf (nonetale) cohomology because X is Noetherian and separated (cf. Hartshorne, ch. III theorem 4.5.). Particularly, H(X,F) ∼= Γ(F , X). However Γ(F , X) ∼= HomX(OX ,F). ...

In this study, we compute simplicial cohomology groups with different coefficients of a connected sum of certain minimal simple surfaces by using the universal coefficient theorem for cohomology groups. The method used in this paper is a different way to compute digital cohomology groups of minimal simple surfaces. We also prove some theorems related to degree properties of a map on digital sph...

A Lie superalgebra is called quasi-Frobenius if it admits a closed anti-symmetric non-degenerate bilinear form. We study the notion of double extensions when form either orthosymplectic or periplectic. show that every satisfies certain conditions can be obtained as extension smaller superalgebra. classify all 4-dimensional superalgebras, and such superalgebras must solvable. [Formula: see text]...