By work of Farinati, Solberg, and Taillefer, it is known that the Hopf algebra cohomology a quasi-triangular algebra, as graded Lie under Gerstenhaber bracket, abelian. Motivated by question whether this holds for nonquasi-triangular algebras, we show brackets on can be expressed via an arbitrary projective resolution using Volkov's homotopy liftings generalized to some exact monoidal categorie...

Given a finite dimensional Lie algebra $L$ let $I$ be the augmentation ideal in universal enveloping $U(L)$. We study conditions on under which $Ext$-groups $Ext (k,k)$ for trivial $L$-module $k$ are same when computed category of all $U(L)$-modules or $I$-torsion $U(L)$-modules. also prove that Rees $\oplus _{n\geq 0}I^n$ is Noetherian if and only nilpotent. An application to cohomology equiva...

To any Hopf algebra H we associate two commutative Hopf algebras Hl1(H) and Hl2(H), which we call the lazy homology Hopf algebras of H . These Hopf algebras are built in such a way that they are “predual” to the lazy cohomology groups based on the so-called lazy cocycles. We establish two universal coefficient theorems relating the lazy cohomology to the lazy homology and we compute the lazy ho...

Developing foundational notions in the theory of Riemann surfaces, we prove the Riemann-Roch Theorem and Abel’s Theorem. These notions include sheaf cohomology, with particular focus on the zeroth and first cohomology groups, exact cohomology sequences induced by short exact sequences of sheaves, divisors, the Jacobian variety, and the Abel-Jacobi map. The general method of proof involves basic...

— The goal of this article is to make explicit a structured complex computing the cohomology of a profinite completion of the n-fold loop space of a sphere of dimension d < n. Our construction involves: the free complete algebra in one variable associated to any fixed En-operad; an element in this free complete algebra determined by a morphism from the operad of L∞-algebras to an operadic suspe...

We construct a cubical CW-complex CK(M3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M3. We construct CK(R3) by attaching cells to CK(R3) for every degenerate 1-singular and 2-singular knot, and we show that π1(CK(R 3)) = 1 and π2(CK(R 3)) = Z. We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R3 we show...

The (classical, small quantum, equivariant) cohomology ring of the grassmannianG(k, n) is generated by certain derivations operating on an exterior algebra of a free module of rank n (Schubert Calculus on a Grassmann Algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. It also provides, by results of Laksov and Thor...

We explore the relationship between de Rham and Lie algebra cohomologies in the finite dimensional and affine settings. In particular, given a ĝκ-module that arises as the global sections of a twisted D-module on the affine flag manifold, we show how to compute its untwisted BRST reduction modulo n(K) using the de Rham cohomology of the restrictions to N(K) orbits. A similar relationship holds ...